1. Introduction
Recently, many mathematicians have paid attention to Lie algebra-like structures. In particular, they seek category theoretic analogs of them in [
1,
2,
3]. A kind of algebra, strict Lie 2-algebra, has appeared in some parts of the articles. Lie 2-algebras play a part in studying algebraic structures on Lie 2-groups, string theory, higher categorical structures, and multisymplectic structures, Courant algebroids, Dirac structures, omni-Lie 2-algebras, and Hom-Lie 2-algebras, and so on [
4,
5]. For example, Omni-Lie 2-algebras are a kind of special weak Lie 2-algebra. Weak Lie 2-algebras are a categorification of Lie algebras, or an internal category of Lie algebras. This paper is going to study extensions of crossed modules of Lie algebras. Meanwhile, crossed modules of Lie algebras can be identified with strict Lie 2-algebras. The extending structures problem for some algebra objects such as Lie algebras, Hopf algebras, Leibniz algebras, associative algebras, left-symmetric algebras and Lie conformal algebras have been studied in [
6,
7,
8,
9,
10,
11] respectively.
Lie 2-algebras were first introduced by J. C. Baez and A. S. Crans in 2004. It is a new kind algebra and more sophisticated than the usual Lie algebras. By now, many Lie 2-algebra theories have not been developed. The extension theory of Lie 2-algebras has been characterized by cohomological groups in [
12]. However, in [
12], one needs the subalgebras to be abelian. This paper will abandon the commutative conditions in [
12]. More explicitly, the paper studies the following extending structures problem of strict Lie 2-algebras.
Problem 1. Let be a strict Lie 2-algebra, a 2-vector space and a 2-vector space such that is a 2-vector subspace. Suppose that as vector spaces for . Describe and classify all strict Lie 2-algebra structures on up to an isomorphism of Lie 2-algebras that stabilizes .
In fact, this problem generalizes two important algebra problems. One is the extension problem for strict Lie 2-algebras.
Problem 2. Given two Lie 2-algebras , . Describe and classify all extensions of V by which are strict Lie 2-algebras up to an isomorphism of Lie 2-algebras that stabilizes .
Here, an extension of
V by
is a Lie 2-algebras
which satisfies short exact sequence
where
are linear maps for
, i.e.,
,
and
(Ref. [
13]). When
is an abelian Lie 2-algebra, all extensions of
V by
, which are strict Lie 2-algebras up to an isomorphism of Lie 2-algebras that stabilizes
, can be characterized by the second cohomology group
defined in [
12]. When
is not abelian, all extensions of
V by
, which are strict Lie 2-algebras that stabilizes
, are exactly the non-abelian extensions of
V by
defined in [
13]. The other problem is the factorization problem for strict Lie 2-algebras.
Problem 3. Let , be two strict Lie 2-algebras and a 2-vector space such that is a 2-vector subspace. Suppose that as vector spaces for . Describe and classify all strict Lie 2-algebra structures on such that and V are two sub-Lie 2-algebras of up to an isomorphism of Lie 2-algebras that stabilizes .
Therefore, the study of the extending structures problem is of signification and will be useful for investigating the structure theory of Lie 2-algebras. The paper always assumes that . Two cohomological type objects are constructed by introducing the unified products. Using this unified product, the extension problem and the factorization problem for strict Lie 2-algebras are studied in detail.
An outline of this paper is as follows. In
Section 2, the paper provides some preliminaries. The unified product
of a strict Lie 2-algebra
by 2-vector space
V associated with an extending datum
is introduced in
Section 3. Then, the paper presents the sufficient and necessary conditions to ensure that
is a Lie 2-algebra. Next, the paper shows that any strict Lie 2-algebra
satisfying the condition in extending structures problem is isomorphic to a unified product of
by
V. Finally, the paper constructs two cohomological type objects, where one is isomorphic to the classification of the extending structures problem. Some special cases of unified products such as crossed product and bicrossed product are introduced in
Section 4. Using the crossed product and bicrossed product, the paper describes the extension problem and factorization problem, respectively.
2. Preliminaries
In this section, some definitions and results about strict Lie 2-algebras are provided. A Lie 2-algebra is an object of an internal category of Lie algebras. It has been noted in [
12,
14,
15] and elsewhere that the category of strict Lie 2-algebras is equivalent to the category of crossed modules of Lie algebras. A crossed module of Lie algebras is defined as follows.
Definition 1. A crossed module of Lie algebras is a quadruple where for are Lie algebras, is a Lie algebra homomorphism and is a Lie algebra action by derivations, such that for any and , These equations are called equivariance and infinitesimal Peiffer, respectively.
The homomorphism of two crossed modules of Lie algebras is given by the following definition.
Definition 2. Let and be two strict Lie 2-algebras. A Lie 2-algebra homomorphism consists of linear maps for , such that the following equalities hold for all ,
If are invertible, then Φ is an isomorphism.
All crossed modules of Lie algebras and homomorphisms between them form a category. Given a 2-vector space
. One can construct a strict Lie 2-algebra
, where
,
,
,
,
and
for
and
(Ref. [
12]). If there is a homomorphism
from a Lie 2-algebra
to the Lie 2-algebra
, then
V is called a representation of the Lie 2-algebra
. Suppose that
V is a representation of a Lie 2-algebra
. Then
is a representation of
, and
is a representation of
. Moreover, the paper introduces the following concept.
Definition 3. Let be a strict Lie 2-algebra, a 2-vector space and a 2-vector space such that the diagramcommutates, where are the canonical projections of and are the inclusion maps for . For linear functor , consider the diagram: The paper calls that φ stabilizes (resp. co-stabilizes ) if the left cube (resp. the right cube) of the diagram (2) is commutative.
Let and be two strict Lie 2-algebra structures on both containing as a sub-Lie 2-algebra. If there exists a Lie 2-algebra homomorphism φ which stabilizes , then and are called equivalent, which is denoted by If there exists a Lie 2-algebra isomorphism φ which stabilizes and co-stabilizes , i.e., the diagram (2) commutates, then and are called cohomologous, which is denoted by It is easy to see that ≡ and ≈ are equivalence relations on the set of all strict Lie 2-algebra structures on containing as a sub-Lie 2-algebra. The set of all equivalence classes via ≡ and ≈ are denoted by and respectively. In addition, it is easy to show that there exists a canonical projection .
Proposition 1. Let be a strict Lie 2-algebra, a 2-vector space and a 2-vector space such that the diagram (1) commutates. Then is 2-vector space , where is a linear map.
Proof. By the definition of and , . Let for any . Since the left square of diagram (1) commutates, . Thus, . Similarly, if for , then as the right square of the diagram (1) commutates. Define by for any , where is the canonical projection. Then for any . Hence for and . □
3. Unified Products for Lie 2-Algebras
In this section, a unified product of two Lie 2-algebras is introduced. Using this product, the paper provides the theoretical answer to the extending structure problem. 2-vector space and Lie 2-algebra are simply denoted by V and respectively in the following.
Definition 4. Suppose that is a strict Lie 2-algebra and V is a 2-vector space. An extending datum of by V is a system consisting of one linear map and fourteen bilinear maps Let
be an extending datum. Define a new strict Lie 2-algebra
as follows. As a 2-vector space,
is equal to
, where
for
and
. The bilinear maps
and the linear map
is given by
and
respectively, for
and all
,
. This strict Lie 2-algebra
is called a unified product of
and
V,
is called a Lie 2-extending structure of
by
V. If only one Lie 2-extending structure
of
by
V is considered, then
is usually simplified as
and the Lie 2-extending structure is simply called extending datum. The set of all Lie 2-extending structures of
by
V is denoted by
.
Theorem 1. Suppose that is a strict Lie 2-algebra, V is a 2-vector space. Then is an extending datum of by V such that is a strict Lie 2-algebra if and only if the following conditions hold for any , and :
- (L1)
- (L2)
is a right -module;
- (L3)
;
- (L4)
- (L5)
- (L6)
- (L7)
- (L8)
- (L9)
- (L10)
- (L11)
- (L12)
- (L13)
- (L14)
- (L15)
- (L16)
- (L17)
- (L18)
- (L19)
- (L20)
- (L21)
- (L22)
- (L23)
- (L24)
- (L25)
- (L26)
- (L27)
- (L28)
- (L29)
- (L30)
- (L31)
- (L32)
- (L33)
- (L34)
- (L35)
- (L36)
- (L37)
- (L38)
- (L39)
- (L40)
- (L41)
- (L42)
- (L43)
Proof. By ([
6] Theorem 2.2),
is a Lie algebra if and only if the conditions
–
hold. Thus,
is a strict Lie 2-algebra if and only if
is a Lie algebra homomorphism,
is a Lie algebra action by derivations and satisfying equivariance and infinitesimal Peiffer, i.e.,
for
and all
,
. Since
in
, Equations (
5)–(
9) hold if and only if they hold for the set
. First, Equation (
5) holds for
as
Equation (
5) holds for
if and only if
and
hold. Equation (
5) holds for
if and only if
and
hold, since
Then, Equation (
6) holds for
as
Since, for
,
Equation (
6) holds for
if and only if
and
hold. Equation (
6) holds for
if and only if
and
hold, since
Equation (
6) holds for
if and only if
and
hold, as
Equation (
6) holds for
if and only if
and
hold. Indeed,
for
. Notice that
for
. Thus, Equation (
6) holds for
if and only if
and
hold. Now Equation (
7) holds for
as
Since
for
, Equation (
7) holds for
if and only if
and
hold. As
Equation (
7) holds for
if and only if
and
hold. Indeed,
for
. Then Equation (
7) holds for
if and only if
and
hold. Because
for
, Equation (
7) holds for
if and only if
and
hold. Since
Equation (
7) holds for
if and only if
and
hold. Next Equation (
8) holds for
as
Equation (
8) holds for
if and only if
and
hold. Indeed,
for
. Thus, Equation (
8) holds for
if and only if
and
hold. Equation (
8) holds for
if and only if
and
hold, since
Finally, Equation (
9) holds for
as
Indeed,
for
. Thus, Equation (
9) holds for
if and only if
and
hold. Since, for
,
Equation (
9) holds for
if and only if
and
hold. Equation (
9) holds for
if and only if
and
hold, since
By now the proof is completed. □
Example 1. Let be an extending datum of Lie 2-algebra by 2-vector space V such that for are trivial maps, i.e., , , , , , , , , , , for and , . Then is a Lie 2-algebra with , the brackets and derivation given by and respectively, for and all , . The paper calls this Lie 2-extending structure the trivial extending structure of by V.
In the sequel, the paper uses the following convention: if one of the maps for of an extending datum is trivial then the paper will omit it from .
Example 2. Let be an extending datum of Lie 2-algebra by 2-vector space V such that for are trivial maps. Then is a Lie 2-extending structure of by V if and only if is a representation of Lie 2-algebra on 2-vector space V, where , and for , . In this case, the associative unified product is the semi-direct product , wherefor and , . Let
be a Lie 2-extending structure and
be the associated unified product. Then the canonical inclusion
is an injective Lie 2-algebra homomorphism. Hence,
can be viewed as a sub-Lie 2-algebra of
by the identification
. Conversely, the paper will prove that any strict Lie 2-algebra structure on 2-vector space
containing
as a sub-Lie 2-algebra is isomorphic to a unified product.
Theorem 2. Let be a strict Lie 2-algebra and a 2-vector space containing as a 2-vector subspace. Suppose that is a strict Lie 2-algebra structure on such that is a sub-Lie 2-algebra in of by a 2-vector space V. Then there is an isomorphism of Lie 2-algebras which stabilizes and co-stabilizes V.
Proof. Let
be linear maps such that
for
and
. Then
is a subspace of
and a complement of
in
. Define the extending datum of
by
V by the following formulas:
for any
,
and
. First, the above maps are all well defined. This paper shall prove that
is a Lie 2-extending structure of
by
V and
is an isomorphism of Lie 2-algebras that stabilizes
and co-stabilizes
V. It is easy to verify that for
and
,
is an inverse of
as 2-vector space. Therefore, there is a unique strict Lie 2-algebra structure on
such that
is an isomorphism of strict Lie 2-algebras and this unique Lie 2-algebra structure is given by
for all
and
. Then the proof is sufficient to prove that this Lie 2-algebra structure coincides with the one defined by (
3) and (
4) associated with the system
. Indeed, for any
and
,
Moreover, the following diagram is commutative
The proof is completed now. □
By Theorem 2, the classification of all strict Lie 2-algebra structure on that containing as a sub-Lie 2-algebra reduces to the classification of all unified products associated with all Lie 2-extending structures , for a given 2-vector space V such that is a complement of in .
Lemma 1. Suppose that and are two Lie 2-extending structures of by V and , are the associated unified products. Then there exists a bijection between the set of all morphisms of Lie 2-algebras which stabilizes and the set of , where and are linear maps satisfying the following compatibility conditions for and any , :
- (M1)
- (M2)
- (M3)
- (M4)
- (M5)
- (M6)
- (M7)
- (M8)
- (M9)
- (M10)
- (M11)
- (M12)
Under the above bijection the morphism of Lie 2-algebras corresponding to is given by:for any , and . Moreover, is an isomorphism if and only if is an isomorphism and co-stabilizes V if and only if . Proof. Suppose that
is a linear functor such that
commutates. Then it is uniquely determined by linear maps
and
such that
for
and all
,
. In fact, let
for all
. Then
. Next, the paper proves that
is a morphism of strict Lie 2-algebras if and only if the compatibility conditions
–
hold. It is sufficient to prove the equations
hold for the set
and
. By ([
6], Lemma 2.5), Equation (
12) holds if and only if conditions
–
hold. First, consider Equation (
13). It is easy to see that Equation (
13) holds for
. Equation (
13) holds for
if and only if
and
hold, since
Now consider Equation (
14). It is easy to see that Equation (
14) holds for
. Since, for
and
,
Equation (
14) holds for
and
if and only if
and
hold. Similarly, it is easy to check that Equation (
14) holds for
and
if and only if
and
hold; Equation (
14) holds for
and
if and only if
and
hold.
Assume that
is bijective. Then
is an isomorphism of Lie 2-algebras with the inverse given by
, where
for
and
. Conversely, assume that
is isomorphic. Then
is an isomorphism of Lie algebras for
. By the proof of ([
6], Lemma 2.5),
is a bijection for
. The last assertion is trivial, and the proof is completed now. □
Definition 5. Let be a strict Lie 2-algebra and V a 2-vector space. If there exists linear maps and for such that Lie 2-algebra extending structure can be yield from another Lie 2-algebra extending structure using via:for and any , , then and are said to be equivalent, which is denoted by . In particular, if for , then and are called cohomologous, which is denoted by . The paper concludes this section by the following theorem, which provides an answer to the extending structures problem of strict Lie 2-algebras.
Theorem 3. Suppose that is a strict Lie 2-algebra, V is a 2-vector space and is a 2-vector space which contains as a 2-subspace and is a complement of in for . Then
- 1.
the relation ≡ is an equivalence relation on the set of all Lie 2-extending structures of by V, and the map , given byis bijective, where is the equivalence class of under the equivalent relation ≡. - 2.
the relation ≈ is an equivalent relation on the set of all Lie 2-extending structures of by V, and the mapping given byis a bijection, where is the equivalence class of under the equivalent relation ≈.
Proof. It follows from Theorem 1, Theorem 2 and Lemma 1. □
4. Special Cases of Unified Products
In this section, two special cases of unified products are studied. One corresponds to the extension problem and the other corresponds to the factorization problem.
4.1. Crossed Products and the Extension Problem
Let be the extending datum of by V such that are trivial maps for . Then is a Lie 2-extending structure of by V if and only if is a strict Lie 2-algebra and the following compatibilities hold for and any , :
In this case, the associated unified product
is called the crossed product of the Lie 2-algebras
and
V. A system
consisting of two strict Lie 2-algebras
, seven bilinear maps
,
,
,
,
for
and one linear map
satisfying the above compatibility conditions will be called a crossed system of Lie 2-algebras. The crossed product associated with the crossed system
is the strict Lie 2-algebra
with the brackets and derivation given by:
for
and
,
. Then
is an ideal of the Lie 2-algebra
since
and
. Conversely, crossed products describe all Lie 2-algebra structures on the 2-vector space
such that a given strict Lie 2-algebra
is an ideal of
.
Corollary 1. Let be a strict Lie 2-algebra, a 2-vector space containing as a 2-vector subspace. Then any strict Lie 2-algebra structure on that contains as an ideal is isomorphic to a crossed product of Lie 2-algebras .
Proof. Let be a strict Lie 2-algebra structure on such that is an ideal of . In particular, is a 2-vector subalgebra of . By Theorem 2, the paper obtains the Lie 2-extending structure , where the action for are trivial. Indeed, for any and , , and and hence , and . Thus, the unified product is the crossed product of the Lie 2-algebras and 2-vector space . □
Remark 1. By Corollary 1, all crossed products of Lie 2-algebras give the theoretical answer to Problem 2. In fact, a crossed product of Lie 2-algebras also corresponds to the non-abelian extension structure defined in [13]. Define for , and , where are linear maps defined in [13]. The result follows. In particular, if is an abelian Lie 2-algebra, i.e., is a 2-vector space. Then the set of all extending structures of the abelian Lie 2-algebra by the 2-vector space V is parameterized by the set of all , such that is a strict Lie 2-algebra, is a left V-module and , are bilinear maps such that and
for
and
,
. For such
, the brackets and the derivation of the extending structure on
are given by
respectively, where
and
,
.
By ([
12], Theorem 5.6), the second cohomology group
classifies 2-extensions. To study the relation between
and Lie 2-algebra
, the paper needs a result of [
12].
Lemma 2 ([
12], Proposition 5.3)
. Let ρ be a 2-representation of strict Lie 2-algebra on V. Given a triple . Thenwithis a 2-extensions if and only if the following equations are satisfied- (i)
;
- (ii)
;
- (iii)
;
- (iv)
- (v)
;
- (vi)
For the contraction seen as a 1-cocycle with values in ;
for and , . Here, ⥀ stands for cyclic permutations.
Proposition 2. If is an abelian Lie 2-algebra, then Lie 2-algebra is the Lie 2-algebra defined by a second cohomology group as in Lemma 2.
Proof. Given a Lie 2-algebra . Let , , . Then the paper obtains a Lie 2-algebra determined by the . Conversely, suppose that is a Lie 2-algebra determined by a the . Let , , . Then the paper obtains a Lie 2-algebra . Thus, the conclusion follows. □
Moreover, this case Lie 2-algebra is also correspondence with the abelian extension of
V by
which is defined in [
14].
4.2. Bicrossed Products and the Factorization Problem
Let be the extending datum of by V such that and are trivial maps for . Then is a Lie 2-extending structure of by V if and only if is a strict Lie 2-algebra, is a left V-module under , V is a right -module under and the following compatibilities hold for and any , :
;
In this case, the associated unified product
is called the bicrossed product
of
of the Lie 2-algebras
and
V. The brackets and derivation on
are given by:
for
and
,
.
Theorem 4. Suppose that , are two Lie 2-algebras and is a 2-vector space such that diagram (1) satisfies. Assume that is a strict Lie 2-algebra structure on such that and V are sub-Lie 2-algebras in . Then Lie 2-algebra is isomorphic to of and V.
Proof. It follows from Theorem 2. □
5. Conclusions
This paper contains information on how to construct a strict Lie 2-algebra from one strict Lie 2-algebra by another strict Lie 2-algebra. A Lie 2-algebra can be obtained from a Lie 2-group. A strict Lie 2-group is usually called a crossed module of Lie groups. It is also an internal object in the category of Lie groups. Similarly, one can study extension structures of a Lie 2-group by another Lie 2-group. One natural avenue of further exploration is to consider the relations between these two kinds of extension, and, furthermore, the relations between the cohomological groups.
Please note that this paper is limited to consideration of extensions of algebras. The paper has neglected the possible topology and geometry of the Lie 2-groups, more generalization 2-gerbes, of which the fields are flat connections—classical, in the sense familiar to physicists.
Another avenue is to discuss algebraic 2-groups over a field of characteristic non-zero p. Then, the Frobenius maps give the corresponding Lie algebra -maps. A kind of Lie algebra over a field of characteristic p with a -map is called restricted Lie algebra. Hence one can study a similar question about strict restricted Lie 2-algebras and algebraic 2-groups.
A strict Lie 2-algebra is similar to another extension algebra of Lie algebras, namely the two-term algebra. It is well known that an algebra is the same thing as a semi-free graded-commutative differential graded algebra. Suppose M is a smooth manifold. Then the algebra of differential forms is a graded-commutative differential graded algebra. One can construct a semi-free graded-commutative differential graded algebra from any algebra L, which is called Chevalley–Eilenberg algebra. It is a complicated but straightforward exercise to check that the nilpotence for Chevalley–Eilenberg algebra differential is exactly equivalent to the homotopy Jacobi identities. Using this ideal, the authors hope to seek similar graded-commutative differential graded algebras to reduce a large number of the equations in Theorem 1. All of these need to be further explored.