Search Results (20)

Let X be a compact Riemann surface of genus $g\ge 2$. An octonionic bundle over X is a fiber bundle whose fiber is the non-associative algebra of complex octonions, equivalently a principal $G_2\left(\mathbb{C}\right)$-bundle, where $G_2\left(\mathbb{C}\right)$ is the exceptional Lie group of automorphisms of the octonions. We prove that the natural inclusion $G_2\left(\mathbb{C}\right)\hookrightarrow \mathrm{SO}(7,\mathbb{C})$ induces a closed embedding of the moduli space ${\mathcal{M}}_{\mathrm{Oct}}\left(X\right)$ into the moduli space ${\mathcal{M}}_{\mathrm{SO}(7,\mathbb{C})}\left(X\right)$ of $\mathrm{SO}(7,\mathbb{C})$-bundles. We further analyze the normal bundle to this embedding, computing its rank as $7(g-1)$ and providing an explicit cohomological description of its fibers, which enables explicit computations of tangent spaces and provides a foundation for deformation theory. As applications of the embedding, we prove that the image is a closed irreducible subvariety not contained in the singular locus of the ambient space, and we derive the Whitney formula $c\left(T_{\mathrm{amb}}\right)=c\left(T\right)·c\left(N\right)$ relating the Chern classes of the tangent bundle of ${\mathcal{M}}_{\mathrm{Oct}}\left(X\right)$, the pullback of the ambient tangent bundle, and the normal bundle over the smooth locus. Full article

Linear vector fields on Lie groups constitute a fundamental class of dynamical systems, as their flows are one-parameter subgroups of automorphisms and their infinitesimal behavior is entirely determined by derivations of the Lie algebra. When a Lie group is endowed with a Hamiltonian-type geometric structure, a natural problem is to determine whether such linear dynamics admit a global variational realization, and how such realizations can be interpreted in terms of reduced models of fluid motion. In the even-dimensional case, where the Lie group carries a symplectic structure, we establish a complete global criterion for the existence of Hamiltonians generating linear symplectic vector fields. The problem reduces to a single global obstruction: the de Rham cohomology class of the 1-form ${\iota }_{\mathcal{X}}\omega$. Thus, every linear symplectic vector field on a simply connected Lie group is globally Hamiltonian, and when the obstruction vanishes, we provide an explicit constructive procedure to recover the Hamiltonian. On the affine group ${\mathrm{Aff}}^{+}\left(1\right)$, this yields a fully explicit, finite-dimensional Hamiltonian model of a 1D ideal fluid with affine symmetries. We then treat odd-dimensional Lie groups, where symplectic geometry is unavailable. Using contact geometry as the canonical replacement, we prove a Hamiltonian lifting theorem ensuring the existence and uniqueness of the associated dynamics. The Reeb vector field appears as a distinguished vertical direction resolving the ambiguities of degenerate Hamiltonian systems. On the Heisenberg group $H_3$, this gives a fully explicit contact Hamiltonian model of an effective non-conservative fluid mode. Finally, we interpret symplectic and contact theories within a unified geometric framework and discuss their relevance to geometric formulations of ideal (symplectic) and effective (contact) fluid equations on Lie groups. Full article

The explanation of thermodynamics through geometric models was initiated by seminal figures such as Carnot, Gibbs, Duhem, Reeb, and Carathéodory. Only recently, however, has the symplectic foliation model, introduced within the domain of geometric statistical mechanics, provided a geometric definition of entropy as an invariant Casimir function on symplectic leaves—specifically, the coadjoint orbits of the Lie group acting on the system, where these orbits are interpreted as level sets of entropy. We present a symplectic foliation interpretation of thermodynamics, based on Jean-Marie Souriau’s Lie group thermodynamics. This model offers a Lie algebra cohomological characterization of entropy, viewed as an invariant Casimir function in the coadjoint representation. The dual space of the Lie algebra is foliated into coadjoint orbits, which are identified with the level sets of entropy. Within the framework of thermodynamics, dynamics on symplectic leaves—described by the Poisson bracket—are associated with non-dissipative phenomena. Conversely, on the transversal Riemannian foliation (defined by the level sets of energy), the dynamics, characterized by the metric flow bracket, induce entropy production as transitions occur from one symplectic leaf to another. Full article

Wigner’s classification has led to the insight that projective unitary representations play a prominent role in quantum mechanics. The physics literature often states that the theory of projective unitary representations can be reduced to the theory of ordinary unitary representations by enlarging the group of physical symmetries. Nevertheless, the enlargement process is not always described explicitly: it is unclear in which cases the enlargement has to be conducted on the universal cover, a central extension, or a central extension of the universal cover. On the other hand, in the mathematical literature, projective unitary representations have been extensively studied, and famous theorems such as the theorems of Bargmann and Cassinelli have been achieved. The present article bridges the two: we provide a precise, step-by-step guide on describing projective unitary representations as unitary representations of the enlarged group. Particular focus is paid to the difference between algebraic and topological obstructions. To build the bridge mentioned above, we present a detailed review of the difference between group cohomology and Lie group cohomology. This culminates in classifying Lie group central extensions by smooth cocycles around the identity. Finally, the take-away message is a hands-on algorithm that takes the symmetry group of a given quantum theory as input and provides the enlarged group as output. This algorithm is applied to several cases of physical interest. We also briefly outline a generalization of Bargmann’s theory to time-dependent phases using Hilbert bundles. Full article

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota–Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota–Baxter pre-Lie 2-algebras. We show that skeletal modified Rota–Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota–Baxter pre-Lie algebras. Full article

In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order $n+1$ deformation if and only if the obstruction class in the second cohomology group is trivial. Full article

The main goal of this paper was to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, and ${\alpha }^k$-derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from the one given by Guan and Chen. This definition is inspired by the process of the queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we developed a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we established a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology. Full article

In this paper, the notion of generalized Reynolds operators on Lie-Yamaguti algebras is introduced, and the cohomology of a generalized Reynolds operator is established. The formal deformations of a generalized Reynolds operator are studied using the first cohomology group. Then, we show that a Nijenhuis operator on a Lie-Yamaguti algebra gives rise to a representation of the deformed Lie-Yamaguti algebra and a 2-cocycle. Consequently, the identity map will be a generalized Reynolds operator on the deformed Lie-Yamaguti algebra. We also introduce the notion of a Reynolds operator on a Lie-Yamaguti algebra, which can serve as a special case of generalized Reynolds operators on Lie-Yamaguti algebras. Full article

It is shown that for any integers $k\ge 2$, $q\ge 2k$ and $N\ge k+q+2$, there exists a real solvable Lie algebra of the first rank with a maximal torus of derivations $\mathfrak{t}$ possessing the eigenvalue spectrum $\mathrm{spec}\left(\mathfrak{t}\right)=\left(1,2,\cdots ,k,q,q+1\cdots ,N\right)$, a nilradical of the nilpotence index $N-k$ and a characteristic sequence $(N-k,1^k)$. Full article

In this paper, we denote the Lie algebra of smooth vector fields on ${\mathbb{RP}}^1$ by $\mathcal{V}\left({\mathbb{RP}}^1\right)$. This paper focuses on two parts. In the beginning, we determine the cohomology space of $\mathfrak{aff}\left(1\right)$ with coefficients in ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$. Afterward, we classify $\mathfrak{aff}\left(1\right)$-invariant fourth-linear differential operators from $\mathcal{V}\left({\mathbb{RP}}^1\right)$ to ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$ vanishing on $\mathfrak{aff}\left(1\right)$. This result enables us to compute the $\mathfrak{aff}\left(1\right)$-relative cohomology of $\mathcal{V}\left({\mathbb{RP}}^1\right)$ with coefficients in ${\mathcal{D}}_{\tau ,\lambda ,\mu ;\nu }$. Full article

A Lie bialgebra is a vector space endowed simultaneously with the structure of a Lie algebra and the structure of a Lie coalgebra, and some compatibility condition. Moreover, Lie brackets have skew symmetry. Because of the close relation between Lie bialgebras and quantum groups, it is interesting to consider the Lie bialgebra structures on the Lie algebra $\mathfrak{L}$ related to the Virasoro algebra. In this paper, the Lie bialgebras on $\mathfrak{L}$ are investigated by computing $\mathrm{Der}(\mathfrak{L}, \mathfrak{L}\otimes \mathfrak{L})$. It is proved that all such Lie bialgebras are triangular coboundary, and the first cohomology group $H^1(\mathfrak{L}, \mathfrak{L}\otimes \mathfrak{L})$ is trivial. Full article

With the help of symbolic computer packages, the study of the cohomological rigidity of real solvable Lie algebras of rank one with a maximal torus of derivations $\mathfrak{t}$ and the eigenvalue spectrum $\mathrm{spec}\left(\mathfrak{t}\right)=\left(1,,,,k,,,,k,+,1,,,\cdots ,,,n,+,k,-,2\right)$ initiated in a previous work is continued for arbitrary values $k\ge 2$, obtaining new hierarchies of solvable rigid Lie algebras. Full article

In this paper, we study the restricted cohomology of Lie algebras of semisimple and simply connected algebraic groups in positive characteristics with coefficients in simple restricted modules and their applications in studying the connections between these cohomology with the corresponding ordinary cohomology and cohomology of algebraic groups. Let $G$ be a semisimple and simply connected algebraic group $G$ over an algebraically closed field of characteristic $p>h,$ where $h$ is a Coxeter number. Denote the first Frobenius kernel and Lie algebra of $G$ by $G_1$ and $\mathfrak{g}$, respectively. First, we calculate the restricted cohomology of $\mathfrak{g}$ with coefficients in simple modules for two families of restricted simple modules. Since in the restricted region the restricted cohomology of $\mathfrak{g}$ is equivalent to the corresponding cohomology of $G_1,$ we describe them as the cohomology of $G_1$ in terms of the cohomology for $G_1$ with coefficients in dual Weyl modules. Then, we give a necessary and sufficient condition for the isomorphisms $H^n\left(G_1,V\right)\cong H^n\left(G,V\right)$ and $H^n\left(\mathfrak{g},V\right)\cong H^n\left(G,V\right),$ and a necessary condition for the isomorphism $H^n\left(\mathfrak{g},V\right)\cong H^n\left(G_1,V\right),$ where $V$ is a simple module with highest restricted weight. Using these results, we obtain all non-trivial isomorphisms between the cohomology of $G$, $G_1$, and $\mathfrak{g}$ with coefficients in the considered simple modules. Full article

In this article, we obtain some cohomology of classical Lie algebras over an algebraically closed field of characteristic $p>h,$ where $h$ is a Coxeter number, with coefficients in simple modules. We assume that these classical Lie algebras are Lie algebras of semisimple and simply connected algebraic groups. To describe the cohomology of simple modules, we will use the properties of the connections between ordinary and restricted cohomology of restricted Lie algebras. Full article