Geometry of Tangent Poisson–Lie Groups

Abstract

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle $TG$ of $G.$ In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle $TG$, and we express in different cases the contravariant Levi-Civita connection and curvature of $TG$ in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms ${\Omega }^{*}(G)$ on G is a differential graded Poisson algebra if, and only if, ${\Omega }^{*}(TG)$ is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group $TG$ is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given.

1. Introduction

The Riemannian geometry of tangent bundles and cotangent bundles of smooth manifolds is an important area in physics, classical mechanics and geometrical optics. If M is the configuration space of a mechanical system, then each point of the cotangent bundle $T^{*}M$ of M determines a state of the system and $T^{*}M$ is called the phase space [1]. Moreover, Poisson manifolds play a fundamental role in Hamiltonian dynamics, where they serve as a phase space. For this reason, there is some interest on how structures and, more generally, properties of M carry down to $T^{*}M$. Furthermore, if M is equipped with a pseudo-Riemannian metric compatible with the Poisson structure on M [2,3], it would be interesting to see if the compatibility remains fulfilled on the tangent bundle $TM$. First, recall that the notion of compatibility between a Poisson structure ${\Pi }_M$ and a contravariant pseudo-Riemannian metric ${\langle ,\rangle }_M^{*}$ on a smooth manifold M was first introduced by M.Boucetta in [2]. A triplet $(M,{\Pi }_M,{\langle ,\rangle }_M^{*})$ is compatible in the sense of M.Boucetta [2,4] and is a so-called pseudo-Riemannian Poisson manifold if, for any $\alpha ,\beta ,\gamma \in {\Omega }^1(M)$:

$${\mathcal{D}}^M{\Pi }_M(\alpha ,\beta ,\gamma )={\Pi }_M^{♯}(\alpha ){\Pi }_M(\beta ,\gamma )-{\Pi }_M({\mathcal{D}}_{\alpha }^M\beta ,\gamma )-{\Pi }_M(\beta ,{\mathcal{D}}_{\alpha }^M\gamma )=0,$$

where ${\mathcal{D}}^M$ is the contravariant Levi-Civita connection associated with the couple $({\Pi }_M,{\langle ,\rangle }_M^{*})$.

In [3,5], Hawkins showed that, if a deformation of the graded algebra ${\Omega }^{*}(M)$ of differential forms on a pseudo-Riemannian manifold $(M,{\langle ,\rangle }_M)$ comes from a spectral triple describing the pseudo-Riemannian structure, then the Poisson tensor ${\Pi }_M$ on M (which characterizes the deformation) and the pseudo-Riemannian metric ${\langle ,\rangle }_M$ satisfy the following compatibility conditions:

$(H_1)$

The metric contravariant connection ${\mathcal{D}}^M$ associated with $({\Pi }_M,{\langle ,\rangle }_M)$ is flat.

$(H_2)$

The metacurvature ${\mathcal{M}}^M$ of ${\mathcal{D}}^M$ is zero, i.e., the connection ${\mathcal{D}}^M$ is metaflat.

The metric contravariant connection ${\mathcal{D}}^M$ naturally associated with $({\Pi }_M,{\langle ,\rangle }_M)$ is exactly the Levi-Civita contravariant connection.

A triplet $(M,{\Pi }_M,{\langle ,\rangle }_M)$ satisfying conditions $H_1$ and $H_2$ is said to be compatible in the sense of Hawkins. A deformation of the differential graded algebra of differential forms ${\Omega }^{*}(M)$ defines a generalized Poisson bracket on this space. Moreover, a generalized Poisson bracket making ${\Omega }^{*}(M)$ a differential graded Poisson algebra exists if, and only if, $(M,{\Pi }_M,{\langle ,\rangle }_M)$ is compatible in the sense of Hawkins [3].

An important class of Poisson manifolds equipped with pseudo-Riemannian metrics is the family of Poisson–Lie groups equipped with left invariant pseudo-Riemannian metrics.

The notion of the Poisson–Lie group was first introduced by Drinfel’d [6,7] and Semenov–Tian–Shansky [8]. Semenov, Kosmann–Schwarzbach and Magri [9] used Poisson–Lie groups to understand the Hamiltonian structure of the group of dressing transformations of certain integrable systems. These Poisson–Lie groups play the role of symmetry groups.

In [10], M.Boumaiza and N.Zaalani showed that if $(G,{\Pi }_G)$ is a Poisson–Lie group, then the tangent bundle $(TG,{\Pi }_{TG})$ of G, with its tangent Poisson structure ${\Pi }_{TG}$ defined in the sense of Sanchez de Alvarez [11], is a Poisson–Lie group. This Poisson–Lie group $(TG,{\Pi }_{TG})$ is called a Sanchez de Alvarez tangent Poisson–Lie group of G [12].

The second author and N. Zaalani [12] have studied the compatibility between the Sanchez de Alvarez Poisson structure and the natural left invariant Riemannian metric. The non-compatibility between the Sanchez de Alvarez Poisson structure and the natural Riemannian metric (except in the trivial case ${\Pi }_G=0$) on $TG$ leads us to define another metric on the tangent Lie group $TG$ which is compatible with the Sanchez de Alvarez Poisson structure.

In this paper, we equip G with a Poisson structure and a pseudo-Riemannian metric. Then, we lift these structures on the tangent bundle $TG$ of G, and we study the Riemannian geometry of G and its relations with the geometry of $TG$.

This paper is organized as follows: In Section 2, we recall basic definitions and facts about contravariant connections, curvatures, metacurvatures, generalized Poisson brackets and pseudo-Riemannian Poisson–Lie groups. In Section 3, we induce a left invariant contravariant pseudo-Riemannian metric ${\langle ,\rangle }_{TG}^{*}$ on the tangent Poisson–Lie group $(TG,{\Pi }_{TG})$ and we express in different cases the Levi-Civita connection and curvature of $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ in terms of the Levi-Civita connection and curvature of $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$. In the case where the tangent bundle $TG$ is equipped with the Sanchez de Alvarez Poisson structure, we show that the space of differential forms ${\Omega }^{*}(TG)$ on $TG$ is a differential graded Poisson algebra if, and only if, ${\Omega }^{*}(G)$ is a differential graded Poisson algebra. In Section 4, we show that $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ is also a pseudo-Riemannian Poisson–Lie group. In Section 5, we give some examples of pseudo-Riemannian tangent Poisson–Lie groups.

2. Preliminaries

2.1. Contravariant Connections and Curvatures

Contravariant connections on Poisson manifolds were defined by Vaisman [13] and studied in detail by Fernandes [14]. This notion appears extensively in the context of noncommutative deformations [3,5].

Let $(M,{\Pi }_M)$ be a Poisson manifold. We associate the Poisson tensor ${\Pi }_M$ with the anchor map ${\Pi }_M^{♯}:T^{*}M\to TM$ defined by $\beta ({\Pi }_M^{♯}(\alpha ))={\Pi }_M(\alpha ,\beta )$ and the Koszul bracket ${[,]}_M$ on the space of differential 1-forms ${\Omega }^1(M)$ given by:

$$\begin{array}{ccc}\hfill {[\alpha ,\beta ]}_M& =& {\mathcal{L}}_{{\Pi }_M^{♯}(\alpha )}\beta -{\mathcal{L}}_{{\Pi }_M^{♯}(\beta )}\alpha -d({\Pi }_M(\alpha ,\beta )).\hfill \end{array}$$

A contravariant connection on $M,$ with respect to ${\Pi }_M$, is an $\mathbb{R}$-bilinear map

$${\mathcal{D}}^M:{\Omega }^1(M)\times {\Omega }^1(M)\to {\Omega }^1(M),(\alpha ,\beta )\mapsto {\mathcal{D}}_{\alpha }^M\beta ,$$

such that for all $f\in {\mathcal{C}}^{\infty }(M)$,

$${\mathcal{D}}_{f\alpha }^M\beta =f{\mathcal{D}}_{\alpha }^M\beta \mathrm{and}{\mathcal{D}}_{\alpha }^M(f\beta )=f{\mathcal{D}}_{\alpha }^M\beta +{\Pi }_M^{♯}(\alpha )(f)\beta .$$

The torsion ${\mathcal{T}}^M$ and the curvature $R^M$ of a contravariant connection ${\mathcal{D}}^M$ are formally identical to the usual ones:

$${\mathcal{T}}^M(\alpha ,\beta )={\mathcal{D}}_{\alpha }^M\beta -{\mathcal{D}}_{\beta }^M\alpha -{[\alpha ,\beta ]}_M$$

$$R^M(\alpha ,\beta )\gamma ={\mathcal{D}}_{\alpha }^M{\mathcal{D}}_{\beta }^M\gamma -{\mathcal{D}}_{\beta }^M{\mathcal{D}}_{\alpha }^M\gamma -{\mathcal{D}}_{{[\alpha ,\beta ]}_M}^M\gamma .$$

(1)

These are (2,1) and (3,1)-type tensor fields, respectively. When ${\mathcal{T}}^M\equiv 0$ (resp., $R^M\equiv 0$), ${\mathcal{D}}^M$ is called torsion-free (resp., flat).

Let $(M,{\Pi }_M)$ be a Poisson manifold. Let ${\langle ,\rangle }_M$ be a covariant Riemannian metric on M and ${\langle ,\rangle }_M^{*}$ the contravariant Riemannian metric associated with ${\langle ,\rangle }_M$. The metric contravariant connection associated with $({\Pi }_M,{\langle ,\rangle }_M^{*})$ is the unique contravariant connection ${\mathcal{D}}^M$ such that ${\mathcal{D}}^M$ is torsion-free and the metric ${\langle ,\rangle }_M^{*}$ is parallel with respect to ${\mathcal{D}}^M$, i.e.,

$${\Pi }_M^{♯}(\alpha ){\langle \beta ,\gamma \rangle }_M^{*}={\langle {\mathcal{D}}_{\alpha }^M\beta ,\gamma \rangle }_M^{*}+{\langle \beta ,{\mathcal{D}}_{\alpha }^M\gamma \rangle }_M^{*}.$$

(2)

The connection ${\mathcal{D}}^M$ is called the Levi-Civita contravariant connection associated with $({\Pi }_M,{\langle ,\rangle }_M^{*})$ and can be defined by the Koszul formula:

$$\begin{array}{ccc}\hfill 2{\langle {\mathcal{D}}_{\alpha }^M\beta ,\gamma \rangle }_M^{*}& =& {\Pi }_M^{♯}(\alpha ){\langle \beta ,\gamma \rangle }_M^{*}+{\Pi }_M^{♯}(\beta ){\langle \alpha ,\gamma \rangle }_M^{*}-{\Pi }_M^{♯}(\gamma ){\langle \alpha ,\beta \rangle }_M^{*}\hfill \\ & +& {\langle {[\alpha ,\beta ]}_M,\gamma \rangle }_M^{*}+{\langle {[\gamma ,\alpha ]}_M,\beta \rangle }_M^{*}+{\langle {[\gamma ,\beta ]}_M,\alpha \rangle }_M^{*}.\hfill \end{array}$$

(3)

We say that ${\mathcal{D}}^M$ is locally symmetric if ${\mathcal{D}}^M{R}^M=0,$ i.e., if for any $\alpha ,\beta ,\gamma ,\delta \in {\Omega }^1(M),$ we have:

$$\begin{array}{ccc}\hfill ({\mathcal{D}}_{\alpha }^M{R}^M)(\beta ,\gamma )\delta & :=& {\mathcal{D}}_{\alpha }^M(R^M(\beta ,\gamma )\delta )-R^M({\mathcal{D}}_{\alpha }^M\beta ,\gamma )\delta -R^M(\beta ,\gamma ){\mathcal{D}}_{\alpha }^M\delta \hfill \\ & -& R^M(\beta ,{\mathcal{D}}_{\alpha }^M\gamma )\delta =0.\hfill \end{array}$$

(4)

2.2. Generalized Poisson Bracket on the Space of Differential forms ${\Omega }^{*}(M)$

Let $(M,{\Pi }_M)$ be a Poisson manifold and ${\mathcal{D}}^M$ a torsion-free and flat connection with respect to ${\Pi }_M$. In [3], E.Hawkins showed that such a connection defines an $\mathbb{R}$-bilinear bracket on the space of differential forms ${\Omega }^{*}(M)$, also denoted by ${\{,\}}_M$, such that:

• The bracket ${\{,\}}_M$ is antisymmetric, i.e.,

  $${\{\sigma ,\upsilon \}}_M=-{(-1)}^{deg(\sigma )deg(\upsilon )}{\{\upsilon ,\sigma \}}_M;$$
• ${\{,\}}_M$ satisfies the product rule, i.e.,

  $${\{\sigma ,\upsilon \wedge \nu \}}_M={\{\sigma ,\upsilon \}}_M\wedge \nu +{(-1)}^{deg(\sigma )deg(\upsilon )}\upsilon \wedge {\{\sigma ,\nu \}}_M;$$
• The exterior differential d is a derivation with respect to ${\{,\}}_M$, i.e.,

  $$d{\{\sigma ,\upsilon \}}_M={\{d\sigma ,\upsilon \}}_M+{(-1)}^{deg(\sigma )}{\{\sigma ,d\upsilon \}}_M;$$
• For any $f_1,f_2\in {\mathcal{C}}^{\infty }(M)$ and for any $\sigma \in {\Omega }^{*}(M)$, the bracket ${\{f_1,f_2\}}_M$ coincides with the initial Poisson bracket on M and
  $${\{f_1,\sigma \}}_M={\mathcal{D}}_{df}^M\sigma .$$

This bracket is given for any $\alpha ,\beta \in {\Omega }^1(M)$ by [15]:

$${\{\alpha ,\beta \}}_M=-{\mathcal{D}}_{\alpha }^{M}d\beta -{\mathcal{D}}_{\beta }^{M}d\alpha +d{\mathcal{D}}_{\beta }^M\alpha +{[\alpha ,d\beta ]}_M,$$

(5)

where ${[,]}_M$ is the generalized Koszul bracket on ${\Omega }^{*}(M)$ satisfying the Leibnuz identity, i.e.,

$${[\sigma ,\upsilon \wedge \nu ]}_M={[\sigma ,\upsilon ]}_M\wedge \nu +{(-1)}^{(deg(\sigma )-1)deg(\upsilon )}\upsilon \wedge {[\sigma ,\nu ]}_M.$$

(6)

Note that the generalized Koszul bracket for the differential forms is analogous to the Schouten–Nijenhuis bracket for the multivector fields (for more details, see [16] p. 44).

We call this bracket ${\{,\}}_M$ a generalized pre-Poisson bracket associated with the contravariant connection ${\mathcal{D}}^M.$ E.Hawkins showed that there exists a (2,3) tensor ${\mathcal{M}}^M$ that is symmetrical in the contravariant indices and antisymmetrical in the covariant indices such that the generalized pre-Poisson bracket satisfies the graded Jacobi identity

$${\{\sigma ,{\{\upsilon ,\nu \}}_M\}}_M-{\{{\{\sigma ,\upsilon \}}_M,\nu \}}_M-{(-1)}^{deg(\sigma )deg(\upsilon )}{\{\upsilon ,{\{\sigma ,\nu \}}_M\}}_M=0,$$

if, and only if, ${\mathcal{M}}^M$ is identically zero.

${\mathcal{M}}^M$ is called metacurvature of ${\mathcal{D}}^M$ and is given by

$${\mathcal{M}}^M(d{f}_1,\alpha ,\beta )={\{f_1,{\{\alpha ,\beta \}}_M\}}_M-{\{{\{f_1,\alpha \}}_M,\beta \}}_M-{\{{\{f_1,\beta \}}_M,\alpha \}}_M.$$

(7)

If ${\mathcal{M}}^M$ vanishes identically, the contravariant connection ${\mathcal{D}}^M$ is called metaflat and the bracket ${\{,\}}_M$ is called the generalized Poisson bracket associated with ${\mathcal{D}}^M$, making ${\Omega }^{*}(M)$ a differential graded Poisson algebra (for more details, see [3]).

2.3. Pseudo-Riemannian Poisson–Lie Group

An important class of Poisson manifolds is the family of Poisson–Lie groups. A Lie group G is called a Poisson–Lie group if it is also a Poisson manifold such that the product

$$m:G\times G\to G:(g,h)\mapsto gh$$

is a Poisson map, where $G\times G$ is equipped with the product Poisson structure.

Let G be a Poisson Lie group with Lie algebra $(\mathfrak{g},{[,]}_{\mathfrak{g}})$ and ${\Pi }_G$ the Poisson tensor on G. Pulling ${\Pi }_G$ back to the identity element e of G by the left translations, we obtain a map ${\Pi }_G^l:G\to \mathfrak{g}\wedge \mathfrak{g}$, defined by ${\Pi }_G^l(g)={(L_{g^{-1}})}_{*}{\Pi }_G(g),$ where ${(L_g)}_{*}$ denotes the tangent map of the left translation $L_g$ of G by g. The intrinsic derivative

$$\xi :=d_e{\Pi }_G^l:\mathfrak{g}\to \mathfrak{g}\wedge \mathfrak{g}$$

of ${\Pi }_G^l$ at e is a 1-cocycle relative to the adjoint representation of $\mathfrak{g}$ on $\mathfrak{g}\wedge \mathfrak{g}$. The dual map of $\xi$ is a Lie bracket ${[,]}_{{\mathfrak{g}}^{*}}:{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}\to {\mathfrak{g}}^{*}$ on ${\mathfrak{g}}^{*}.$ It is well-known that $(\mathfrak{g},{\mathfrak{g}}^{*})$ is a Lie bialgebra.

Let $(G,{\Pi }_G)$ be a Poisson–Lie group with Lie bialgebra $(\mathfrak{g},{\mathfrak{g}}^{*})$. Let ${\langle ,\rangle }_{{\mathfrak{g}}^{*}}$ be a bilinear, symmetric and non-degenerate form on ${\mathfrak{g}}^{*}$ and let ${\langle ,\rangle }_G^{*}$ be the contravariant pseudo-Riemannian given by ${\langle ,\rangle }_G^{*}={(L_g)}_{*}{\langle ,\rangle }_{{\mathfrak{g}}^{*}}$. We say that $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ is a pseudo-Riemannian Poisson–Lie group if, and only if, the Poisson tensor ${\Pi }_G$ and the metric ${\langle ,\rangle }_G^{*}$ are compatible in the sense given by M.Boucetta in [4,17], as follows:

$${[A{d}_g^{*}(A_{\alpha }\gamma +a{d}_{{\Pi }_G^l(g)(\alpha )}^{*}\gamma ),A{d}_g^{*}(\beta )]}_{{\mathfrak{g}}^{*}}+{[A{d}_g^{*}(\alpha ),A{d}_g^{*}(A_{\beta }\gamma +a{d}_{{\Pi }_G^l(g)(\beta )}^{*}\gamma )]}_{{\mathfrak{g}}^{*}}=0,$$

(8)

for any $g\in G$ and for any $\alpha ,\beta ,\gamma \in {\mathfrak{g}}^{*}$, where A is the infinitesimal Levi-Civita connection associated with $({[,]}_{{\mathfrak{g}}^{*}},{\langle ,\rangle }_{{\mathfrak{g}}^{*}})$.

Note that the infinitesimal Levi-Civita connection A is the restriction of the Levi-Civita contravariant connection ${\mathcal{D}}^G$ to ${\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$ and is given for any $\alpha ,\beta ,\gamma \in {\mathfrak{g}}^{*},$ by:

$$2{\langle A_{\alpha }\beta ,\gamma \rangle }_{{\mathfrak{g}}^{*}}={\langle {[\alpha ,\beta ]}_{{\mathfrak{g}}^{*}},\gamma \rangle }_{{\mathfrak{g}}^{*}}+{\langle {[\gamma ,\alpha ]}_{{\mathfrak{g}}^{*}},\beta \rangle }_{{\mathfrak{g}}^{*}}+{\langle {[\gamma ,\beta ]}_{{\mathfrak{g}}^{*}},\alpha \rangle }_{{\mathfrak{g}}^{*}}.$$

(9)

In [4], M.Boucetta showed that if $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ is a pseudo-Riemannian Poisson–Lie group, then its dual Lie algebra $({\mathfrak{g}}^{*},{[,]}_{{\mathfrak{g}}^{*}},{\langle ,\rangle }_{{\mathfrak{g}}^{*}})$ equipped with the form ${\langle ,\rangle }_{{\mathfrak{g}}^{*}}$ is a pseudo-Riemannian Lie algebra, i.e, for any $\alpha ,\beta ,\gamma \in {\mathfrak{g}}^{*}$, we have

$${[A_{\alpha }\beta ,\gamma ]}_{{\mathfrak{g}}^{*}}+{[\alpha ,A_{\gamma }\beta ]}_{{\mathfrak{g}}^{*}}=0.$$

(10)

3. Pseudo-Riemannian Geometry of Tangent Poisson–Lie Group

Let G be a n-dimensional Lie group with multiplication $m:G\times G\to G:(g,h)\mapsto gh$ and with Lie algebra $(\mathfrak{g},{[,]}_{\mathfrak{g}})$. We denote by $L_g:G\to G:h\mapsto gh,$ the left translation and $R_g:G\to G:h\mapsto hg,$ the right translation of G by g.

The tangent map $Tm$ of $m,$

$$Tm:TG\times TG\mapsto TG:(X_g,Y_h)\mapsto T_h{L}_g{Y}_h+T_g{R}_h{X}_g,$$

(11)

defines a Lie group structure on $TG$ with identity element $(e,0)$ and with Lie algebra the semi-direct product of Lie algebra $\mathfrak{g}⋊\mathfrak{g}$, with bracket [10,18]:

$${[(X,Y),(X^{{}^{\prime }},Y^{{}^{\prime }})]}_{\mathfrak{g}⋊\mathfrak{g}}=({[X,X^{{}^{\prime }}]}_{\mathfrak{g}},{[X,Y^{{}^{\prime }}]}_{\mathfrak{g}}+{[Y,X^{{}^{\prime }}]}_{\mathfrak{g}}),$$

(12)

where $(X,X^{{}^{\prime }}),(Y,Y^{{}^{\prime }})\in \mathfrak{g}⋊\mathfrak{g}.$

Let $(G,{\Pi }_G)$ be a Poisson–Lie group with Lie bilagebra $(\mathfrak{g},{\mathfrak{g}}^{*})$ and let $TG$ be the tangent bundle of G. According to M.Boumaiza and N.Zaalani [10], the tangent bundle $TG$ of G with the multiplication (11) and with its tangent Poisson structure ${\Pi }_{TG}$, defined in the sense of Sanchez de Alvarez [11], is a Poisson–Lie group with Lie bialgebra $(\mathfrak{g}⋊\mathfrak{g},{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*})$, where ${\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}$ is the semi-direct product Lie algebra with bracket:

$${[(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime })]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}=({[\alpha ,{\beta }^{\prime }]}_{{\mathfrak{g}}^{*}}+{[\beta ,{\alpha }^{\prime }]}_{{\mathfrak{g}}^{*}},{[\beta ,{\beta }^{\prime }]}_{{\mathfrak{g}}^{*}}),$$

(13)

where $(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime })\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$.

On the other hand, if $(G,{\Pi }_G)$ is a Poisson–Lie group, there exists a linear Poisson structure ${\Pi }_{\mathfrak{g}}$ on $\mathfrak{g}$, whose value at $X\in \mathfrak{g}$ is given by ${\Pi }_{\mathfrak{g}}(X)=d_e{\Pi }_G(X).$ The linear Poisson structure ${\Pi }_{\mathfrak{g}}$ on $\mathfrak{g}=T_{e}G$ makes $(\mathfrak{g},{\Pi }_{\mathfrak{g}})$ an abelian Poisson–Lie group with Lie bialgebra $(\mathfrak{g},{\mathfrak{g}}^{*})$ such that the Lie bracket of $\mathfrak{g}$ is zero and the Lie bracket of ${\mathfrak{g}}^{*}$ is ${[,]}_{{\mathfrak{g}}^{*}}$.

If we identify the tangent bundle $TG\equiv G\times \mathfrak{g}$ with the direct product Poisson–Lie group of $(G,{\Pi }_G)$ and $(\mathfrak{g},{\Pi }_{\mathfrak{g}})$; then, $(TG\equiv G\times \mathfrak{g},{\Pi }_{G\times \mathfrak{g}})$ is a Poisson–Lie group, with Lie-bialgebra $(\mathfrak{g}\times \mathfrak{g},{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*})$, where $\mathfrak{g}\times \mathfrak{g}$ is the direct product Lie algebra with bracket:

$${[(X,Y),(X^{\prime },Y^{\prime })]}_{\mathfrak{g}\times \mathfrak{g}}=({[X,X^{\prime }]}_{\mathfrak{g}},0),(X,Y),(X^{\prime },Y^{\prime })\in \mathfrak{g}\times \mathfrak{g},$$

(14)

and ${\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$ is the direct product Lie algebra with bracket:

$${[(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime })]}_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}}=({[\alpha ,{\alpha }^{\prime }]}_{{\mathfrak{g}}^{*}},{[\beta ,{\beta }^{\prime }]}_{{\mathfrak{g}}^{*}}),(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime })\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}.$$

(15)

Now, we equip G with a left invariant pseudo-Riemannian metric, and we lift this metric to the tangent bundle $TG.$

Let $\pi :TG\to G:(g,X)\mapsto g,$ be the natural projection. The differential mapping $d\pi (e,0)$ at the point $(e,0)$ is given by:

$$d\pi (e,0):\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}:(X,Y)\mapsto X,$$

and the vertical subspace ${\mathcal{V}}_{(e,0)}$ of $\mathfrak{g}\times \mathfrak{g}$ is given by ${\mathcal{V}}_{(e,0)}=ker(d\pi (e,0))=\left\{0\right\}\times \mathfrak{g}.$

It has been shown that the complete and vertical lifts of any left invariant vector fields of G are left invariant fields on the tangent Lie group $TG$ (see proposition 1.3 page 183 of [19] or theorems 1.2.2 and 1.2.3 of [20]). In fact, if $(X_1,\dots ,X_n)$ is a basis for the Lie algebra $\mathfrak{g}$ of G, then $\{X_1^v=(0,X_1),\dots ,X_n^v=(0,X_n),X_1^c=(X_1,0),\dots ,X_n^c=(X_n,0)\}$ is a basis for the Lie algebra $\mathfrak{g}\times \mathfrak{g}$ of $TG,$ where $X_1^v=(0,X_1)$ (resp., $X_1^c=(X_1,0)$) is the vertical lift (resp., the complete lift) of the vector field $X_1$ on G to $TG.$

Let ${\langle ,\rangle }_G$ be a left invariant pseudo-Riemannian metric on G. Then, we define a left invariant pseudo-Riemannian metric ${\langle ,\rangle }_{TG}$ on $TG$ as follows:

$$\begin{array}{ccc}\hfill {\langle (0,Y),(0,Y^{{}^{\prime }})\rangle }_{TG}(e,0)& =& 0,\hfill \\ \hfill {\langle (X,0),(0,Y^{{}^{\prime }})\rangle }_{TG}(e,0)& =& {\langle X,Y^{{}^{\prime }}\rangle }_G(e)\hfill \\ \hfill {\langle (X,0),(X^{{}^{\prime }},0)\rangle }_{TG}(e,0)& =& {\langle X,X^{{}^{\prime }}\rangle }_G(e),\hfill \end{array}$$

(16)

where $(X,Y),(X^{{}^{\prime }},Y^{{}^{\prime }})\in \mathfrak{g}\times \mathfrak{g}$.

The left invariant contravariant pseudo-Riemannian metric ${\langle ,\rangle }_{TG}^{*}$ on $TG$ associated with ${\langle ,\rangle }_{TG}$ is given for any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$ by:

$$\begin{array}{ccc}\hfill {\langle (\alpha ,0),({\alpha }^{{}^{\prime }},0)\rangle }_{TG}^{*}(e,0)& =& 0,\hfill \\ \hfill {\langle (\alpha ,0),(0,{\beta }^{{}^{\prime }})\rangle }_{TG}^{*}(e,0)& =& {\langle \alpha ,{\beta }^{{}^{\prime }}\rangle }_G^{*}(e)\hfill \\ \hfill {\langle (0,\beta ),(0,{\beta }^{{}^{\prime }})\rangle }_{TG}^{*}(e,0)& =& {\langle \beta ,{\beta }^{{}^{\prime }}\rangle }^{*}(e),\hfill \end{array}$$

(17)

where ${\alpha }^v=(\alpha ,0)$ (resp., ${\alpha }^c=(0,\alpha )$) is the vertical lift (resp., the complete lift) of the 1-form $\alpha$ on G to $TG.$ (for more details on lift tensor fields, see [19]).

3.1. Pseudo-Riemannian Geometry of Product Poisson Structure on $TG$

In this subsection, we consider the left invariant contravariant pseudo-Riemannian metric ${\langle ,\rangle }_{TG}^{*}$ defined as above on the tangent bundle $(TG,{\Pi }_{G\times \mathfrak{g}})$ equipped with the product Poisson structure. Then, we study the geometry of the triplet $(TG,{\Pi }_{G\times \mathfrak{g}},{\langle ,\rangle }_{TG}^{*})$ and its relations with the geometry of $(G,{\Pi }_G,{\langle ,\rangle }_G^{*}).$

First of all, we note that if we denote by ${\mathcal{D}}^g$ the Levi-Civita connection associated with $({\Pi }_{\mathfrak{g}},{\langle ,\rangle }_G(e))$ and by $R^{\mathfrak{g}}$ the curvature of ${\mathcal{D}}^{\mathfrak{g}}$, then the restriction of ${\mathcal{D}}^G$ to ${\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$ coincides with ${\mathcal{D}}^{\mathfrak{g}}$ and the restriction of the curvature $R^G$ of ${\mathcal{D}}^G$ to ${\mathfrak{g}}^{*}$ coincides with $R^{\mathfrak{g}}$, i.e.,

$${\mathcal{D}}_{\alpha }^G\beta ={\mathcal{D}}_{\alpha }^{\mathfrak{g}}\beta ,R^G(\alpha ,\beta )\gamma =R^{\mathfrak{g}}(\alpha ,\beta )\gamma ,$$

for any $\alpha ,\beta ,\gamma \in {\mathfrak{g}}^{*}.$

Proposition 1. 

Let ${\mathcal{D}}^G$ and ${\mathcal{D}}^{G\times \mathfrak{g}}$ be the Levi-Civita contravariant connections associated with $({\Pi }_G,{\langle ,\rangle }_G^{*})$ and $({\Pi }_{G\times \mathfrak{g}},{\langle ,\rangle }_{TG}^{*})$, respectively. Then, for any $(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime }),({\alpha }^{″},{\beta }^{″})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*},$ we have:

1. 

${\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}({\alpha }^{\prime },0),({\alpha }^{″},0)\rangle }_{TG}^{*}=0,$

2. 

${\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}({\alpha }^{\prime },0),(0,{\beta }^{″})\rangle }_{TG}^{*}=\frac{1}{2}{\langle ({[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},0),(0,{\beta }^{″})\rangle }_{TG}^{*},$

3. 

${\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime }),({\alpha }^{″},0)\rangle }_{TG}^{*}=-\frac{1}{2}{\langle (0,{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}),({\alpha }^{″},0)\rangle }_{TG}^{*},$

4. 

${\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime }),(0,{\beta }^{″})\rangle }_{TG}^{*}=-\frac{1}{2}{\langle ({\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t\alpha ,0),(0,{\beta }^{{}^{″}})\rangle }_{TG}^{*},$

5. 

${\langle {\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}({\alpha }^{\prime },0),({\alpha }^{″},0)\rangle }_{TG}^{*}=-\frac{1}{2}{\langle (0,{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t\beta ),({\alpha }^{″},0)\rangle }_{TG}^{*},$

6. 

${\langle {\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}({\alpha }^{\prime },0),(0,{\beta }^{″})\rangle }_{TG}^{*}=-\frac{1}{2}{\langle ({\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{\prime }},0),(0,{\beta }^{{}^{″}})\rangle }_{TG}^{*},$

7. 

${\langle {\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}(0,{\beta }^{\prime }),({\alpha }^{″},0)\rangle }_{TG}^{*}=\frac{1}{2}{\langle (0,{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}),({\alpha }^{″},0)\rangle }_{TG}^{*},$

8. 

${\langle {\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}(0,{\beta }^{\prime }),(0,{\beta }^{″})\rangle }_{TG}^{*}={\langle (0,{\mathcal{D}}_{\beta }^G{\beta }^{\prime }),(0,{\beta }^{″})\rangle }_{TG}^{*},$

where ${\mathrm{ad}}_{\alpha }^t$ denotes the transpose of ${\mathrm{ad}}_{\alpha }$ with respect to ${\langle ,\rangle }_G^{*}.$

Proof. 

According to Equations (9), (15) and (17), for example for (5) we obtain:

$$\begin{array}{ccc}\hfill 2{\langle {\mathcal{D}}_{(0,\beta )}^{TG}({\alpha }^{\prime },0),({\alpha }^{″},0)\rangle }_{TG}^{*}& =& {\langle {[(0,\beta ),({\alpha }^{\prime },0)]}_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}},({\alpha }^{″},0)\rangle }_{TG}^{*}\hfill \\ & +& {\langle {[({\alpha }^{″},0),(0,\beta )]}_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}},({\alpha }^{\prime },0)\rangle }_{TG}^{*}\hfill \\ & +& {\langle {[({\alpha }^{″},0),({\alpha }^{\prime },0)]}_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}},(0,\beta )\rangle }_{TG}^{*}\hfill \\ & =& {\langle {[{\alpha }^{{}^{″}},{\alpha }^{\prime }]}_{{\mathfrak{g}}^{*}},(\beta ,0)\rangle }_G^{*}\hfill \\ & =& -{\langle {\alpha }^{{}^{″}},{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t\beta \rangle }_G^{*}\hfill \\ & =& -{\langle (0,{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t\beta ),({\alpha }^{{}^{″}},0)\rangle }_{TG}^{*}.\hfill \end{array}$$

□

Lemma 1. 

For any $(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime })\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*},$ we have:

1. 

${\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}({\alpha }^{\prime },0)=\frac{1}{2}({[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},0),$

2. 

${\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime })=\frac{1}{2}({\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}-{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t\alpha ,-{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}),$

3. 

${\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}({\alpha }^{\prime },0)=\frac{1}{2}({\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t\beta -{\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{\prime }},-{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t\beta ),$

4. 

${\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}(0,{\beta }^{\prime })=\frac{1}{2}({\mathcal{D}}_{\beta }^G{\beta }^{{}^{\prime }}+{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G\beta ,{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}).$

Proof. 

Using the previous proposition we obtain:

(1)

$$\begin{array}{ccc}\hfill {\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}({\alpha }^{\prime },0),({\alpha }^{″},{\beta }^{″})\rangle }_{TG}^{*}& =& {\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}({\alpha }^{\prime },0),({\alpha }^{″},0)\rangle }_{TG}^{*}+{\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}({\alpha }^{\prime },0),(0,{\beta }^{″})\rangle }_{TG}^{*}\hfill \\ & =& \frac{1}{2}{\langle ({[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},0),(0,{\beta }^{″})\rangle }_{TG}^{*}\hfill \\ & =& {\langle \frac{1}{2}({[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},0),({\alpha }^{″},{\beta }^{″})\rangle }_{TG}^{*},\hfill \end{array}$$

then, ${\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}({\alpha }^{\prime },0)=\frac{1}{2}({[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},0)$.

(2)

$$\begin{array}{ccc}\hfill {\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime }),({\alpha }^{″},{\beta }^{″})\rangle }_{TG}^{*}& =& {\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime }),({\alpha }^{″},0)\rangle }_{TG}^{*}+{\langle {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime }),(0,{\beta }^{″})\rangle }_{TG}^{*}\hfill \\ & =& -\frac{1}{2}{\langle (0,{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}),({\alpha }^{″},0)\rangle }_{TG}^{*}+-\frac{1}{2}{\langle ({\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t\alpha ,0),(0,{\beta }^{{}^{″}})\rangle }_{TG}^{*}\hfill \\ & =& {\langle \frac{1}{2}({\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}-{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t\alpha ,-{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}),({\alpha }^{″},{\beta }^{″})\rangle }_{TG}^{*},\hfill \end{array}$$

then, ${\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime })=\frac{1}{2}({\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}-{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t\alpha ,-{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}).$

In the same way, we can obtain $(3)$ and $(4)$.

□

Theorem 1. 

Let $R^G$ and $R^{G\times \mathfrak{g}}$ be the curvatures of ${\mathcal{D}}^G$ and ${\mathcal{D}}^{G\times \mathfrak{g}}$ respectively. Then for any $(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime }),({\alpha }^{″},{\beta }^{″})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*},$ we have:

1. 

$$\begin{array}{ccc}\hfill R^{G\times \mathfrak{g}}((\alpha ,0),({\alpha }^{\prime },0))({\alpha }^{″},0)& =& \frac{1}{4}({[\alpha ,{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}]}_{{\mathfrak{g}}^{*}}+{[{\alpha }^{{}^{\prime }},{[{\alpha }^{{}^{″}},\alpha ]}_{{\mathfrak{g}}^{*}}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +& 2{[{\alpha }^{{}^{″}},{[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}]}_{{\mathfrak{g}}^{*}},0).\hfill \end{array}$$

2. 

$$\begin{array}{cc}\hfill R^{G\times \mathfrak{g}}((\alpha ,0),({\alpha }^{\prime },0))(0,{\beta }^{″})& =\frac{1}{4}({[\alpha ,{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}-{\mathrm{ad}}_{{\beta }^{{}^{″}}}^t{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}-{[{\alpha }^{{}^{\prime }},{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{″}}-{\mathrm{ad}}_{{\beta }^{{}^{″}}}^t\alpha ]}_{{\mathfrak{g}}^{*}}\hfill \\ & -{\mathrm{ad}}_{\alpha }^t{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}+{\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}}^t\alpha +{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{″}}-{\mathrm{ad}}_{{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{″}}}^t{\alpha }^{{}^{\prime }}\hfill \\ & -2{\mathrm{ad}}_{{[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^t{\beta }^{{}^{″}}+2{\mathrm{ad}}_{{\beta }^{{}^{″}}}^t{[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},{\mathrm{ad}}_{\alpha }^t{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}-{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{″}}\hfill \\ & +2{\mathrm{ad}}_{{[\alpha ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^t{\beta }^{{}^{″}}).\hfill \end{array}$$

3. 

$$\begin{array}{cc}\hfill R^{G\times \mathfrak{g}}((0,\beta ),({\alpha }^{\prime },0))({\alpha }^{″},0)& =\frac{1}{4}({\mathrm{ad}}_{{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}}^t\beta -{\mathrm{ad}}_{\beta }^t{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}-{[{\alpha }^{{}^{\prime }},{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta }^t{\alpha }^{{}^{\prime }},-{\mathrm{ad}}_{{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}}^t\beta -{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta ).\hfill \end{array}$$

4. 

$$\begin{array}{cc}\hfill R^{G\times \mathfrak{g}}((0,\beta ),({\alpha }^{\prime },0))(0,{\beta }^{″})& =\frac{1}{4}({\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}-{\mathrm{ad}}_{{\beta }^{{}^{″}}}^t{\alpha }^{{}^{\prime }}}^t\beta -{\mathrm{ad}}_{\beta }^t{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}+{\mathrm{ad}}_{\beta }^t{\mathrm{ad}}_{{\beta }^{{}^{″}}}^t{\alpha }^{{}^{\prime }}-{\mathcal{D}}_{\beta }^G{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}\hfill \\ & -{\mathcal{D}}_{{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}}^G\beta -{[{\alpha }^{{}^{\prime }},{\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{″}}}^G\beta ]}_{{\mathfrak{g}}^{*}}-{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{[\beta ,{\beta }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +{\mathrm{ad}}_{{[\beta ,{\beta }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}}^t{\alpha }^{{}^{\prime }},-{\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{\prime }}-{\mathrm{ad}}_{{\beta }^{{}^{″}}}^t{\alpha }^{{}^{\prime }}}^t\beta -{[\beta ,{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\beta }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}\hfill \\ & -{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}).\hfill \end{array}$$

5. 

$$\begin{array}{cc}\hfill R^{G\times \mathfrak{g}}((0,\beta ),(0,{\beta }^{\prime }))({\alpha }^{″},0)& =\frac{1}{4}({\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t{\beta }^{{}^{\prime }}-{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{\beta }^t{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t{\beta }^{{}^{\prime }}+{\mathrm{ad}}_{\beta }^t{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t{\alpha }^{{}^{″}}-{\mathcal{D}}_{\beta }^G{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t{\beta }^{{}^{\prime }}\hfill \\ & -{\mathcal{D}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta }^G{\beta }^{{}^{\prime }}-{\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{″}}}^t{\beta }^{{}^{\prime }}+{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t{\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{″}}\hfill \\ & +{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta +{\mathcal{D}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta }^G{\beta }^{{}^{\prime }}-2{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +2{\mathrm{ad}}_{{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^t{\alpha }^{{}^{″}},-{\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t{\beta }^{{}^{\prime }}-{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t{\alpha }^{{}^{″}}}^t\beta -{[\beta ,{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +{\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{″}}}^t{\beta }^{{}^{\prime }}+{[{\beta }^{{}^{\prime }},{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta ]}_{{\mathfrak{g}}^{*}}-2{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}).\hfill \end{array}$$

6. 

$$\begin{array}{cc}\hfill R^{G\times \mathfrak{g}}((0,\beta ),(0,{\beta }^{\prime }))(0,{\beta }^{″})& =\frac{1}{4}(R^G(\beta ,{\beta }^{{}^{\prime }}){\beta }^{{}^{″}}-{\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{\beta }^{{}^{\prime }}+{\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{\beta }^{{}^{\prime }}-{\mathcal{D}}_{{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^G{\beta }^{{}^{″}}\hfill \\ & +{\mathrm{ad}}_{({\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{\beta }^{{}^{\prime }})}^t\beta -{\mathrm{ad}}_{\beta }^t({\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{\beta }^{{}^{\prime }})\hfill \\ & -{\mathrm{ad}}_{({\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{″}}}^G\beta )}^t{\beta }^{{}^{\prime }}+{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t({\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{″}}}^G\beta )-{\mathcal{D}}_{{[\beta ,{\beta }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}}^G{\beta }^{{}^{\prime }}\hfill \\ & +2{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},-{\mathrm{ad}}_{({\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{\beta }^{{}^{\prime }})}^t\beta \hfill \\ & +{\mathrm{ad}}_{({\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{″}}}^G\beta )}^t{\beta }^{{}^{\prime }}+{[{\beta }^{{}^{″}},{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}]}_{{\mathfrak{g}}^{*}}).\hfill \end{array}$$

Proof. 

Using the Equation (1) and the Lemma 1, for example, for (3), we find:

$$\begin{array}{ccc}\hfill R^{G\times \mathfrak{g}}((0,\beta ),({\alpha }^{\prime },0))({\alpha }^{″},0)& =& {\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}{\mathcal{D}}_{({\alpha }^{{}^{\prime }},0)}^{G\times \mathfrak{g}}({\alpha }^{{}^{″}},0)-{\mathcal{D}}_{({\alpha }^{{}^{\prime }},0)}^{G\times \mathfrak{g}}{\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}({\alpha }^{{}^{″}},0)\hfill \\ & -& {\mathcal{D}}_{{[(0,\beta ),({\alpha }^{{}^{\prime }},0)]}_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}}}^{G\times \mathfrak{g}}({\alpha }^{{}^{″}},0)\hfill \\ & =& {\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}\frac{1}{2}({[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}},0)-{\mathcal{D}}_{({\alpha }^{{}^{\prime }},0)}^{G\times \mathfrak{g}}\frac{1}{2}({\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{″}},-{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta )\hfill \\ & =& \frac{1}{4}\left({\mathrm{ad}}_{{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}}\beta -{\mathrm{ad}}_{\beta }^t{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}},-{\mathrm{ad}}_{{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}}^t\beta \right)\hfill \\ & -& \frac{1}{4}\left({[{\alpha }^{{}^{\prime }},{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}},0\right)\hfill \\ & +& \frac{1}{4}\left({\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta }^t{\alpha }^{{}^{\prime }},-{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta \right)\hfill \\ & =& \frac{1}{4}({\mathrm{ad}}_{{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}}^t\beta -{\mathrm{ad}}_{\beta }^t{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}-{[{\alpha }^{{}^{\prime }},{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{\beta }^t{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +& {\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta -{\mathrm{ad}}_{{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta }^t{\alpha }^{{}^{\prime }},-{\mathrm{ad}}_{{[{\alpha }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}}^t\beta -{\mathrm{ad}}_{{\alpha }^{{}^{\prime }}}^t{\mathrm{ad}}_{{\alpha }^{{}^{″}}}^t\beta ).\hfill \end{array}$$

□

If ${\langle ,\rangle }_G^{*}$ is a bi-invariant pseudo-Riemannian metric on a Poisson–Lie group $(G,{\Pi }_G)$, then as a consequence of Formula (9), we have ${\mathcal{D}}_{\alpha }^G\beta =\frac{1}{2}{[\alpha ,\beta ]}_{{\mathfrak{g}}^{*}}$ and ${\mathrm{ad}}_{\alpha }^t=-{\mathrm{ad}}_{\alpha },$ for any $\alpha ,\beta \in {\mathfrak{g}}^{*}.$ (For more details in the covariant case, see [21]).

Corollary 1. 

If we let ${\langle ,\rangle }_G^{*}$ be a bi-invariant contravariant pseudo-Riemannian metric on a Poisson–Lie group $(G,{\Pi }_G)$, then for any $(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime })\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*},$ we have:

1. 

${\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}({\alpha }^{\prime },0)=({\mathcal{D}}_{\alpha }^G{\alpha }^{{}^{\prime }},0);$

2. 

${\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime })=(2{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G\alpha ,{\mathcal{D}}_{\alpha }^G{\beta }^{{}^{\prime }});$

3. 

${\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}({\alpha }^{\prime },0)=(2{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{\prime }},{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G\beta );$

4. 

${\mathcal{D}}_{(0,\beta )}^{G\times \mathfrak{g}}(0,{\beta }^{\prime })=(0,{\mathcal{D}}_{\beta }^G{\beta }^{{}^{\prime }}).$

Proof. 

Since ${\langle ,\rangle }_G^{*}$ is bi-invariant, then using Lemma 1, for example, for (2), we find:

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{(\alpha ,0)}^{G\times \mathfrak{g}}(0,{\beta }^{\prime })& =& \frac{1}{2}({\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }}-{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}^t\alpha ,-{\mathrm{ad}}_{\alpha }^t{\beta }^{{}^{\prime }})\hfill \\ & =& \frac{1}{2}(-{\mathrm{ad}}_{\alpha }{\beta }^{{}^{\prime }}+{\mathrm{ad}}_{{\beta }^{{}^{\prime }}}\alpha ,{\mathrm{ad}}_{\alpha }{\beta }^{{}^{\prime }})\hfill \\ & =& ({[{\beta }^{{}^{\prime }},\alpha ]}_{{\mathfrak{g}}^{*}},\frac{1}{2}{[\alpha ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}})\hfill \\ & =& (2{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G\alpha ,{\mathcal{D}}_{\alpha }^G{\beta }^{{}^{\prime }}).\hfill \end{array}$$

□

Corollary 2. 

If we let ${\langle ,\rangle }_G^{*}$ be a bi-invariant contravariant pseudo-Riemannian metric on a Poisson–Lie group $(G,{\Pi }_G)$, then for any $(\alpha ,\beta ),({\alpha }^{\prime },{\beta }^{\prime }),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*},$ we have:

1. 

$R^{G\times \mathfrak{g}}((\alpha ,0),({\alpha }^{\prime },0))({\alpha }^{″},0)=(R^G(\alpha ,{\alpha }^{{}^{\prime }}){\alpha }^{{}^{″}},0)$;

2. 

$$\begin{array}{ccc}\hfill R^{G\times \mathfrak{g}}((\alpha ,0),({\alpha }^{\prime },0))(0,{\beta }^{{}^{″}})& =& (2(R^G(\alpha ,{\beta }^{{}^{″}}){\alpha }^{{}^{\prime }}+R^G({\beta }^{{}^{″}},{\alpha }^{{}^{\prime }})\alpha )-{\mathcal{D}}_{{\mathcal{D}}_{{\beta }^{{}^{″}}}^G\alpha }^G{\alpha }^{{}^{\prime }}\hfill \\ & +& {\mathcal{D}}_{{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{\alpha }^{{}^{\prime }}}^G\alpha ,R^G(\alpha ,{\alpha }^{{}^{\prime }}){\beta }^{{}^{″}})\hfill \end{array};$$

3. 

$$\begin{array}{ccc}\hfill R^{G\times \mathfrak{g}}((0,\beta ),({\alpha }^{\prime },0))({\alpha }^{{}^{″}},0)& =& (2({\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G{\alpha }^{{}^{″}}-{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G{\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}-{\mathcal{D}}_{{\mathcal{D}}_{{\alpha }^{{}^{\prime \prime }}}^G\beta }^G{\alpha }^{{}^{\prime }}),{\mathcal{D}}_{{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G{\alpha }^{{}^{″}}}^G\beta \hfill \\ & -& {\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G{\mathcal{D}}_{{\alpha }^{{}^{″}}}^G\beta )\hfill \end{array};$$

4. 

$R^{G\times \mathfrak{g}}((0,\beta ),({\alpha }^{\prime },0))(0,{\beta }^{{}^{″}})=\left(4{\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{\alpha }^{{}^{\prime }}-2{\mathcal{D}}_{{\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}}^G{\alpha }^{{}^{\prime }},2{\mathcal{D}}_{{\mathcal{D}}_{{\beta }^{{}^{″}}}^G{\alpha }^{{}^{\prime }}}^G\beta -{\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G{\beta }^{{}^{″}}\right);$

5. 

$$\begin{array}{ccc}\hfill R^{G\times \mathfrak{g}}((0,\beta ),(0,{\beta }^{{}^{\prime }}))({\alpha }^{{}^{″}},0)& =& (4R^G(\beta ,{\beta }^{{}^{\prime }}){\alpha }^{{}^{″}}+2{\mathcal{D}}_{{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^G{\alpha }^{{}^{″}},{\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\alpha }^{{}^{″}}}^G{\beta }^{{}^{\prime }}\hfill \\ & -& {\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\mathcal{D}}_{{\alpha }^{{}^{″}}}^G\beta +2{\mathcal{D}}_{{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\alpha }^{{}^{″}}}^G\beta -2{\mathcal{D}}_{{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{″}}}^G{\beta }^{{}^{\prime }}-{\mathcal{D}}_{{\alpha }^{{}^{″}}}^G{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}})\hfill \end{array};$$

6. 

$R^{G\times \mathfrak{g}}((0,\beta ),(0,{\beta }^{{}^{\prime }}))(0,{\beta }^{{}^{″}})=\left(0,R^G(\beta ,{\beta }^{{}^{\prime }}){\beta }^{{}^{″}}\right).$

According to the Theorem 1 if the connection $D^G$ is flat, then the connection $D^{G\times \mathfrak{g}}$ is not necessarily flat. So, in this case, we cannot study the generalized Poisson bracket on the space of differential forms ${\Omega }^{*}(TG)$. For this reason, we focus on the Sanchez de Alvarez Poisson structure on the tangent bundle $TG$ in the following subsection.

3.2. Pseudo-Riemannian Geometry of Sanchez de Alvarez Tangent Poisson–Lie Group

In this subsection, we consider the left invariant contravariant pseudo-Riemannian metric ${\langle ,\rangle }_{TG}^{*}$ on the Sanchez de Alvarez Poisson–Lie group $(TG,{\Pi }_{TG})$, and we study the geometry of the triplet $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ and its relations with the geometry of $(G,{\Pi }_G,{\langle ,\rangle }_G^{*}).$

Proposition 2. 

Let $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ be a Poisson–Lie group equipped with the left invariant contravariant pseudo-Riemannian metric ${\langle ,\rangle }_G^{*}$ and $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ the Sanchez de Alvarez tangent Poisson–Lie group of G equipped with the left invariant pseudo-Riemannian metric ${\langle ,\rangle }_{TG}^{*}$ associated with ${\langle ,\rangle }_G^{*}$. Let ${\mathcal{D}}^{TG}$ and ${\mathcal{D}}^G$ be the Levi-Civita contravariant connections associated with $({\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ and $({\Pi }_G,{\langle ,\rangle }_G^{*})$, respectively. Then, for any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, we have:

$${\mathcal{D}}_{(\alpha ,\beta )}^{TG}({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})=\left({\mathcal{D}}_{\alpha }^G{\beta }^{{}^{\prime }}+{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{\prime }},{\mathcal{D}}_{\beta }^G{\beta }^{{}^{\prime }}\right).$$

Proof. 

According to Equations (9), (13) and (17), we obtain:

$$\begin{array}{ccc}\hfill 2{\langle {\mathcal{D}}_{(\alpha ,\beta )}^{TG}({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\rangle }_{TG}^{*}& =& {\langle {[(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}},({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\rangle }_{TG}^{*}\hfill \\ & +& {\langle {[({\alpha }^{{}^{″}},{\beta }^{{}^{″}}),(\alpha ,\beta )]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}},({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\rangle }_{TG}^{*}\hfill \\ & +& {\langle {[({\alpha }^{{}^{″}},{\beta }^{{}^{″}}),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}},(\alpha ,\beta )\rangle }_{TG}^{*}\hfill \\ & =& {\langle ({[\alpha ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[\beta ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\rangle }_{TG}^{*}\hfill \\ & +& {\langle ({[{\alpha }^{{}^{″}},\beta ]}_{{\mathfrak{g}}^{*}}+{[{\beta }^{{}^{″}},\alpha ]}_{{\mathfrak{g}}^{*}},{[{\beta }^{{}^{″}},\beta ]}_{{\mathfrak{g}}^{*}}),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\rangle }_{TG}^{*}\hfill \\ & +& \langle {[{\alpha }^{{}^{″}},{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[{\beta }^{{}^{″}},{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},{[{\beta }^{{}^{″}},{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}),(\alpha ,\beta ){\rangle }_{TG}^{*}\hfill \\ & =& 2{\langle ({\mathcal{D}}_{\alpha }^G{\beta }^{{}^{\prime }}+{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{\prime }},{\mathcal{D}}_{\beta }^G{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\rangle }_{TG}^{*}.\end{array}$$

□

Lemma 2. 

Let $R^{TG}$ and $R^G$ be the curvatures of ${\mathcal{D}}^{TG}$ and ${\mathcal{D}}^G$, respectively. Then, for any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, we have:

$$R^{TG}((\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}))({\alpha }^{{}^{″}},{\beta }^{{}^{″}})=\left(R^G(\alpha ,{\beta }^{{}^{\prime }}){\beta }^{{}^{″}}+R^G(\beta ,{\alpha }^{{}^{\prime }}){\beta }^{{}^{″}}+R^G(\beta ,{\beta }^{{}^{\prime }}){\alpha }^{{}^{″}},R^G(\beta ,{\beta }^{{}^{\prime }}){\beta }^{{}^{″}}\right).$$

Proof. 

Using the definition of the curvature tensor (1) and Proposition 2, we obtain:

$$\begin{array}{ccc}\hfill R^{TG}((\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}))({\alpha }^{{}^{″}},{\beta }^{{}^{″}})& =& {\mathcal{D}}_{(\alpha ,\beta )}^{TG}{\mathcal{D}}_{({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})}^{TG}({\alpha }^{{}^{″}},{\beta }^{{}^{″}})-{\mathcal{D}}_{({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})}^{TG}{\mathcal{D}}_{(\alpha ,\beta )}^{TG}({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\hfill \\ & -& {\mathcal{D}}_{{[(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}}^{TG}({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\hfill \\ & =& {\mathcal{D}}_{(\alpha ,\beta )}^{TG}({\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\alpha }^{{}^{″}},{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\beta }^{{}^{″}})\hfill \\ & -& {\mathcal{D}}_{({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})}^{TG}({\mathcal{D}}_{\alpha }^G{\beta }^{{}^{″}}+{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{″}},{\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}})\hfill \\ & -& {\mathcal{D}}_{({[\alpha ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[\beta ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}})}^{TG}({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\hfill \\ & =& \left({\mathcal{D}}_{\alpha }^G{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\beta }^{{}^{″}}+{\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G{\beta }^{{}^{″}}+{\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\alpha }^{{}^{″}},{\mathcal{D}}_{\beta }^G{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\beta }^{{}^{″}}\right)\hfill \\ & -& \left({\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G{\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\mathcal{D}}_{\alpha }^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{″}},{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G{\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}\right)\hfill \\ & -& \left({\mathcal{D}}_{{[\alpha ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{[\beta ,{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^G{\beta }^{{}^{″}}+{\mathcal{D}}_{{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^G{\alpha }^{{}^{″}},{\mathcal{D}}_{{[\beta ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}}^G{\beta }^{{}^{″}}\right)\hfill \\ & =& \left(R^G(\alpha ,{\beta }^{{}^{\prime }}){\beta }^{{}^{″}}+R^G(\beta ,{\alpha }^{{}^{\prime }}){\beta }^{{}^{″}}+R^G(\beta ,{\beta }^{{}^{\prime }}){\alpha }^{{}^{″}},R^G(\beta ,{\beta }^{{}^{\prime }}){\beta }^{{}^{″}}\right)\hfill \end{array}$$

□

Proposition 3. 

The Levi-Civita contravariant connection ${\mathcal{D}}^G$ is locally symmetric if and only if the connection ${\mathcal{D}}^{TG}$ is locally symmetric.

Proof. 

For any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}}),({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$ we obtain

$H=({\mathcal{D}}_{(\alpha ,\beta )}^{TG}{R}^{TG})(({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}}))({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})$. According to Equation (4), Proposition 2 and Lemma 2, we obtain:

$$\begin{array}{ccc}\hfill H& =& {\mathcal{D}}_{(\alpha ,\beta )}^{TG}(R^{TG}(({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}}))({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})-R^{TG}({\mathcal{D}}_{(\alpha ,\beta )}^{TG}({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}}))({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})\hfill \\ & -& R^{TG}(({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})){\mathcal{D}}_{(\alpha ,\beta )}^{TG}({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})-R^{TG}(({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),{\mathcal{D}}_{(\alpha ,\beta )}^{TG}({\alpha }^{{}^{″}},{\beta }^{{}^{″}}))({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})\hfill \\ & =& {\mathcal{D}}_{(\alpha ,\beta )}^{TG}\left((R^G({\alpha }^{{}^{\prime }},{\beta }^{{}^{″}}){\beta }^{{}^{‴}}+R^G({\beta }^{{}^{\prime }},{\alpha }^{{}^{″}}){\beta }^{{}^{‴}}+R^G({\beta }^{{}^{\prime }},{\beta }^{{}^{″}}){\alpha }^{{}^{‴}},R^G({\beta }^{{}^{\prime }},{\beta }^{{}^{″}}){\beta }^{{}^{‴}}\right)\hfill \\ & -& R^{TG}(({\mathcal{D}}_{\alpha }^G{\beta }^{{}^{\prime }}+{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{\prime }},{\mathcal{D}}_{\beta }^G{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}}))({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})\hfill \\ & -& R^{TG}(({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}}))({\mathcal{D}}_{\alpha }^G{\beta }^{{}^{‴}}+{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{‴}},{\mathcal{D}}_{\beta }^G{\beta }^{{}^{‴}})\hfill \\ & -& R^{TG}(({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\mathcal{D}}_{\alpha }^G{\beta }^{{}^{″}}+{\mathcal{D}}_{\beta }^G{\alpha }^{{}^{″}},{\mathcal{D}}_{\beta }^G{\beta }^{{}^{″}}))({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})\hfill \end{array}$$

By developing again with Proposition 2 and Lemma 2, we obtain:

$$\begin{array}{ccc}\hfill ({\mathcal{D}}_{(\alpha ,\beta )}^{TG}{R}^{TG})(({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}}))({\alpha }^{{}^{‴}},{\beta }^{{}^{‴}})& =& (({\mathcal{D}}_{\alpha }^G{R}^G)({\beta }^{{}^{\prime }},{\beta }^{{}^{″}}){\beta }^{{}^{‴}}+({\mathcal{D}}_{\beta }^G{R}^G)({\alpha }^{{}^{\prime }},{\beta }^{{}^{″}}){\beta }^{{}^{‴}}\hfill \\ & +& ({\mathcal{D}}_{\beta }^G{R}^G)({\beta }^{{}^{\prime }},{\alpha }^{{}^{″}}){\beta }^{{}^{‴}}\hfill \\ & +& ({\mathcal{D}}_{\beta }^G{R}^G)({\beta }^{{}^{\prime }},{\beta }^{{}^{″}}){\alpha }^{{}^{‴}},({\mathcal{D}}_{\beta }^G{R}^G)({\beta }^{{}^{\prime }},{\beta }^{{}^{″}}){\beta }^{{}^{‴}})\hfill \end{array}$$

If ${\mathcal{D}}^G{R}^G=0,$ then ${\mathcal{D}}^{TG}{R}^{TG}=(0,0).$ Conversely, if ${\mathcal{D}}^{TG}{R}^{TG}=(0,0)$, then for any $\beta ,{\beta }^{{}^{\prime }},{\beta }^{{}^{″}},{\beta }^{{}^{‴}}\in {\mathfrak{g}}^{*}$, we have

$${\mathcal{D}}_{\beta }^G{R}^G({\beta }^{{}^{\prime }},{\beta }^{{}^{″}}){\beta }^{{}^{‴}}=0.$$

Hence, ${\mathcal{D}}^G$ is locally symmetric. □

Lemma 3. 

Let ${[,]}_G$ and ${[,]}_{TG}$ be the generalized Koszul brackets on ${\Omega }^{*}(G)$ and ${\Omega }^{*}(TG)$, respectively. Then, for any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, we have:

$$\begin{array}{ccc}\hfill {[(\alpha ,\beta ),d({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})]}_{TG}& =& ({[\alpha ,d{\beta }^{{}^{\prime }}]}_G+{[\beta ,d{\alpha }^{{}^{\prime }}]}_G,{[\beta ,d{\beta }^{{}^{\prime }}]}_G)\hfill \\ & =& {({[\alpha ,d{\beta }^{{}^{\prime }}]}_G+{[\beta ,d{\alpha }^{{}^{\prime }}]}_G)}^v+{({[\beta ,d{\beta }^{{}^{\prime }}]}_G)}^c,\hfill \end{array}$$

where ${({[\alpha ,d{\beta }^{{}^{\prime }}]}_G)}^v$ (resp., ${({[\beta ,d{\beta }^{{}^{\prime }}]}_G)}^c$) is the vertical lift (resp., the complete lift) of the 2-form ${[\alpha ,d{\beta }^{{}^{\prime }}]}_G$ (resp., ${[\beta ,d{\beta }^{{}^{\prime }}]}_G$) on G to $TG.$

Proof. 

Let $(x_i)$ be local coordinates of G in a neighborhood of e and $(x_i,y_i)$ be the correspondent local coordinates of $TG,$ in a neighborhood of $(e,0)$. Let $\alpha ={\displaystyle {\displaystyle \sum _i}}{\alpha }_{i}d{x}_i$ and ${\beta }^{{}^{\prime }}={\displaystyle {\displaystyle \sum _i}}{\beta }_i^{{}^{\prime }}d{x}_i$ be elements of ${\mathfrak{g}}^{*}$. We write $(\alpha ,0)={\displaystyle {\displaystyle \sum _i}}{\alpha }_{i}d{x}_i$ and $(0,{\beta }^{{}^{\prime }})={\displaystyle {\displaystyle \sum _i}}{\beta }_i^{{}^{\prime }}d{y}_i$. Then, using Equations (6) and (13), for example, for ${[(\alpha ,0),d(0,{\beta }^{{}^{\prime }})]}_{TG}$, we have:

$$\begin{array}{ccc}\hfill {[(\alpha ,0),d(0,{\beta }^{{}^{\prime }})]}_{TG}& =& {\displaystyle \sum _i}{[(\alpha ,0),d(0,{\beta }_i^{{}^{\prime }}d{x}_i)]}_{TG}={\displaystyle \sum _i}{[(\alpha ,0),(0,d{\beta }_i^{{}^{\prime }}\wedge d{x}_i)]}_{TG}\hfill \\ & =& {\displaystyle \sum _i}\left({[(\alpha ,0),(0,d{\beta }_i^{{}^{\prime }})\wedge (d{x}_i,0)+(d{\beta }_i^{{}^{\prime }},0)\wedge (0,d{x}_i)]}_{TG}\right)\hfill \\ & =& {\displaystyle \sum _i}({[(\alpha ,0),(0,d{\beta }_i^{{}^{\prime }})]}_{TG}\wedge (d{x}_i,0)+(0,d{\beta }_i^{{}^{\prime }})\wedge {[(\alpha ,0),(d{x}_i,0)]}_{TG}\hfill \\ & +& {[(\alpha ,0),(d{\beta }_i^{{}^{\prime }},0)]}_{TG}\wedge (0,d{x}_i)+(d{\beta }_i^{{}^{\prime }},0)\wedge {[(\alpha ,0),(0,d{x}_i)]}_{TG})\hfill \\ & =& {\displaystyle \sum _i}\left(({[\alpha ,d{\beta }_i^{{}^{\prime }}]}_G,0)\wedge (d{x}_i,0)+(d{\beta }_i^{{}^{\prime }},0)\wedge ({[\alpha ,d{x}_i]}_G,0)\right)\hfill \\ & =& {\displaystyle \sum _i}\left(({[\alpha ,d{\beta }_i^{{}^{\prime }}]}_G\wedge d{x}_i,0)+(d{\beta }_i^{{}^{\prime }}\wedge {[\alpha ,d{x}_i]}_G,0)\right)\hfill \\ & =& {\displaystyle \sum _i}({[\alpha ,d{\beta }_i^{{}^{\prime }}\wedge d{x}_i]}_G,0)\hfill \\ & =& ({[\alpha ,d{\beta }^{{}^{\prime }}]}_G,0)\hfill \\ & =& {({[\alpha ,d{\beta }^{{}^{\prime }}]}_G)}^v.\hfill \end{array}$$

Considering all the possible cases

$$({[(0,\beta ),d(0,{\beta }^{{}^{\prime }})]}_{TG},{[(0,\beta ),d({\alpha }^{{}^{\prime }},0)]}_{TG},{[(\alpha ,0),d(0,{\beta }^{{}^{\prime }})]}_{TG},{[(\alpha ,0),d({\alpha }^{{}^{\prime }},0)]}_{TG}),$$

we obtain the following lemma. □

Proposition 4. 

Let ${\{,\}}_{TG}$ and ${\{,\}}_G$ be the Hawkins generalized pre-Poisson brackets of the Levi-Civita contravariant connections ${\mathcal{D}}^{TG}$ and ${\mathcal{D}}^G$, respectively. Then, for any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, we have:

$$\begin{array}{ccc}\hfill {\{(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\}}_{TG}& =& ({\{\alpha ,{\beta }^{{}^{\prime }}\}}_G+{\{\beta ,{\alpha }^{{}^{\prime }}\}}_G,{\{\beta ,{\beta }^{{}^{\prime }}\}}_G)\hfill \\ & =& {({\{\alpha ,{\beta }^{{}^{\prime }}\}}_G+{\{\beta ,{\alpha }^{{}^{\prime }}\}}_G)}^v+{({\{\beta ,{\beta }^{{}^{\prime }}\}}_G)}^c.\hfill \end{array}$$

Proof. 

Note that the Levi-Civita contravariant connections ${\mathcal{D}}^G$ and ${\mathcal{D}}^{TG}$ naturally extend to ${\Omega }^2(G)$ and ${\Omega }^2(TG)$, respectively. Using Equation (5), Proposition 2 and Lemma 3, we obtain

$$\begin{array}{ccc}\hfill {\{(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\}}_{TG}& =& -{\mathcal{D}}_{(\alpha ,\beta )}^{TG}d({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})-{\mathcal{D}}_{({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})}^{TG}d(\alpha ,\beta )+d{\mathcal{D}}_{({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})}^{TG}(\alpha ,\beta )\hfill \\ & +& {[(\alpha ,\beta ),d({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})]}_{TG}\hfill \\ & =& -({\mathcal{D}}_{\alpha }^{G}d{\beta }^{{}^{\prime }}+{\mathcal{D}}_{\beta }^{G}d{\alpha }^{{}^{\prime }},{\mathcal{D}}_{\beta }^{G}d{\beta }^{{}^{\prime }})-({\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^{G}d\beta +{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^{G}d\alpha ,{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^{G}d\beta )\hfill \\ & +& (d{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G\beta +d{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G\alpha ,d{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G\beta )+({[\alpha ,d{\beta }^{{}^{\prime }}]}_G+{[\beta ,d{\alpha }^{{}^{\prime }}]}_G,{[\beta ,d{\beta }^{{}^{\prime }}]}_G)\hfill \\ & =& (-{\mathcal{D}}_{\alpha }^{G}d{\beta }^{{}^{\prime }}-{\mathcal{D}}_{\beta }^{G}d{\alpha }^{{}^{\prime }}-{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^{G}d\beta -{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^{G}d\alpha +d{\mathcal{D}}_{{\alpha }^{{}^{\prime }}}^G\beta +d{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G\alpha \hfill \\ & +& {[\alpha ,d{\beta }^{{}^{\prime }}]}_G+{[\beta ,d{\alpha }^{{}^{\prime }}]}_G,-{\mathcal{D}}_{\beta }^{G}d{\beta }^{{}^{\prime }}-{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^{G}d\beta +d{\mathcal{D}}_{{\beta }^{{}^{\prime }}}^G\beta +{[\beta ,d{\beta }^{{}^{\prime }}]}_G)\hfill \\ & =& \left({\{\alpha ,{\beta }^{{}^{\prime }}\}}_G+{\{\beta ,{\alpha }^{{}^{\prime }}\}}_G,{\{\beta ,{\beta }^{{}^{\prime }}\}}_G\right)\hfill \\ & =& {({\{\alpha ,{\beta }^{{}^{\prime }}\}}_G+{\{\beta ,{\alpha }^{{}^{\prime }}\}}_G)}^v+{({\{\beta ,{\beta }^{{}^{\prime }}\}}_G)}^c.\hfill \end{array}$$

□

Lemma 4. 

Let ${\mathcal{M}}^G$ and ${\mathcal{M}}^{TG}$ be the metacurvatures of the Levi-Civita contravariant connections ${\mathcal{D}}^G$ and ${\mathcal{D}}^{TG}$, respectively. Then, for any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, we have:

1. 

${\mathcal{M}}^{TG}((\alpha ,0),({\alpha }^{{}^{\prime }},0),({\alpha }^{{}^{″}},0))=0;$

2. 

${\mathcal{M}}^{TG}((\alpha ,0),({\alpha }^{{}^{\prime }},0),(0,{\beta }^{{}^{″}}))=0;$

3. 

${\mathcal{M}}^{TG}((\alpha ,0),(0,{\beta }^{{}^{\prime }}),(0,{\beta }^{{}^{″}}))=({\mathcal{M}}^G(\alpha ,{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}),0)$;

4. 

${\mathcal{M}}^{TG}((0,\beta ),(0,{\beta }^{{}^{\prime }}),(0,{\beta }^{{}^{″}}))=(0,{\mathcal{M}}^G(\beta ,{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}));$

5. 

${\mathcal{M}}^{TG}((0,\beta ),(0,{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},0))=({\mathcal{M}}^G(\beta ,{\beta }^{{}^{\prime }},{\alpha }^{{}^{″}}),0);$

6. 

${\mathcal{M}}^{TG}((0,\beta ),({\alpha }^{{}^{\prime }},0),({\alpha }^{{}^{″}},0))=0.$

Proof. 

Let $(x_i)$ be local coordinates of G in a neighborhood of e and let $(x_i,y_i)$ be the correspondent local coordinates of $TG$ in a neighborhood of $(e,0)$. Let $\alpha ={\displaystyle {\displaystyle \sum _i}}{\alpha }_{i}d{x}_i$ and $\beta ={\displaystyle {\displaystyle \sum _i}}{\beta }_{i}d{x}_i$ be elements of ${\mathfrak{g}}^{*}$. We write $(\alpha ,0)={\displaystyle {\displaystyle \sum _i}}{\alpha }_{i}d{x}_i$ and $(0,\beta )={\displaystyle \sum _i}{\beta }_{i}d{y}_i$. Using Equation (7) and Propositions 2 and 4, then—for example, for (3)—we obtain:

$$\begin{array}{ccc}\hfill {\mathcal{M}}^{TG}((\alpha ,0),(0,{\beta }^{{}^{\prime }}),(0,{\beta }^{{}^{″}}))& =& {\displaystyle \sum _i}{\alpha }_i({\{x_i\circ \pi ,{\{(0,{\beta }^{{}^{\prime }}),(0,{\beta }^{{}^{″}})\}}_{TG}\}}_{TG}\hfill \\ & -& {\{{\{x_i\circ \pi ,(0,{\beta }^{{}^{\prime }})\}}_{TG},(0,{\beta }^{{}^{″}})\}}_{TG}\hfill \\ & -& {\{{\{x_i\circ \pi ,(0,{\beta }^{{}^{″}})\}}_{TG},(0,{\beta }^{{}^{\prime }})\}}_{TG})\hfill \\ & =& {\displaystyle \sum _i}{\alpha }_i({\mathcal{D}}_{(d{x}_i,0)}^{TG}(0,{\{{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}\}}_G)-{\{{\mathcal{D}}_{(d{x}_i,0)}^{TG}(0,{\beta }^{{}^{\prime }}),(0,{\beta }^{{}^{″}})\}}_{TG}\hfill \\ & -& {\{{\mathcal{D}}_{(d{x}_i,0)}^{TG}(0,{\beta }^{{}^{″}}),(0,{\beta }^{{}^{\prime }})\}}_{TG})\hfill \\ & =& {\displaystyle \sum _i}{\alpha }_i(({\mathcal{D}}_{d{x}_i}^G{\{{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}\}}_G,0)-({\{{\mathcal{D}}_{d{x}_i}^G{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}\}}_G,0)\hfill \\ & -& ({\{{\mathcal{D}}_{d{x}_i}^G{\beta }^{{}^{″}},{\beta }^{{}^{\prime }}\}}_G,0))\hfill \\ & =& {\displaystyle \sum _i}{\alpha }_i(({\{x_i,{\{{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}\}}_G\}}_G,0)-({\{{\{x_i,{\beta }^{{}^{\prime }}\}}_G,{\beta }^{{}^{″}}\}}_G,0)\hfill \\ & -& ({\{{\{x_i,{\beta }^{{}^{″}}\}}_G,{\beta }^{{}^{\prime }}\}}_G,0))\hfill \\ & =& ({\mathcal{M}}^G(\alpha ,{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}),0).\hfill \end{array}$$

□

Theorem 2. 

Let $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ be a Poisson–Lie group equipped with the left invariant contravariant pseudo-Riemannian metric ${\langle ,\rangle }_G^{*}$ and $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ the Sanchez de Alvarez tangent Poisson–Lie group of G equipped with the left invariant pseudo-Riemannian metric ${\langle ,\rangle }_{TG}^{*}$ associated with ${\langle ,\rangle }_G^{*}$. Then, the space of the differential form ${\Omega }^{*}(G)$ is a differential graded Poisson algebra if, and only if, ${\Omega }^{*}(TG)$ is a differential graded Poisson algebra.

Proof. 

According to Lemma 2, if $R^G=0,$ then $R^{TG}=(0,0).$ We now assume that $R^{TG}=(0,0)$; then, for any $\beta ,{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}\in {\mathfrak{g}}^{*}$, we have

$$R^G(\beta ,{\beta }^{{}^{\prime }}){\beta }^{{}^{″}}=0.$$

Then, ${\mathcal{D}}^G$ is flat if, and only if, ${\mathcal{D}}^{TG}$ is flat.

Moreover, According to Lemma 4, for any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, we obtain:

$$\begin{array}{ccc}\hfill {\mathcal{M}}^{TG}((\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}}))& =& ({\mathcal{M}}^G(\alpha ,{\beta }^{{}^{\prime }},{\beta }^{{}^{″}})+{\mathcal{M}}^G(\beta ,{\alpha }^{{}^{\prime }},{\beta }^{{}^{″}})\hfill \\ & +& {\mathcal{M}}^G(\beta ,{\beta }^{{}^{\prime }},{\alpha }^{{}^{″}}),{\mathcal{M}}^G(\beta ,{\beta }^{{}^{\prime }},{\beta }^{{}^{″}})).\hfill \end{array}$$

So, if ${\mathcal{M}}^G=0,$ then ${\mathcal{M}}^{TG}=(0,0).$ We now assume that ${\mathcal{M}}^{TG}=(0,0)$; then, for any $\beta ,{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}\in {\mathfrak{g}}^{*}$, we have

$${\mathcal{M}}^G(\beta ,{\beta }^{{}^{\prime }}){\beta }^{{}^{″}}=0.$$

Then, ${\mathcal{D}}^G$ is metaflat if, and only if, ${\mathcal{D}}^{TG}$ is metaflat.

Hence, we deduce that the connection ${\mathcal{D}}^G$ defines a generalized Poisson bracket ${\{,\}}_G$ on ${\Omega }^{*}(G)$ if, and only if, the connection ${\mathcal{D}}^{TG}$ defines a generalized Poisson bracket ${\{,\}}_{TG}$ on ${\Omega }^{*}(TG)$. □

4. Pseudo-Riemannian Sanchez de Alvarez Tangent Poisson–Lie Group

The second author and N.Zaalani [12] showed that the Sanchez de Alvarez tangent Poisson–Lie group $(TG,{\Pi }_{TG})$ equipped with the natural left invariant Riemannian metric is a Riemannian Poisson–Lie group if, and only if, $(G,{\Pi }_G)$ is a trivial Poisson–Lie group. In this section, we study the compatibility in the sense of M.Boucetta between the Sanchez de Alvarez Poisson–Lie structure ${\Pi }_{TG}$ and the pseudo-Riemannian metric ${\langle ,\rangle }_{TG}^{*}$ given in (17).

Let ${\langle ,\rangle }_{{\mathfrak{g}}^{*}}$ be a bilinear, symmetric and non-degenerate form on ${\mathfrak{g}}^{*}$. We define a bilinear, symmetric and non-degenerate form ${\langle ,\rangle }_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}}$ on ${\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, which is analogous to (17), as follows:

$$\begin{array}{ccc}\hfill {\langle (\alpha ,0),({\alpha }^{{}^{\prime }},0)\rangle }_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}}& =& 0,\hfill \\ \hfill {\langle (\alpha ,0),(0,{\beta }^{{}^{\prime }})\rangle }_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}}& =& {\langle \alpha ,{\beta }^{{}^{\prime }}\rangle }_{{\mathfrak{g}}^{*}}\hfill \\ \hfill {\langle (0,\beta ),(0,{\beta }^{{}^{\prime }})\rangle }_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}}& =& {\langle \beta ,{\beta }^{{}^{\prime }}\rangle }_{{\mathfrak{g}}^{*}}.\hfill \end{array}$$

where $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$.

Let ${\langle ,\rangle }_G^{*}$ be the left invariant contravariant pseudo-Riemannian metric associated with ${\langle ,\rangle }_{{\mathfrak{g}}^{*}}$ and let ${\langle ,\rangle }_{TG}^{*}$ be the metric associated with ${\langle ,\rangle }_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}}$.

Remark 1.

If $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ is a pseudo-Riemannian Poisson Lie group, then its dual Lie algebra $({\mathfrak{g}}^{*},{[,]}_{{\mathfrak{g}}^{*}},{\langle ,\rangle }_{{\mathfrak{g}}^{*}})$ equipped with the form ${\langle ,\rangle }_{{\mathfrak{g}}^{*}}$ is a pseudo-Riemannian Lie algebra and the abelian Poisson–Lie group $(\mathfrak{g},{\Pi }_{\mathfrak{g}},{\langle ,\rangle }_{\mathfrak{g}})$ equipped with the form ${\langle ,\rangle }_{\mathfrak{g}}$ associated with ${\langle ,\rangle }_{{\mathfrak{g}}^{*}}$ is a pseudo-Riemannian Poisson–Lie group [4].

Theorem 3. 

Let $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ be a Poisson–Lie group equipped with the left invariant contravariant pseudo-Riemannian metric ${\langle ,\rangle }_G^{*}$ and let $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ be the Sanchez de Alvarez tangent Poisson–Lie group of G equipped with the left invariant pseudo-Riemannian metric ${\langle ,\rangle }_{TG}^{*}$. Then, $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ is a pseudo-Riemannian Poisson–Lie group if, and only if, $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ is a pseudo-Riemannian Poisson–Lie group.

Proof. 

Note that the linear transformation $A{d}_g^{*}:{\mathfrak{g}}^{*}\to {\mathfrak{g}}^{*}$ is a Lie algebra automorphism [22].

The infinitesimal Levi-Civita connection B associated with $({[,]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}},{\langle ,\rangle }_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}})$ is given for any $(\alpha ,{\alpha }^{{}^{\prime }}),(\gamma ,{\gamma }^{{}^{\prime }})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$ by:

$$B_{(\alpha ,{\alpha }^{{}^{\prime }})}(\gamma ,{\gamma }^{{}^{\prime }})=(A_{\alpha }{\gamma }^{{}^{\prime }}+A_{{\alpha }^{{}^{\prime }}}\gamma ,A_{{\alpha }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}),$$

where A is the infinitesimal Levi-Civita connection associated with $({[,]}_{{\mathfrak{g}}^{*}},{\langle ,\rangle }_{{\mathfrak{g}}^{*}})$, respectively.

For any $(X,Y)\in \mathfrak{g}⋊\mathfrak{g}$ and $(\gamma ,{\gamma }^{{}^{\prime }})\in {\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*},$

$$a{d}_{(X,Y)}^{*}(\gamma ,{\gamma }^{{}^{\prime }})=(a{d}_X^{*}\gamma +a{d}_Y^{*}{\gamma }^{{}^{\prime }},a{d}_X^{*}{\gamma }^{{}^{\prime }}).$$

Let $(x_i)$ be local coordinates of G in a neighborhood of e and let $(x_i,y_i)$ be the correspondent local coordinates of $TG$. The Poisson tensors of G and $TG$ are expressed by [10]:

$${\Pi }_G=\sum _{i,j}{\Pi }_G^{ij}\frac{\partial }{{\partial }_{x_i}}\wedge \frac{\partial }{{\partial }_{x_j}},$$

and

$${\Pi }_{TG}=\sum _{i,j,k}{\Pi }_G^{ij}\frac{\partial }{{\partial }_{x_i}}\wedge \frac{\partial }{{\partial }_{y_j}}+y_k\frac{\partial {\Pi }_G^{ij}}{{\partial }_{x_k}}\frac{\partial }{{\partial }_{y_i}}\wedge \frac{\partial }{{\partial }_{y_j}},$$

(18)

respectively. Then, for any $(g,X)\in TG$ and for any $(\alpha ,{\alpha }^{{}^{\prime }})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, we have

$${\Pi }_{TG}^l(g,X)(\alpha ,{\alpha }^{{}^{\prime }})=({\Pi }_G^l(g)({\alpha }^{{}^{\prime }}),{\Pi }_G^l(g)(\alpha )+{\Pi }_{\mathfrak{g}}(X)({\alpha }^{{}^{\prime }})),$$

where ${\Pi }_{\mathfrak{g}}$ is the linear Poisson structure on $\mathfrak{g}$ associated with ${\Pi }_G$.

Then, for any $(\alpha ,{\alpha }^{{}^{\prime }}),(\beta ,{\beta }^{{}^{\prime }}),(\gamma ,{\gamma }^{{}^{\prime }})\in {\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}$, we obtain:

$$\begin{array}{ccc}& & {[B_{(\alpha ,{\alpha }^{{}^{\prime }})}(\gamma ,{\gamma }^{{}^{\prime }})+a{d}_{{\Pi }_{TG}^l(g,X)(\alpha ,{\alpha }^{{}^{\prime }})}^{*}(\gamma ,{\gamma }^{{}^{\prime }}),(\beta ,{\beta }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}\hfill \\ & +& {[(\alpha ,{\alpha }^{{}^{\prime }}),B_{(\beta ,{\beta }^{{}^{\prime }})}(\gamma ,{\gamma }^{{}^{\prime }})+a{d}_{{\Pi }_{TG}^l(g,X)(\beta ,{\beta }^{{}^{\prime }})}^{*}(\gamma ,{\gamma }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}\hfill \\ & =& {[(A_{\alpha }{\gamma }^{{}^{\prime }}+A_{{\alpha }^{{}^{\prime }}}\gamma ,A_{{\alpha }^{{}^{\prime }}}{\gamma }^{{}^{\prime }})+a{d}_{({\Pi }_G^l(g)({\alpha }^{{}^{\prime }}),{\Pi }_G^l(g)(\alpha )+{\Pi }_{\mathfrak{g}}(X)({\alpha }^{{}^{\prime }}))}^{*}(\gamma ,{\gamma }^{{}^{\prime }}),(\beta ,{\beta }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}\hfill \\ & +& {[(\alpha ,{\alpha }^{{}^{\prime }}),(A_{\beta }{\gamma }^{{}^{\prime }}+A_{{\beta }^{{}^{\prime }}}\gamma ,A_{{\beta }^{{}^{\prime }}}{\gamma }^{{}^{\prime }})+a{d}_{({\Pi }_G^l(g)({\beta }^{{}^{\prime }}),{\Pi }_G^l(g)(\beta )+{\Pi }_{\mathfrak{g}}(X)({\beta }^{{}^{\prime }}))}^{*}(\gamma ,{\gamma }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}\hfill \\ & =& [(A_{\alpha }{\gamma }^{{}^{\prime }}+A_{{\alpha }^{{}^{\prime }}}\gamma +a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}\gamma +a{d}_{{\Pi }_G^l(g)(\alpha )+{\Pi }_{\mathfrak{g}}(X)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},A_{{\alpha }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}\hfill \\ & +& a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }}{),(\beta ,{\beta }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}\hfill \\ & +& [(\alpha ,{\alpha }^{{}^{\prime }}),(A_{\beta }{\gamma }^{{}^{\prime }}+A_{{\beta }^{{}^{\prime }}}\gamma +a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}\gamma +a{d}_{{\Pi }_G^l(g)(\beta )+{\Pi }_{\mathfrak{g}}(X)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},A_{{\beta }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}\hfill \\ & +& a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }}{)]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}\hfill \\ & =& ({[A_{\alpha }{\gamma }^{{}^{\prime }}+A_{{\alpha }^{{}^{\prime }}}\gamma +a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}\gamma +a{d}_{{\Pi }_G^l(g)(\alpha )}^{*}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_{\mathfrak{g}}(X)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +& {[A_{{\alpha }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},\beta ]}_{{\mathfrak{g}}^{*}},{[A_{{\alpha }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},\beta ]}_{{\mathfrak{g}}^{*}})\hfill \\ & +& ({[\alpha ,A_{{\beta }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+[\alpha ,A_{\beta }{\gamma }^{{}^{\prime }}+A_{{\beta }^{{}^{\prime }}}\gamma +a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}\gamma +a{d}_{{\Pi }_G^l(g)(\beta )}^{*}{\gamma }^{{}^{\prime }}\hfill \\ & +& a{d}_{{\Pi }_{\mathfrak{g}}(X)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }}{]}_{{\mathfrak{g}}^{*}},[{\alpha }^{{}^{\prime }},A_{{\beta }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }})\hfill \\ & =& ({[A_{\alpha }{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)(\alpha )}^{*}{\gamma }^{{}^{\prime }},{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[\alpha ,A_{{\beta }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[A_{{\alpha }^{{}^{\prime }}}\gamma +a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}\gamma ,{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +& {[{\alpha }^{{}^{\prime }},A_{{\beta }^{{}^{\prime }}}\gamma +a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}\gamma ]}_{{\mathfrak{g}}^{*}}+{[A_{{\alpha }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},\beta ]}_{{\mathfrak{g}}^{*}}+{[{\alpha }^{{}^{\prime }},A_{\beta }{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)(\beta )}^{*}{\gamma }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +& {[a{d}_{{\Pi }_{\mathfrak{g}}(X)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[{\alpha }^{{}^{\prime }},a{d}_{{\Pi }_{\mathfrak{g}}(X)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}},{[A_{{\alpha }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}\hfill \\ & +& {[{\alpha }^{{}^{\prime }},A_{{\beta }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}).\hfill \end{array}$$

Then, using Remark 1, if $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ is a pseudo-Riemannian Poisson–Lie group, then $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ is a pseudo-Riemannian Poisson–Lie group. Conversely, if $(TG,{\Pi }_{TG},{\langle ,\rangle }_{TG}^{*})$ is a pseudo-Riemannian Poisson–Lie group, then for any $x\in G$ and for any ${\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }},{\gamma }^{{}^{\prime }}\in {\mathfrak{g}}^{*}$, we have

$${[A_{{\alpha }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\alpha }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }},{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[{\alpha }^{{}^{\prime }},A_{{\beta }^{{}^{\prime }}}{\gamma }^{{}^{\prime }}+a{d}_{{\Pi }_G^l(g)({\beta }^{{}^{\prime }})}^{*}{\gamma }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}=0.$$

Therefore, $(G,{\Pi }_G,{\langle ,\rangle }_G^{*})$ is a pseudo-Riemannian Poisson–Lie group. □

Corollary 3. 

The semi-direct product Lie algebra $({\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*},{[,]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}},{\langle ,\rangle }_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}})$ equipped with the form ${\langle ,\rangle }_{{\mathfrak{g}}^{*}\times {\mathfrak{g}}^{*}}$ is a pseudo-Riemannian Lie algebra if, and only if, $({\mathfrak{g}}^{*},{[,]}_{{\mathfrak{g}}^{*}},{\langle ,\rangle }_{{\mathfrak{g}}^{*}})$ is a pseudo-Riemannian Lie algebra.

Proof. 

According to Equation (10), for any $(\alpha ,\beta ),({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})\in {\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*},$ we obtain:

$$\begin{array}{ccc}& & {[B_{(\alpha ,\beta )}({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }}),({\alpha }^{{}^{″}},{\beta }^{{}^{″}})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}+{[(\alpha ,\beta ),B_{({\alpha }^{{}^{″}},{\beta }^{{}^{″}})}({\alpha }^{{}^{\prime }},{\beta }^{{}^{\prime }})]}_{{\mathfrak{g}}^{*}⋉{\mathfrak{g}}^{*}}\hfill \\ & =& ({[A_{\alpha }{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}+{[\alpha ,A_{{\beta }^{{}^{″}}}{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[A_{\beta }{\alpha }^{{}^{\prime }},{\beta }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}+{[\beta ,A_{{\beta }^{{}^{″}}}{\alpha }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}+{[A_{\beta }{\beta }^{{}^{\prime }},{\alpha }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}\hfill \\ & & +{[\beta ,A_{{\alpha }^{{}^{″}}}{\beta }^{{}^{\prime }}]}_{\mathfrak{g}},{[A_{\beta }{\beta }^{{}^{\prime }},{\beta }^{{}^{″}}]}_{{\mathfrak{g}}^{*}}+{[\beta ,A_{{\beta }^{{}^{″}}}{\beta }^{{}^{\prime }}]}_{{\mathfrak{g}}^{*}}),\hfill \end{array}$$

Then we obtain the corollary. □

5. Examples

Let $(x_i)$ be local coordinates of G in a neighborhood of e and let $(x_i,y_i)$ be the correspondent local coordinates of $TG$. The pseudo-Riemannian metrics on G and $TG$ are expressed by:

$${\langle ,\rangle }_G=\sum _{i,j}{g}^{ij}d{x}_i\otimes d{x}_j$$

and

$${\langle ,\rangle }_{TG}=\sum _{i,j,k}{y}_k\frac{\partial g^{ij}}{{\partial }_{x^k}}d{x}_i\otimes d{x}_j+g^{ij}d{x}_i\otimes d{y}_j+g^{ij}d{y}_i\otimes d{x}_j,$$

(19)

respectively.

• Let $(e_1,e_2,e_3)$ be an orthonormal basis of ${\mathbb{R}}^3$. The Lie algebra ${\mathbb{R}}^3$ with the bracket

  $${[e_1,e_2]}_{{\mathbb{R}}^3}=\lambda e_3,{[e_1,e_3]}_{{\mathbb{R}}^3}=-\lambda e_2,{[e_2,e_3]}_{{\mathbb{R}}^3}=0,\lambda <0,$$

  is a Riemannian Lie algebra [4]. The infinitesimal situation can be integrated, and we obtain that the triplet $({\mathbb{R}}^3,{\Pi }_{{\mathbb{R}}^3},{\langle ,\rangle }_{{\mathbb{R}}^3})$ is a Riemannian Poisson Lie group, where ${\mathbb{R}}^3$ is equipped with its abelian Lie group structure, ${\langle ,\rangle }_{{\mathbb{R}}^3}$ its canonical Euclidian metric and

  $${\Pi }_{{\mathbb{R}}^3}=\lambda \frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_y}-y\frac{\partial }{{\partial }_z}),\lambda <0.$$
  Using Equations (18) and (19), the six-dimensional Sanchez de Alvarez tangent Poisson–Lie group $(T{\mathbb{R}}^3\equiv {\mathbb{R}}^6,{\Pi }_{{\mathbb{R}}^6},{\langle ,\rangle }_{TG})$, where ${\mathbb{R}}^6$ is equipped with its abelian Lie group structure with coordinate $(x,y,z,u,v,w)$,

  $${\Pi }_{{\mathbb{R}}^6}=\lambda \frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_v}-y\frac{\partial }{{\partial }_w})+\lambda \frac{\partial }{{\partial }_u}\wedge (w\frac{\partial }{{\partial }_v}-v\frac{\partial }{{\partial }_w})$$

  and

  $${\langle ,\rangle }_{{\mathbb{R}}^6}=dxdu+dydv+dzdw+dudx+dvdy+dwdz,$$

  is a pseudo-Riemannian Poisson–Lie group.
• The Poisson–Lie group $({\mathbb{R}}^4,{\Pi }_{{\mathbb{R}}^4},{\langle ,\rangle }_{{\mathbb{R}}^4})$, where

  $${\Pi }_{{\mathbb{R}}^4}=\frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_t}-t\frac{\partial }{{\partial }_z}),{\langle ,\rangle }_{{\mathbb{R}}^4}=d{x}^2+d{y}^2+d{z}^2+d{t}^2,$$

  is compatible in the sense of Hawkins and is also a Riemannian Poisson–Lie group [22]. Then, the eight-dimensional tangent Poisson–Lie group $(T{\mathbb{R}}^4={\mathbb{R}}^8,{\Pi }_{{\mathbb{R}}^8},{\langle ,\rangle }_{{\mathbb{R}}^8})$, with coordinates $(x,y,z,t,u,v,w,s)$,

  $${\Pi }_{{\mathbb{R}}^8}=\frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_s}-t\frac{\partial }{{\partial }_w})+\frac{\partial }{{\partial }_u}\wedge (w\frac{\partial }{{\partial }_s}-s\frac{\partial }{{\partial }_w})$$

  and

  $${\langle ,\rangle }_{{\mathbb{R}}^8}=dxdu+dydv+dzdw+dtds+dudx+dvdy+dwdz+dsdt$$

  is also compatible in the sense of Hawkins and a pseudo-Riemannian Poisson–Lie group.
• By [22], the four-dimensional torus $({\mathbb{T}}^4={\mathbb{R}}^4/{\mathbb{Z}}^4,{\Pi }_{{\mathbb{T}}^4},{\langle ,\rangle }_G)$, is a Riemannian Poisson–Lie group (resp., compatible in the sense of Hawkins), where

  $${\mathbb{T}}^4=\{(e^{ix},e^{iy},e^{iz},e^{it})/x,y,z,t\in [0,2\pi ]\},$$

  $${\Pi }_{{\mathbb{T}}^4}=\frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_t}-t\frac{\partial }{{\partial }_z})\mathrm{and}g=d{x}^2+d{y}^2+d{z}^2+d{t}^2.$$
  Then, the eight-dimensional tangent Poisson–Lie group $(T{\mathbb{T}}^4,{\Pi }_{T{\mathbb{T}}^4},{\langle ,\rangle }_{TG})$, with coordinates $(x,y,z,t,u,v,w,s)$,

  $${\Pi }_{T{\mathbb{T}}^4}=\frac{\partial }{{\partial }_x}\wedge (z\frac{\partial }{{\partial }_s}-t\frac{\partial }{{\partial }_w})+\frac{\partial }{{\partial }_u}\wedge (w\frac{\partial }{{\partial }_s}-s\frac{\partial }{{\partial }_w})$$

  and

  $${\langle ,\rangle }_{T{\mathbb{T}}^4}=dxdu+dydv+dzdw+dtds+dudx+dvdy+dwdz+dsdt$$

  is also a pseudo-Riemannian Poisson–Lie group (resp., compatible in the sense of Hawkins).