Integration of derivations for Lie 2 -algebras
aa r X i v : . [ m a t h . DG ] M a y Integration of derivations for Lie 2-algebras ∗ Honglei Lang , Zhangju Liu , Yunhe Sheng School of Mathematics and LMAM, Peking University, Beijing 100871, China School of Mathematics, Jilin University, Changchun 130012, Chinaemail: [email protected], [email protected], [email protected]
Abstract
In this paper, for a Lie 2-algebra g , we construct the automorphism 2-group Aut( g ), whichturns out to be an integration of the derivation Lie 2-algebra Der( g ). Recently, people have paid a lot of attention to higher categorical structures with motivationfrom string theory. One way to provide higher categorical structures is by categorifying existingmathematical concepts. One of the simplest higher structures is a 2-vector space, which is acategorified vector space. If we further put Lie algebra structures on 2-vector spaces, then weobtain Lie 2-algebras [1]. The Jacobi identity is replaced by a natural transformation, called theJacobiator, which also satisfies some coherence laws of its own. It is well-known that the categoryof Lie 2-algebras is equivalent to the category of 2-term L ∞ -algebras [1]. The concept of an L ∞ -algebra (sometimes called a strongly homotopy (sh) Lie algebra) was originally introduced as amodel for “Lie algebras that satisfy Jacobi identity up to all higher homotopies”. The structureof a Lie 2-algebra appears in many areas such as higher symplectic geometry [3], string theory [4]and Courant algebroids.To study nonabelian extensions of Lie 2-algebras, the authors introduced the notion of a deriva-tion of a Lie 2-algebra g = ( g ⊕ g − , d, [ · , · ] , l ) in [5], and showed that there is a strict Lie 2-algebraDer( g ), in which the degree − − − ( g ), and the degree 0part is the set of degree 0 derivations Der ( g ). Furthermore, one can also construct a Lie 3-algebraDER( g ) (called the derivation Lie 3-algebra), in which the degree − g − , the degree − − ( g ) ⊕ g , and the degree 0 part is Der ( g ). A nonabelian extension of a Lie 2-algebra g by a Lie 2-algebra h can be given by a homomorphism from g to DER( h ).In this paper, we study the integration of the strict Lie 2-algebra Der( g ). Here, “integration”is meant in the sense in which a Lie algebra is integrated to a corresponding Lie group. In recentyears, there are many works about the integration of Lie-algebra-like structures, such as that ofLie algebroids [6, 19], of L ∞ -algebras [8, 9] and of Courant algebroids [12, 14]. In particular,semidirect product Lie 2-algebras are integrated to strict Lie 2-groups via equivalence in [17],and a homomorphism between strict Lie 2-algebras is integrated to a homomorphism between Lie2-groups in [18]. In the current paper, we construct a strict Lie 2-group Aut( g ) associated toautomorphisms of g , and show that Aut( g ) is an integration of the strict Lie 2-algebra Der( g ). Keyword: Lie -algebras, derivations, automorphisms, integration MSC : 17B40, 18B40. ∗ Research supported by NSFC (11101179,11471139) and NSF of Jilin Province (20140520054JH). g is a strict Lie 2-algebra, with the crossed module of Lie groups G as its integration,we can construct a sub-Lie 2-group SAut( g ) of Aut( g ). This turns out to be the differential of theautomorphism Lie 2-group Aut( G ) given in [13]; In case g is abelian, i.e. both [ · , · ] and l are zero,End( g ) is a sub-Lie 2-algebra of Der( g ), where End( g ) is a generalization of the general linear Liealgebra from a vector space to a 2-term complex of vector spaces. In this case, our result alsogives the integration of End( g ), which was already given in [17]. See [15] for more details aboutautomorphisms of strict 2-groups.The paper is organized as follows. In Section 2, we recall some basic definitions regarding Lie2-algebras, crossed modules of Lie algebras (Lie groups). In Section 3, we work out the deriva-tion Lie 2-algebra for the string Lie 2-algebra. We also compare our derivations with homotopyderivations introduced in [7] recently. In Section 4, we construct a crossed module of Lie groups(Aut − ( g ) , Aut ( g ) , ∂, ⊲ ) associated to automorphisms of a Lie 2-algebra g (Theorem 4.4). In Sec-tion 5, we prove that the strict Lie 2-group Aut( g ) is an integration of the strict Lie 2-algebraDer( g ) (Theorem 5.1). In addition, we give the expressions of exponential maps from Der i ( g ) toAut i ( g ) for i = 0 , −
1, which commute with the differentials. In Section 6, we first consider a sub-Lie 2-algebra SDer( g ) of Der( g ) and a sub-crossed module SAut( g ) of Aut( g ). Then we give therelation between SAut( g ) and Aut( G ) for a strict Lie 2-algebra g , where G is the crossed moduleintegrating g . Finally, we explore the inner automorphism Lie 2-group of a Lie 2-algebra. Acknowledgement
We give our warmest thanks to referees for very helpful comments that improve the paper. • Lie 2-algebras and crossed modules of Lie algebras
A Lie 2-algebra is equivalent to a 2-term L ∞ -algebra [1]. The notion of L ∞ -algebras wasintroduced by Schlessinger and Stasheff in [16]. See [10] for more information. Definition 2.1. A Lie -algebra is a graded vector space g = g − ⊕ g , together with a differential d : g − −→ g , a skew-symmetric bilinear map [ · , · ] : g i ∧ g j −→ g i + j , − ≤ i + j ≤ and a skew-symmetric trilinear map l : ∧ g −→ g − satisfying the following equalities (a) d [ x, a ] = [ x, da ] , [ da, b ] = [ a, db ] , (b) [[ x, y ] , z ] + c.p. = − dl ( x, y, z ) , [[ x, y ] , a ] + c.p. = − l ( x, y, da ) , (c) l ([ x, y ] , z, w ) + c.p. = [ l ( x, y, z ) , w ] + c.p. ,for all x, y, z, w ∈ g , a, b ∈ g − , where c.p. means cyclic permutation. If l = 0 , g is called a strict Lie -algebra . We denote a Lie 2-algebra by ( g ⊕ g − , d, [ · , · ] , l ), or simply by g if there is no confusion. Definition 2.2.
Let g = ( g ⊕ g − , d, [ · , · ] , l ) and g ′ = ( g ′ ⊕ g ′− , d ′ , [ · , · ] ′ , l ′ ) be Lie -algebras. A Lie -algebra homomorphism A : g −→ g ′ consists of • two linear maps A : g −→ g ′ and A : g − −→ g ′− , satisfying d ′ ◦ A = A ◦ d , In this paper, all the Lie 2-algebras are semistrict. one bilinear map A : g ∧ g −→ g ′− ,such that the following equalities hold for all x, y, z ∈ g , a ∈ g − : (i) A [ x, y ] − [ A ( x ) , A ( y )] ′ = d ′ A ( x, y ) , (ii) A [ x, a ] − [ A ( x ) , A ( a )] ′ = A ( x, da ) , (iii) [ A ( x ) , A ( y, z )] ′ + c.p. + l ′ ( A ( x ) , A ( y ) , A ( z )) = A ([ x, y ] , z ) + c.p. + A ( l ( x, y, z )) .It is called a strict homomorphism if A = 0 . Definition 2.3. A crossed module of Lie algebras is a quadruple ( h , h , ϕ, φ ) , which wedenote by h , where h and h are Lie algebras, ϕ : h −→ h is a Lie algebra homomorphism and φ : h −→ Der( h ) is an action of h on h by derivations, such that ϕ ( φ x ( a )) = [ x, ϕ ( a )] h , φ ϕ ( a ) ( b ) = [ a, b ] h . It is well-known that there is a one-to-one correspondence between strict Lie 2-algebras andcrossed modules of Lie algebras. In short, the formula for the correspondence can be given asfollows: A strict Lie 2-algebra g − d −→ g gives rise to a crossed module with h = g − and h = g , where the Lie brackets are given by:[ a, b ] h = [ d ( a ) , b ] , [ x, y ] h = [ x, y ] , ∀ x, y ∈ g , a, b ∈ g − and ϕ = d , φ : h −→ Der( h ) is given by φ x ( a ) = [ x, a ] . The strict Lie 2-algebra structure givesthe Jacobi identities for [ · , · ] h and [ · , · ] h , and various other conditions for crossed modules.Conversely, a crossed module ( h , h , ϕ, φ ) gives rise to a strict Lie 2-algebra with d = ϕ , g − = h and g = h , and [ · , · ] is given by:[ a, b ] , , [ x, y ] , [ x, y ] h , [ x, a ] = − [ a, x ] , φ x ( a ) . • Lie 2-groups and crossed modules of Lie groups
A group is a monoid where every element has an inverse. A 2-group is a monoidal categorywhere every object has a weak inverse and every morphism has an inverse. Denote the categoryof smooth manifolds and smooth maps by Diff, a (semistrict) Lie 2-group is a 2-group in DiffCat,where DiffCat is the 2-category consisting of categories, functors, and natural transformations inDiff. For more details, see [2]. Here we only give the definition of strict Lie 2-groups.
Definition 2.4. A strict Lie -group is a Lie groupoid C such that (1) The space of morphisms C and the space of objects C are Lie groups. (2) The source and the target s, t : C −→ C , the identity assigning function i : C −→ C andthe composition ◦ : C × C C −→ C are all Lie group morphisms. It is known that strict Lie 2-groups can be described by crossed modules of Lie groups.
Definition 2.5. A crossed module of Lie groups is a quadruple ( H , H , ∂, ⊲ ) , which wedenote simply by H , where H and H are Lie groups, ∂ : H −→ H is a homomorphism, and ⊲ : H −→ Aut( H ) is an action of H on H by automorphisms such that ∂ is H -equivariant: ∂ ( g ⊲ h ) = g∂ ( h ) g − , ∀ g ∈ H , h ∈ H , (1) and satisfies the so called Peiffer identity: ∂ ( h ) ⊲ ( h ′ ) = hh ′ h − , ∀ h, h ′ ∈ H . (2)3t is well-known that there is a one-to-one correspondence between crossed modules of Liegroups and strict Lie 2-groups. Roughly speaking, given a crossed module ( H , H , ∂, ⊲ ), there isa strict Lie 2-group for which C = H and C = H × H . In this strict Lie 2-group, the sourceand target maps s, t : C −→ C are given by s ( g, h ) = g, t ( g, h ) = ∂ ( h ) g, the groupoid multiplication · v is given by( g ′ , h ′ ) · v ( g, h ) = ( g, h ′ h ) , where g ′ = ∂ ( h ) g, (3)the group multiplication · h is given by( g, h ) · h ( g ′ , h ′ ) = ( gg ′ , h ( g ⊲ h ′ )) . (4) Let V : V − −→ V be a 2-term complex of vector spaces. We can form a new 2-term complexof vector spaces End( V ) : End − ( V ) δ −→ End ( V ) by defining δ (Θ) = d ◦ Θ + Θ ◦ d for anyΘ ∈ End − ( V ), where End − ( V ) = Hom( V , V − ) andEnd ( V ) = { X = ( X , X ) ∈ End( V ) ⊕ End( V − ) | X ◦ d = d ◦ X } . There is a natural bracket operation [ · , · ] C on End( V ) given by the commutator:[( X , X ) , ( Y , Y )] C = ( X ◦ Y − Y ◦ X , X ◦ Y − Y ◦ X ) , [( X , X ) , Θ] C = X ◦ Θ − Θ ◦ X . Consequently, (End( V ) , δ, [ · , · ] C ) is a strict Lie 2-algebra. It plays the same role as End( V ) for avector space V . Its integration is given in [17].Let g = ( g ⊕ g − , d, [ · , · ] , l ) be a Lie 2-algebra. Definition 3.1. A derivation of degree of g is a triple ( X , X , l X ) , where X = ( X , X ) ∈ End d ( g ) and l X : g ∧ g → g − is a linear map, such that for all x, y, z ∈ g , a ∈ g − , (a) dl X ( x, y ) = X [ x, y ] − [ X x, y ] − [ x, X y ] , (b) l X ( x, da ) = X [ x, a ] − [ X x, a ] − [ x, X a ] , (c) X l ( x, y, z ) = l X ( x, [ y, z ]) + [ x, l X ( y, z )] + l ( X x, y, z ) + c.p. ( x, y, z ) . Denote by Der ( g ) the set of derivations of degree 0 of g . Then we can obtain a 2-vector space:Der( g ) : Der − ( g ) , End − ( g ) ¯ d −−−−→ Der ( g ) , where ¯ d is given by ¯ d (Θ) = ( δ (Θ) , l δ (Θ) ), in which δ (Θ) = ( d ◦ Θ , Θ ◦ d ) and l δ (Θ) ( x, y ) = Θ[ x, y ] − [ x, Θ y ] − [Θ x, y ] . Strictly speaking, we should use the terminology “a weak derivation” since the derivation conditions hold up tohomotopy. To be succinct, we just say “a derivation”.
4n addition, define { ( X, l X ) , Θ } = [ X, Θ] C , (5) { ( X, l X ) , ( Y, l Y ) } = ([ X, Y ] C , L X ( l Y ) − L Y ( l X )) , (6)where [ · , · ] C is the commutator bracket and for all X = ( X , X ) ∈ End( g ) ⊕ End( g − ), L X :Hom( ∧ k g , g − ) −→ Hom( ∧ k g , g − ) is given by L X ω ( x , · · · , x k ) = X ω ( x , · · · , x k ) − k X i =1 ω ( x , · · · , X ( x i ) , · · · , x k ) . Theorem 3.2. [5]
With the notations above, (Der( g ) , {· , ·} ) is a strict Lie -algebra. We call itthe derivation Lie -algebra of g . Remark 3.3.
Recently, the notion of a homotopy derivation was introduced in [7] using the theoryof operads. See [7, Proposition 4.1] for precise formulas. Restricting to the 2-term case, for a Lie2-algebra g = ( g ⊕ g − , d, [ · , · ] , l ) , we could obtain that a homotopy derivation of degree 0 is thesame as the one given in Definition 3.1, and a homotopy derivation of degree − ∈ End − ( g )such that d ◦ Θ = 0 , Θ ◦ d = 0 , (7)Θ[ x, y ] = [Θ( x ) , y ] + [ x, Θ( y )] . (8)There are also homotopy derivations of other degrees. However, since we are dealing with Lie2-algebras, homotopy derivations of degree 0 and − X, l X ) of degree 0 canhave a nonzero homotopy term “ l X ”, while the strict derivation of degree 0 requires that l X = 0.Moreover, a homotopy derivation of degree − − g ) , {· , ·} ) be the derivation Lie 2-algebra of g , and denote the corresponding crossedmodule of Lie algebras by (Der − ( g ) , Der ( g ) , ¯ d, φ ), where the action φ is given by (5), and the Liebracket on Der − ( g ) is given by { Θ , Θ ′ } = { ¯ d (Θ) , Θ ′ } = [ δ (Θ) , Θ ′ ] C = Θ ◦ d ◦ Θ ′ − Θ ′ ◦ d ◦ Θ . (9)Now we work out the derivation Lie 2-algebra of the string Lie 2-algebra string( k ). Let( k , [ · , · ] k )be a semisimple Lie algebra and K the Killing form. Consider the corresponding string Lie 2-algebra string( k ) = ( R −→ k , [ · , · ] , l ). More precisely,[ x, y ] = [ x, y ] k , [ x, r ] = 0 , l ( x, y, z ) = K ([ x, y ] k , z ) , ∀ x, y, z ∈ k , r ∈ R . In [7], the authors use an alternative definition of L ∞ -algebras. For their convention, g is of degree − g − is of degree − roposition 3.4. For the string Lie -algebra string( k ) , Der (string( k )) is isomorphic to thesemidirect product Lie algebra k ⋉ ad ∗ k ∗ . Furthermore, Der − (string( k )) = k ∗ , which is abelian,and the differential ¯ d is given by ¯ d (Θ) = (0 , , − D Θ) , where D is the coboundary operator on k with the coefficients in the trivial representation.Proof. A straightforward computation shows that ( X , t, l X ), where X ∈ End( k ) , t ∈ R , l X ∈ ∧ k ∗ ,is a derivation of degree 0 if and only if X ∈ Der( k ) , D l X ( x, y, z ) = tl ( x, y, z ) − l ( X x, y, z ) − l ( x, X y, z ) − l ( x, y, X z ) . Since k is semisimple, every derivation is an inner derivation, and thus skew-symmetric. Then wehave l ( X x, y, z ) + c.p. = K ([ X x, y ] k , z ) + K ([ x, X y ] k , z ) + K ([ x, y ] k , X z )= K ([ X x, y ] k + [ x, X y ] k , z ) − K ( X [ x, y ] k , z )= 0 . Now the second condition reduces to D l X ( x, y, z ) = tl ( x, y, z ) . However, the Cartan 3-form l could not be exact, which implies that t = 0. Therefore, we haveDer (string( k )) = { ( X , , ω ) | X ∈ Der( k ) , ω ∈ ∧ k ∗ is a 2-cocycle } . Note that every 2-cocycle is exact. For (ad x , , D ( ξ )) , (ad y , , D ( η )) ∈ Der (string( k )), we have { (ad x , , D ( ξ )) , (ad y , , D ( η )) } = (ad [ x,y ] k , , D (ad ∗ x η − ad ∗ y ξ )) , which implies the map ( x, ξ ) (ad x , , D ( ξ )) is an isomorphism between Lie algebras k ⋉ ad ∗ k ∗ and Der (string( k )). Other conclusions are obvious. The proof is finished.More generally, we have Example 3.5.
Let ( k , [ · , · ] k ) be a Lie algebra and φ : k −→ End( V ) a representation of k on V . Let C k ( k , V ) be the set of k -cochains, i.e. C k ( k , V ) = { f : ∧ k k −→ V } . Given a Lie algebra 3-cocycle l ∈ C ( k , V ), we get a skeletal Lie 2-algebra g = ( V ⊕ k , d = 0 , [ · , · ] , l ), where [ · , · ] is defined by[ x, y ] = [ x, y ] k , [ x, u ] = − [ u, x ] = φ x ( u ) , ∀ x, y ∈ k , u ∈ V. For X = ( X , X ) ∈ End( k ) ⊕ End( V ), and l X ∈ C ( k , V ), it is easy to check that( X, l X ) ∈ Der ( g ) ⇐⇒ (cid:26) X ∈ Der( k ⋉ φ V ) , D l X = L X ( l ) , where D : C k ( k , V ) → C k +1 ( k , V ) is the coboundary operator on k with coefficients in V , andDer( k ⋉ φ V ) is the Lie algebra of derivations of the semidirect product Lie algebra k ⋉ φ V .Moreover, we have Der − ( g ) = C ( k , V ) and the map ¯ d : Der − ( g ) → Der ( g ) is given by¯ d (Θ) = (0 , , − D (Θ)).A representation of a Lie 2-algebra g on a 2-term complex of vector spaces V : V − −→ V is a homomorphism from g to End( V ). The adjoint representation ad = (ad , ad , ad ) of g onitself is given byad ( x ) = [ x, · ] , ad ( a ) = [ a, · ] , ad ( y, z ) = − l ( y, z, · ) , ∀ x, y, z ∈ g , a ∈ g − . g , the derivation Lie 2-algebra Der( g ) is strict. However, there is still aLie 2-algebra homomorphism from g to Der( g ). Define ad : g −→ Der ( g ) , ad : g − −→ Der − ( g ) , ad : ∧ g −→ Der − ( g )by ad ( x ) = (ad ( x ) , l ( x, · , · )) , ad ( a ) = ad ( a ) , ad ( y, z ) = ad ( y, z ) . (10)It is straightforward to obtain that Lemma 3.6.
With the above notations, ad = ( ad , ad , ad ) is a Lie -algebra homomorphismfrom g to Der( g ) . Remark 3.7.
In the classical case, for a semisimple Lie algebra k , ad : k −→ Der( k ) is an isomor-phism of Lie algebras. In the case of Lie 2-algebras, we can not expect such a result. • Even for the simple string Lie 2-algebra string( k ) given in Proposition 3.4, ad : k −→ Der (string( k )) is not surjective. • The image of ad is not closed in Der ( g ). Explicitly, we have { ad ( x ) , ad ( y ) } = ad ([ x, y ]) + ¯ dl ( x, y, · ) . Thus, we can not define the inner derivation inn ( g ) simply by Im ad . Definition 3.8.
Define the set of inner derivations of degree , which is denoted by inn ( g ) , by inn ( g ) = Im ad + Im ¯ d. (11)Then we can get a Lie 2-algebra inn( g ) given byinn( g ) : inn − ( g ) , End − ( g ) ¯ d −−−−→ inn ( g ) , (12)which we call the inner derivation Lie -algebra of g . Remark 3.9.
In this remark, we give some reasons for our definition of inn ( g ). From thehomological viewpoint, Der ( g ) is the set of 1-cocycles of the Lie 2-algebra g with respect to theadjoint representation ad , and inn ( g ) is the set of 1-coboundaries. See [11] for details. For a Liealgebra k , inn( k ) = Im(ad). However, for a Lie 2-algebra g , they are not the same. Furthermore, itis straightforward to see that inn( g ) is an ideal of Der( g ). Remark 3.10.
Parallel to Remark 3.3, we compare our inner derivations with homotopy innerderivations given in [7]. For a semistrict Lie 2-algebra, a homotopy inner derivation of degree 0 isgiven by (ad ( x ) , l ( x, · , · )) for x ∈ g , and a homotopy inner derivation of degree − ( a ) for a ∈ g − satisfying d ( a ) = 0. See [7, Proposition 4.4] for general formulas.The Lie 2-algebra of homotopy inner derivations is a sub-Lie 2-algebra of inn( g ). For a strict Lie2-algebra, homotopy inner derivations of degree − d ( a ) = 0. Our inner derivation Lie 2-algebra inn( g ) can also be viewed as aunification of them. 7 The automorphisms of a Lie 2-algebra
In this section, we construct a Lie 2-group Aut( g ) associated to the automorphisms of a Lie 2-algebra g = ( g ⊕ g − , d, [ · , · ] , l ).Clearly, i = ( I g , I g − ,
0) : g −→ g is a Lie 2-algebra homomorphism, which is called the identityhomomorphism. Let A : g → g ′ and B : g ′ → g ′′ be two Lie 2-algebra homomorphisms. Then theircomposition B ⋄ A : g → g ′′ is also a homomorphism defined as ( B ⋄ A ) = B ◦ A : g → g ′′ ,( B ⋄ A ) = B ◦ A : g − → g ′′− and( B ⋄ A ) = B ◦ ( A × A ) + B ◦ A : g ∧ g → g ′′− . A homomorphism A : g → g ′ is called an isomorphism if there exists a homomorphism A − : g ′ → g such that the compositions A − ⋄ A : g → g and A ⋄ A − : g ′ → g ′ are both identityhomomorphisms. It is easy to show that Lemma 4.1.
Let A = ( A , A , A ) : g → g ′ be a homomorphism. If A , A are invertible, then A is an isomorphism, and A − is given by A − = ( A − , A − , − A − ◦ A ◦ ( A − × A − )) . To define Aut( g ), first denote by Aut ( g ) the set of Lie 2-algebra automorphisms of g . It isevident that (Aut ( g ) , ⋄ ) is a Lie group. Next, define a multiplication on End − ( g ) by τ ⋆ τ ′ = τ + τ ′ + τ ◦ d ◦ τ ′ , ∀ τ, τ ′ ∈ End − ( g ) . (13)It is obvious that ⋆ satisfies the associative law. Thus, (End − ( g ) , ⋆ ) is a monoid, in which thezero map is the identity element. Aut − ( g ) is defined to be the group of units of End − ( g ), whichis a Lie group. Lemma 4.2.
For all τ ∈ End − ( g ) , we have τ ∈ Aut − ( g ) ⇐⇒ I + d ◦ τ ∈ GL ( g ) ⇐⇒ I + τ ◦ d ∈ GL ( g − ) . Proof.
First we prove the equivalence of the first two formulas. Let τ ∈ Aut − ( g ), then thereexists an element τ − ∈ Aut − ( g ) such that τ ⋆ τ − = τ − ⋆ τ = 0 . We have( I + d ◦ τ ) − = I + d ◦ τ − , (14)which follows from( I + d ◦ τ ) ◦ ( I + d ◦ τ − ) = I + d ◦ ( τ + τ − + τ ◦ d ◦ τ − ) = I, ( I + d ◦ τ − ) ◦ ( I + d ◦ τ ) = I + d ◦ ( τ + τ − + τ − ◦ d ◦ τ ) = I. For the inverse direction, if the inverse of I + d ◦ τ exists, then we claim that τ − = − τ ◦ ( I + d ◦ τ ) − . In fact, we have τ ⋆ ( − τ ◦ ( I + d ◦ τ ) − ) = τ − τ ◦ ( I + d ◦ τ ) − − τ ◦ d ◦ τ ◦ ( I + d ◦ τ ) − = τ − τ ◦ ( I + d ◦ τ ) ◦ ( I + d ◦ τ ) − = 0 . Similarly, we have ( − τ ◦ ( I + d ◦ τ ) − ) ⋆ τ = 0 . Strictly speaking, we should use the terminology “a weak automorphism” since it preserves the structure onlyup to homotopy. To be succinct, we just say “an automorphism”. ( g ) , ⋄ ) and(Aut − ( g ) , ⋆ ). The following observation plays an important role in the construction of the homo-morphism from Aut − ( g ) to Aut ( g ). Lemma 4.3.
Let A : g −→ g be a Lie -algebra homomorphism. Then for any τ ∈ End − ( g ) , ( A + d ◦ τ, A + τ ◦ d, A + l Aτ ) is a Lie -algebra homomorphism, where l Aτ ( x, y ) = τ [ x, y ] − [ A x, τ y ] − [ τ x, A y ] − [ τ x, dτ y ] . In particular, ( I + d ◦ τ, I + τ ◦ d, l i τ ) is a Lie -algebra homomorphism, where i = ( I, I, is theidentity homomorphism. Proof.
Firstly, due to A ◦ d = d ◦ A , it is obvious that( A + d ◦ τ ) ◦ d = d ◦ ( A + τ ◦ d ) . (15)Then, by the fact that A is a Lie 2-algebra homomorphism, we have( A + d ◦ τ )[ x, y ] − [( A + d ◦ τ ) x, ( A + d ◦ τ ) y ]= dA ( x, y ) + dτ [ x, y ] − d [ A x, τ y ] − d [ τ x, A y ] − d [ τ x, dτ y ]= d ( A + l Aτ )( x, y ) . (16)Similarly, we can deduce that( A + τ ◦ d )[ x, a ] − [( A + d ◦ τ ) x, ( A + τ ◦ d ) a ] = ( A + l Aτ )( x, da ) . (17)Finally, also by the fact that A is a Lie 2-algebra homomorphism, we have[( A + d ◦ τ ) x, ( A + l Aτ )( y, z )] − ( A + l Aτ )([ x, y ] , z ) + c.p. = [ A x, A ( y, z )] − A ([ x, y ] , z ) + c.p. +[ dτ x, A ( y, z )] + [ dτ x + A x, τ [ y, z ] − [ A y, τ z ] − [ τ y, A z ] − [ τ y, dτ z ]] − τ [[ x, y ] , z ] + [[ A x, A y ] + dA ( x, y ) , τ z ] + [ τ [ x, y ] , A z + dτ z ] + c.p. = A l ( x, y, z ) − l ( A x, A y, A z ) + τ dl ( x, y, z ) − [ dτ x + A x, [ A y, τ z ] + [ τ y, A z ] + [ τ y, dτ z ]] + [[ A x, A y ] , τ z ] + c.p. = ( A + τ ◦ d ) l ( x, y, z ) − l (cid:0) ( A + d ◦ τ ) x, ( A + d ◦ τ ) y, ( A + d ◦ τ ) z (cid:1) . (18)By (15)-(18), we deduce that ( A + d ◦ τ, A + τ ◦ d, A + l Aτ ) is also a Lie 2-algebra homomorphism.This ends the proof.By Lemma 4.2, if τ ∈ Aut − ( g ), then ( I + d ◦ τ, I + τ ◦ d, l i τ ) ∈ Aut ( g ). Thus, we have a smoothmap ∂ : Aut − ( g ) −→ Aut ( g ), which is given by ∂ ( τ ) = ( I + d ◦ τ, I + τ ◦ d, l i τ ) , ∀ τ ∈ Aut − ( g ) . (19)Furthermore, Aut ( g ) acts on Aut − ( g ) naturally: A ⊲ τ = A ◦ τ ◦ A − ∀ A = ( A , A , A ) ∈ Aut ( g ) , τ ∈ Aut − ( g ) . (20) Theorem 4.4.
With the notations above, (Aut − ( g ) , Aut ( g ) , ∂, ⊲ ) is a crossed module of Liegroups, which is called the automorphism -group of g , and denoted by Aut( g ) . roof. For any τ, τ ′ ∈ Aut − ( g ), we have( ∂τ ) ⋄ ( ∂τ ′ ) = ( I + d ◦ τ, I + τ ◦ d, l i τ ) ⋄ ( I + d ◦ τ ′ , I + τ ′ ◦ d, l i τ ′ )= ( I + d ◦ ( τ ⋆ τ ′ ) , I + ( τ ⋆ τ ′ ) ◦ d, F ) , where F is given by F ( x, y ) = l i τ (( I + d ◦ τ ′ ) x, ( I + d ◦ τ ′ ) y ) + ( I + τ ◦ d ) l i τ ′ ( x, y )= τ [ x + dτ ′ x, y + dτ ′ y ] − [ x + dτ ′ x, τ ( y + dτ ′ y )] − [ τ ( x + dτ ′ x ) , y + dτ ′ y ] − [ τ ( x + dτ ′ x ) , dτ ( y + dτ ′ y )] + ( I + τ ◦ d )( τ ′ [ x, y ] − [ τ ′ x, y ] − [ x, τ ′ y ] − [ τ ′ x, dτ ′ y ])= τ [ x, y ] + τ ′ [ x, y ] + τ dτ ′ [ x, y ] − [ τ x + τ ′ x + τ dτ ′ x, y ] − [ x, τ y + τ ′ y + τ dτ ′ y ] − [ τ x + τ ′ x + τ dτ ′ x, dτ y + dτ ′ y + dτ dτ ′ y ]= l i τ⋆τ ′ ( x, y ) . Thus, we have ∂ ( τ ⋆ τ ′ ) = ( ∂τ ) ⋄ ( ∂τ ′ ) , (21)which implies that ∂ is a Lie group homomorphism.For any A ∈ Aut ( g ), we have that A − = ( A − , A − , − A − ◦ A ◦ ( A − × A − )), and A ⋄ ( ∂τ ) ⋄ A − = ( A , A , A ) ⋄ ( I + d ◦ τ, I + τ ◦ d, l i τ ) ⋄ ( A − , A − , − A − ◦ A ◦ ( A − × A − ))= (cid:0) I + d ◦ ( A ⊲ τ ) , I + ( A ⊲ τ ) ◦ d, H ) , where H ( x, y )= (cid:0) A ⋄ ( ∂τ ) (cid:1) ( A − x, A − y ) − A ◦ ( I + τ ◦ d ) ◦ A − ◦ A ◦ ( A − × A − )( x, y )= A (cid:0) ( I + d ◦ τ )( A − x ) , ( I + d ◦ τ )( A − y ) (cid:1) − A ( A − x, A − y ) − A τ dA − A ( A − x, A − y )+ A (cid:0) τ [ A − x, A − y ] − [ τ A − x, A − y ] − [ A − x, τ A − y ] − [ τ A − x, dτ A − y ] (cid:1) = A τ A − [ x, y ] − [ A τ A − x, y ] − [ x, A τ A − y ] − [ A τ A − x, dA τ A − y ]= l i A ◦ τ ◦ A − ( x, y ) = l i A⊲τ ( x, y ) . Thus, we have A ⋄ ( ∂τ ) ⋄ A − = ∂ ( A ⊲ τ ) . (22)By straightforward computations, we have τ ⋆ τ ′ ⋆ τ − = τ ⋆ τ ′ + τ − + ( τ ⋆ τ ′ ) ◦ d ◦ τ − = τ + τ ′ + τ ◦ d ◦ τ ′ + τ − + τ ◦ d ◦ τ − + τ ′ ◦ d ◦ τ − + τ ◦ d ◦ τ ′ ◦ d ◦ τ − = τ ′ + τ ◦ d ◦ τ ′ + τ ′ ◦ d ◦ τ − + τ ◦ d ◦ τ ′ ◦ d ◦ τ − . On the other hand, by (14), we have ∂ ( τ ) ⊲ τ ′ = ( I + τ ◦ d ) ◦ τ ′ ◦ ( I + d ◦ τ ) − = ( I + τ ◦ d ) ◦ τ ′ ◦ ( I + d ◦ τ − )= τ ′ + τ ◦ d ◦ τ ′ + τ ′ ◦ d ◦ τ − + τ ◦ d ◦ τ ′ ◦ d ◦ τ − . Thus, we have ∂ ( τ ) ⊲ τ ′ = τ ⋆ τ ′ ⋆ τ − . (23)By (22) and (23), we deduce that (Aut − ( g ) , Aut ( g ) , ∂, ⊲ ) is a crossed module of Lie groups. Theproof is finished. 10 xample 4.5. ( the string Lie 2-algebra string( k )) For the string Lie 2-algebra string( k ), weexplore its automorphism 2-group Aut(string( k )). For any ( A , t ) ∈ End( k ) ⊕ R and A ∈ ∧ k ∗ , wehave A ∈ Aut (string( k )) ⇐⇒ (cid:26) A ∈ Aut( k ) , D A ( x, y, z ) = tl ( x, y, z ) − l ( A x, A y, A z ) . Since k is semisimple, any automorphism is orthogonal. Then the second condition reduces to D A ( x, y, z ) = tl ( x, y, z ) − l ( x, y, z ) . Also by the fact that the Cartan 3-form is not exact, we deduce that t = 1 and D ( A ) = 0. Thus,any automorphism A ∈ Aut( k ) and a 2-cocycle ω ∈ ∧ k ∗ give rise to an automorphism ( A , , ω )of degree 0 of the string Lie 2-algebra. Parallel to the discussion in Proposition 3.4, we obtainthat Aut (string( k )) is isomorphic to the semidirect product K ⋉ k ∗ , where K is the connected andsimply connected Lie group that integrating k . Obviously, the set of automorphisms of degree − k ∗ , which is a Lie group with the abeliangroup structure. The map ∂ : Aut − (string( k )) → Aut (string( k )) is given by ∂ ( τ ) = ( I, I, − D ( τ )). Example 4.6.
For the skeletal Lie 2-algebra g = k ⊕ V given in Example 3.5, we explore itsautomorphism 2-group Aut( g ). Indeed, for any ( A , A ) ∈ End( k ) ⊕ End( V ) and A ∈ C ( k , V ), itis easy to check that A ∈ Aut ( g ) ⇐⇒ (cid:26) A ⊕ A ∈ Aut( k ⋉ φ V ) , D A A = [ A, l ] , where D A : C k ( k , V ) → C k +1 ( k , V ) is the Lie algebra coboundary operator with respect to therepresentation φ Ax ( v ) = φ A ( x ) ( v ), Aut( k ⋉ φ V ) is the Lie group of automorphisms of k ⋉ φ V andthe bracket [ · , · ] is given by[ A, l ]( x, y, z ) = A l ( x, y, z ) − l ( A x, A y, A z ) , ∀ x, y, z ∈ k . Moreover, we have Aut − ( g ) = C ( k , V ), which is a Lie group with the abelian group structure.The map ∂ : Aut − ( g ) → Aut ( g ) is given by ∂ ( τ ) = ( I, I, − D ( τ )). Der( g ) In this section, we show that for a Lie 2-algebra g , the automorphism 2-group Aut( g ) is an integra-tion of the derivation Lie 2-algebra Der( g ), i.e. the differentiation of Aut( g ) is the crossed moduleof Lie algebras corresponding to Der( g ). In addition, we give the expressions of exponential mapsfrom Der i ( g ) to Aut i ( g ) for i = 0 , −
1, which commute with the differentials ∂ and ¯ d . Theorem 5.1.
The differentiation of the crossed module of Lie groups (Aut − ( g ) , Aut ( g ) , ∂, ⊲ ) is the crossed module of Lie algebras (Der − ( g ) , Der ( g ) , ¯ d, φ ) . We split the proof into three steps. • Step 1: We shall check the Lie algebra of Aut ( g ) is Der ( g ). Define a mapexp : R × (cid:0) End( g ) ⊕ End( g − ) ⊕ Hom( ∧ g , g − ) (cid:1) −→ End( g ) ⊕ End( g − ) ⊕ Hom( ∧ g , g − )by exp( t, ( X , X , l X )) = ( e tX , e tX , e tl X ) , (24)11here e tX i = P ∞ n =0 t n X ni n ! , i = 0 , , is the usual exponential map, and e tl X is given by e tl X ( x, y ) = ∞ X n =1 t n n ! X i + j + k = n − ,i,j,k ≥ C ii + j X k l X ( X i x, X j y ) , ∀ x, y ∈ g . (25)Later, we will see that exp(1 , ( X, l X )) : Der ( g ) −→ Aut ( g ) is the exponential map. Lemma 5.2.
The tangent space of
Aut ( g ) at the identity element i = ( I g , I g − , is Der ( g ) ,i.e. T i Aut ( g ) = Der ( g ) . Proof.
Firstly, we prove T i Aut ( g ) ⊂ Der ( g ). Let c ( t ) = ( c ( t ) , c ( t ) , c ( t )) be any curve inAut ( g ) such that c (0) = ( I g , I g − ,
0) and c ′ (0) = ( X , X , l X ). By the fact that c ( t ) is a curve inAut ( g ), we have ddt | t =0 (cid:16) [ c ( t ) x, c ( t )( y, z )] − c ( t )([ x, y ] , z ) + c.p. + l ( c ( t ) x, c ( t ) y, c ( t ) z ) − c ( t ) l ( x, y, z ) (cid:17) = [ x, l X ( y, z )] − l X ([ x, y ] , z ) + l ( X x, y, z ) + c.p. − X l ( x, y, z ) , which implies that one can obtain Condition (c) in Definition 3.1 from Condition (iii) in Definition2.2 by differentiation. Similarly, we can obtain (a)-(b) from (i)-(ii) by differentiation. Thus, c ′ (0) ∈ Der ( g ). This proves that T i Aut ( g ) ⊂ Der ( g ).On the other hand, for all ( X, l X ) ∈ Der ( g ), we first claim that c ( t ) := exp( t, ( X, l X )) = ( e tX , e tX , e tl X ) ∈ Aut ( g ) , ∀ t ∈ R . It is obvious that e tX and e tX are invertible and commute with d since X ◦ d = d ◦ X . Considerthe following two curves in g : y ( t ) = e tX [ x, y ] , y ( t ) = [ e tX x, e tX y ] + de tl X ( x, y ) . Obviously, we have y (0) = y (0) = [ x, y ] , (26)and y ( t ) satisfies the following ordinary differential equation: y ′ ( t ) = X e tX [ x, y ] = X y ( t ) . (27)For any t , we have ddt (cid:0) e tl X ( x, y )) (cid:1) = ∞ X n =1 t n − ( n − X i + j + k = n − C ii + j X k l X ( X i x, X j y )= ∞ X n =1 t n − ( n − X i + j = n − ( n − i ! j ! l X ( X i x, X j y )+ X ∞ X n =2 t n − ( n − X i + j + k − n − C ii + j X k − l X ( X i x, X j y )= l X ( e tX x, e tX y ) + X e tl X ( x, y ) . Hence, we get y ′ ( t ) = [ X e tX x, e tX y ] + [ e tX x, X e tX y ] + dl X ( e tX x, e tX y ) + X de tl X ( x, y )= X [ e tX x, e tX y ] + X de tl X ( x, y )= X y ( t ) , (28)12here we have used the fact X [ x, y ] = [ X x, y ] + [ x, X y ] + dl X ( x, y ) , ∀ x, y ∈ g . By (26)-(28) and the uniqueness theorem for linear systems of ordinary differential equations, weobtain that y ( t ) = y ( t ), i.e. e tX [ x, y ] = [ e tX x, e tX y ] + de tl X ( x, y ) . (29)Similarly, we can get e tX [ x, a ] = [ e tX x, e tX a ] + e tl X ( x, da ) . (30)At last, consider the following two curves in g − : z ( t ) = − e tX l ( x, y, z ) ,z ( t ) = e tl X ([ x, y ] , z ) + [ e tl X ( x, y ) , e tX z ] + c.p. − l ( e tX x, e tX y, e tX z ) . Obviously, we have z (0) = z (0) = − l ( x, y, z ) , and z ( t ) satisfies the ordinary differential equa-tion: z ′ ( t ) = X z ( t ) . By the fact (
X, l X ) ∈ Der ( g ), we can deduce that z ′ ( t ) = X z ( t ). Thus,we have z ( t ) = z ( t ), i.e. e tl X ([ x, y ] , z ) + [ e tl X ( x, y ) , e tX z ] + c.p. − l ( e tX x, e tX y, e tX z ) = − e tX l ( x, y, z ) . (31)By (29)-(31), we deduce that ( e tX , e tX , e tl X ) ∈ Aut ( g ) for all t ∈ R .Therefore, for all ( X, l X ) ∈ Der ( g ), we get a curve c ( t ) := ( e tX , e tX , e tl X ) in Aut ( g ) satisfying c (0) = i = ( I g , I g − , c ′ (0) = ( X, l X ), which implies that Der ( g ) ⊂ T i Aut ( g ).This ends the proof.For all ( X, l X ) ∈ Der ( g ) = T i Aut ( g ), we get a smooth map σ ( X,l X ) : R −→ Aut ( g )defined by σ ( X,l X ) ( t ) = exp( t, ( X, l X )) = ( e tX , e tX , e tl X ) . Define a map e : Der ( g ) −→ Aut ( g ) by e ( X,l X ) = σ ( X,l X ) (1) . (32) Lemma 5.3.
With the notations above, for all ( X, l X ) ∈ Der ( g ) , the map σ ( X,l X ) ( t ) is a one-parameter subgroup of the Lie group Aut ( g ) determined by ( X, l X ) . Consequently, the map e :Der ( g ) −→ Aut ( g ) given by (32) is the exponential map. Proof.
For simplicity, we omit the subscript (
X, l X ) when there is no confusion. It is obvious that e ( t + s ) X i = e tX i ◦ e sX i . Furthermore, by the equality C ii + j = P mα =0 C αm C i − αi + j − m , for any i + j ≥ m ,13e have e ( t + s ) l X ( x, y ) = ∞ X n =1 ( t + s ) n n ! X i + j + k = n − C ii + j X k l X ( X i x, X j y )= ∞ X n =1 n X m =0 s m t n − m m !( n − m )! X i + j + k = n − C ii + j X k l X ( X i x, X j y )= ∞ X p,m =0 ,p + m ≥ t p p ! s m m ! X i + j + k = p + m − C ii + j X k l X ( X i x, X j y )= ∞ X p =0 t p p ! ∞ X m =1 s m m ! X i + j + k − p = m − C ii + j X p X k − p l X ( X i x, X j y )+ ∞ X p =1 t p p ! ∞ X m =0 X u + v + k = p − m X α =0 s m C uu + v α !( m − α )! X k l X ( X u X α x, X v X m − α y )= e tX e sl X ( x, y ) + e tl X ( e sX x, e sX y )= (cid:0) ( e tX , e tX , e tl X ) ⋄ ( e sX , e sX , e sl X ) (cid:1) ( x, y ) . Therefore, we have( e ( t + s ) X , e ( t + s ) X , e ( t + s ) l X ) = ( e tX , e tX , e tl X ) ⋄ ( e sX , e sX , e sl X ) , which implies that σ ( t + s ) = σ ( t ) ⋄ σ ( s ), and σ ( t ) is a one-parameter subgroup.It is obvious that σ ′ (0) = ddt | t =0 ( e tX , e tX , e tl X ) = ( X, l X ) , which implies that σ ( t ) is a one-parameter subgroup determined by ( X, l X ). Thus, the map e : Der ( g ) −→ Aut ( g ) given by (32) is the exponential map.With regard to the relation between the Lie group structure on K and the Lie bracket of itsLie algebra k , there is a useful formula: e sX e tY e − sX e − tY . = e st [ X,Y ] k , ∀ X, Y ∈ k , (33)where e : k −→ K is the exponential map. Lemma 5.4.
The Lie algebra of the Lie group (Aut ( g ) , ⋄ ) is (Der ( g ) , {· , ·} ) , where the bracket {· , ·} is given by (6) . Proof.
By Lemma 5.2, Der ( g ) is the tangent space of Aut ( g ) at the identity element. To provethat the Lie algebra of the Lie group (Aut ( g ) , ⋄ ) is (Der ( g ) , {· , ·} ), we only need to show that theinduced Lie bracket [ · , · ] ind on Der ( g ) is exactly {· , ·} . By Lemma 5.3, we have the exponentialmap e : Der ( g ) −→ Aut ( g ) given by (32). Thus, we have[( X, l X ) , ( Y, l Y )] ind = ddt dds | t,s =0 ( e sX , e sX , e sl X ) ⋄ ( e tY , e tY , e tl Y ) ⋄ ( e − sX , e − sX , e − sl X ) ⋄ ( e − tY , e − tY , e − tl Y )= ddt dds | t,s =0 ( e sX e tY e − sX e − tY , e sX e tY e − sX e − tY , F )= (cid:0) [ X , Y ] C , [ X , Y ] C , ddt dds | t,s =0 F (cid:1) , F is given by F ( x, y ) = e sl X ( e tY e − sX e − tY x, e tY e − sX e − tY y ) + e sX e tl Y ( e − sX e − tY x, e − sX e − tY y )+ e sX e tY e − sl X ( e − tY x, e − tY y ) + e sX e tY e − sX e − tl Y ( x, y ) . By (24), we get ddt dds | t,s =0 F = X l Y ( x, y ) − l Y ( X x, y ) − l Y ( x, X y ) − Y l X ( x, y ) + l X ( Y x, y ) + l X ( x, Y y )= L X l Y ( x, y ) − L Y l X ( x, y ) . So the induced Lie bracket is[(
X, l X ) , ( Y, l Y )] ind = ([ X, Y ] C , L X l Y − L Y l X ) , which is exactly the Lie bracket (6) on Der ( g ). This ends the proof. • Step 2: We verify that the Lie algebra of Aut − ( g ) is Der − ( g ). Moreover, the two exponentialmaps from Der i ( g ) to Aut i ( g ) for i = 0 , − ∂ and ¯ d . Lemma 5.5.
The Lie algebra of (Aut − ( g ) , ⋆ ) is (Der − ( g ) , {· , ·} ) , where the multiplication ⋆ and the bracket {· , ·} are given by (13) and (9) respectively. Additionally, the exponential map e : Der − ( g ) −→ Aut − ( g ) is given by e Θ = Θ + Θ ◦ d ◦ Θ2! + Θ ◦ d ◦ Θ ◦ d ◦ Θ3! + · · · . (34) Proof.
For all Θ ∈ Der − ( g ), define a one-parameter map σ Θ : R −→ End − ( g ) by σ Θ ( t ) = e t Θ . Then, we have I + d ◦ σ Θ ( t ) = e t ( d ◦ Θ) ∈ GL ( g ) . By Lemma 4.2, σ Θ ( t ) ∈ Aut − ( g ). Itis straightforward to verify that T Aut − ( g ) = Der − ( g ). Furthermore, σ Θ is a one-parametersubgroup of Aut − ( g ) determined by Θ, which follows from σ Θ ( t + s ) = ∞ X n =1 n X m =0 t m s n − m m !( n − m )! (Θ ◦ d ) n − ◦ Θ= ∞ X n =1 s n n ! (Θ ◦ d ) n − ◦ Θ + ∞ X n =1 t n n ! (Θ ◦ d ) n − ◦ Θ+ ∞ X n =1 n − X m =1 t m s n − m m !( n − m )! (Θ ◦ d ) m − ◦ Θ ◦ d ◦ (Θ ◦ d ) n − m − ◦ Θ= σ Θ ( t ) + σ Θ ( s ) + σ Θ ( t ) ◦ d ◦ σ Θ ( s )= σ Θ ( t ) ⋆ σ Θ ( s ) . So we get that e : Der − ( g ) −→ Aut − ( g ) defined by (34) is the exponential map determined by Θ.By (9), (13) and (33), the induced Lie bracket on Der − ( g ) by the Lie group structure on Aut − ( g )15s given by [Θ , Θ ′ ] ind = ddt dds | t,s =0 e s Θ ⋆ e t Θ ′ ⋆ e − s Θ ⋆ e − t Θ ′ = ddt dds | t,s =0 ( s Θ + · · · ) ⋆ ( t Θ ′ + · · · ) ⋆ ( − s Θ + · · · ) ⋆ ( − t Θ ′ + · · · )= ddt dds | t,s =0 ( s Θ + t Θ ′ + st Θ d Θ ′ + · · · ) ⋆ ( − s Θ + · · · ) ⋆ ( − t Θ ′ + · · · )= ddt dds | t,s =0 ( st Θ ◦ d ◦ Θ ′ − ts Θ ′ ◦ d ◦ Θ + · · · )= Θ ◦ d ◦ Θ ′ − Θ ′ ◦ d ◦ Θ= { Θ , Θ ′ } . Thus, the Lie algebra of (Aut − ( g ) , ⋆ ) is (Der − ( g ) , {· , ·} ). This completes the proof.The two exponential maps given by (32) and (34) commute with the differentials ∂ and d . Lemma 5.6.
For any Θ ∈ Der − ( g ) , ∂ ( e Θ ) = e ¯ d (Θ) , i.e. the following diagram is commutative: Der − ( g ) e / / d (cid:15) (cid:15) Aut − ( g ) ∂ (cid:15) (cid:15) Der ( g ) e / / Aut − ( g ) . Proof.
First, by definition, we have I + d ◦ e Θ = I + ∞ X n =1 d ◦ (Θ ◦ d ) n − ◦ Θ n ! = e d ◦ Θ , I + e Θ ◦ d = e Θ ◦ d . Then, by straightforward computation, we get e l δ (Θ) ( x, y )= ∞ X n =1 n ! X i + j + k = n − C ii + j (Θ d ) k l δ (Θ) (( d Θ) i x, ( d Θ) j y )= ∞ X n =1 n ! X i + j + k = n − C ii + j (Θ d ) k (Θ[( d Θ) i x, ( d Θ) j y ] − [Θ( d Θ) i x, ( d Θ) j y ] − [( d Θ) i x, Θ( d Θ) j y ])= e Θ [ x, y ] − [ e Θ x, y ] − [ x, e Θ y ] + ∞ X n =1 n ! { X i ′ + j + k ′ = n − ,k ′ ≥ C i ′ +1 i ′ + j +1 (Θ d ) k ′ [(Θ d ) i ′ Θ x, ( d Θ) j y ] − X i + j + k = n − ,j + k ≥ C ii + j (Θ d ) k [(Θ d ) i Θ x, ( d Θ) j y ] − X i ′ + j ′ + k = n − ,j ′ ≥ C i ′ +1 i ′ + j ′ (Θ d ) k [(Θ d ) i ′ Θ x, ( d Θ) j ′ y ] } = e Θ [ x, y ] − [ e Θ x, y ] − [ x, e Θ y ] − ∞ X n =1 n ! { X i ′ + j = n,i ′ ,j ≥ C i ′ − n − [(Θ d ) i ′ − Θ x, d (Θ d ) j − Θ y ]+ X i ′′ + j ′ = n,i ′′ ,j ′ ≥ C i ′′ n − [(Θ d ) i ′′ − Θ x, d (Θ d ) j ′ − Θ y ] } = e Θ [ x, y ] − [ e Θ x, y ] − [ x, e Θ y ] − [ e Θ x, de Θ y ]= l i e Θ ( x, y ) , where we have used the equalities C p +1 p + q +1 = C pp + q + C p +1 p + q and C p − n − + C pn − = C pn in the third andfourth steps respectively. Therefore, we have ∂ ( e Θ ) = ( I + d ◦ e Θ , I + e Θ ◦ d, l i e Θ ) = ( e d ◦ Θ , e Θ ◦ d , e l δ (Θ) ) = e ¯ d (Θ) . This finishes the proof. • Step 3: The final step is to prove that the differential of ∂ is ¯ d and the induced action of ⊲ on Der( g ) is φ . By Lemma 5.6, we have ∂ ( e t Θ ) = e t ¯ d (Θ) , which implies ∂ ∗ = ¯ d .Denote by e φ the induced action of Der ( g ) on Der − ( g ). By straightforward calculations, wehave e φ ( X,l X ) Θ = ddt dds | t,s =0 ( e tX , e tX , e tl X ) ⊲ e s Θ = ddt dds | t,s =0 e tX ◦ e s Θ ◦ e − tX = X ◦ Θ − Θ ◦ X = { X + l X , Θ } = φ ( X,l X ) Θ . This ends the proof of Theorem 5.1.Consider the Lie algebra Der ( g ) ⋉ Der − ( g ) with brackets defined by (5)-(9) and the Lie groupAut ( g ) ⋉ Aut − ( g ) with group structure (4) defined by( A, τ ) · ( A ′ , τ ′ ) = ( A ⋄ A ′ , τ ⋆ ( A ⊲ τ ′ )) . Corollary 5.7.
The Lie algebra of
Aut ( g ) ⋉ Aut − ( g ) is Der ( g ) ⋉ Der − ( g ) . Moreover, theexponential map e : Der ( g ) ⋉ Der − ( g ) −→ Aut ( g ) ⋉ Aut − ( g ) is given by e (( X,l X ) , Θ) = ( e ( X,l X ) , e Θ ) , where e ( X,l X ) , e Θ are described explicitly by (32) and (34) respectively. Strict automorphisms and Inner automorphisms
In this part, we first consider a sub-Lie 2-algebra SDer( g ) of Der( g ) and a sub-crossed moduleSAut( g ) of Aut( g ). Then, for a strict Lie 2-algebra g seen as a crossed module with the corre-sponding crossed module of Lie groups G = ( G , G , d G , ⊲ ), we derive the relation between Aut( G )introduced in [13] and SAut( g ).For a Lie 2-algebra g , consider a sub-complex SDer( g ) of Der( g ) defined bySDer ( g ) , { X ∈ End( g ) ⊕ End( g − ) | ( X, ∈ Der ( g ) } ;SDer − ( g ) , { Θ ∈ Der − ( g ) = Hom( g , g − ) | Θ[ x, y ] = [ x, Θ y ] + [Θ x, y ] , ∀ x, y ∈ g } . We call them strict derivations of a Lie 2-algebra. Explicitly, X = ( X , X ) ∈ SDer ( g ) if andonly if for any x, y, z ∈ g , a ∈ g − , X ◦ d = d ◦ X ,X [ x, y ] = [ X x, y ] + [ x, X y ] ,X [ x, a ] = [ X x, a ] + [ x, X a ] ,X l ( x, y, z ) = l ( X x, y, z ) + c.p. ( x, y, z ) . Then consider a sub-complex SAut( g ) of Aut( g ) given bySAut ( g ) , { A ∈ End( g ) ⊕ End( g − ) | ( A , A , ∈ Aut ( g ) } ;SAut − ( g ) , { τ ∈ Aut − ( g ) = Hom( g , g − ) | τ [ x, y ] = [ x, τ y ] + [ τ x, y ] + [ τ x, dτ y ] , ∀ x, y ∈ g } . In detail, A = ( A , A ) ∈ SAut ( g ) if and only if A ◦ d = d ◦ A , and A [ x, y ] = [ A x, A y ] , A [ x, a ] = [ A x, A a ] , A l ( x, y, z ) = l ( A x, A y, A z ) . Lemma 6.1. (SAut − ( g ) , SAut ( g ) , ∂, ⊲ ) is a sub-crossed module of Lie groups of Aut( g ) . More-over, the differentiation of SAut( g ) is the crossed module of Lie algebras (SDer − ( g ) , SDer ( g ) , ¯ d, φ ) ,which is a sub-crossed module of Der( g ) . Proof.
It is a straightforward verification.Next, suppose that g is a strict Lie 2-algebra with the corresponding crossed module of Liegroups G = ( G , G , d G , ⊲ ). According to [13], the automorphism 2-group Aut( G ) of G is definedas follows. DefineDer( G , G ) = { χ : G −→ G | χ ( αβ ) = χ ( α )( α ⊲ χ ( β )) , ∀ α, β ∈ G } , with a multiplication χ ∗ χ ′ ( α ) = χ ( d G χ ′ ( α ) α ) χ ′ ( α ) . (35)Denote by Aut − ( G ) the group of units of Der( G , G ). DefineAut ( G ) := { ( F , F ) ∈ Aut( G ) × Aut( G ) | F ◦ d G = d G ◦ F , F ( α ⊲ ξ ) = F ( α ) ⊲ F ( ξ ) } . Indeed, Aut ( G ) is the set of isomorphisms of G . Then Aut( G ) becomes a crossed module of Liegroups Aut( G ) : Aut − ( G ) ˜ ∂ −−−−→ Aut ( G ) , ∂ is defined by ˜ ∂ ( χ )( α ) = d G ( χ ( α )) α, ˜ ∂ ( χ )( ξ ) = χ ( d G ( ξ )) ξ, (36)and the action of Aut ( G ) on Aut − ( G ) is defined by( F , F ) ⊲ χ = F ◦ χ ◦ F − . Lemma 6.2.
For any F ∈ Aut ( G ) , let F ∗ = ( F ∗ , F ∗ ) be the tangent map at the identity. Then, F ∗ ∈ SAut ( g ) , Furthermore, we have ( F ◦ F ′ ) ∗ = F ∗ ⋄ F ′∗ , ∀ F, F ′ ∈ Aut ( G ) . Lemma 6.3.
For any χ ∈ Aut − ( G ) , let χ ∗ be the tangent map at the identity. Then, χ ∗ ∈ SAut − ( g ) . Moveover, we have ( χ ∗ χ ′ ) ∗ = χ ∗ ⋆ χ ′∗ , ∀ χ, χ ′ ∈ Aut − ( G ) . where ∗ and ⋆ are defined by (35) and (13) respectively. Proof.
The condition χ ( αβ ) = χ ( α ) α ⊲ χ ( β ) implies that ( I G , χ ) : G ⋉ G −→ G ⋉ G , whichis given by ( I G , χ )( α, ξ ) = ( α, χ ( α )) , is a Lie group homomorphism. By the fact that the Lie algebra of G ⋉ G is g ⋉ g − , we obtainthat ( I, χ ∗ ) : g ⋉ g − −→ g ⋉ g − is a Lie algebra homomorphism, which implies that χ ∗ [ x, y ] = [ x, χ ∗ y ] + [ χ ∗ x, y ] + [ dχ ∗ x, χ ∗ y ] , i.e. χ ∗ ∈ SAut − ( g ). The other conclusion is obvious.Moveover, by simple calculations, we have Lemma 6.4.
For all F ∈ Aut ( G ) and χ ∈ Aut − ( G ) , we have ( ˜ ∂ ( χ )) ∗ = ∂ ( χ ∗ ) , ( F ⊲ χ ) ∗ = F ∗ ⊲ χ ∗ , where ¯ ∂ and ∂ are defined by (36) and (19) respectively. In summary, by Lemma 6.2-6.4, we have
Theorem 6.5.
The map from
Aut( G ) to SAut( g ) given by ( F, χ ) ( F ∗ , χ ∗ ) , ∀ F ∈ Aut ( G ) , χ ∈ Aut − ( G ) , is a homomorphism between crossed modules of Lie groups. Recall the inner derivation Lie 2-algebra of g defined in Section 3, which is given byinn( g ) : inn − ( g ) , End − ( g ) ¯ d −−−−→ inn ( g ) , where inn ( g ) = Im ad + Im ¯ d is indeed the set of 1-coboundaries of g with the coefficients in theadjoint representation. Moreover, inn( g ) is an ideal of the derivation Lie 2-algebra Der( g ). Thegoal of this subsection is to define the inner automorphism 2-group of a Lie 2-algebra g , which is19 normal sub-crossed module [13] of Aut( g ). The main ingredient used in this subsection is theexponential map given in Corollary 5.7.For the Lie group Aut( g ), we have the adjoint map Ad : Aut( g ) −→ Aut(Aut( g )) defined by Ad( A )( A ′ ) = A · A ′ · A − = A ⋄ A ′ ⋄ A − , Ad( A )( τ ) = A · τ · A − = A ⊲ τ,
Ad( τ )( A ) = τ · A · τ − = ( A, τ ⋆ ( A ⊲ τ − )) , Ad( τ )( τ ′ ) = τ · τ ′ · τ − = τ ⋆ τ ′ ⋆ τ − . We use the same notation to denote the induced adjoint representation of Aut( g ) on its Lie algebraDer( g ). Concretely, we have Lemma 6.6.
The map
Ad : Aut( g ) −→ GL (Der( g )) is formulated as follows: Ad( A )( X, l X ) = ( A ◦ X ◦ A − , A ◦ X ◦ A − , l A ◦ X ◦ A − ) , Ad( A )(Θ) = A ◦ Θ ◦ A − , Ad( τ )( X, l X ) = (( X, l X ) , X ◦ τ − + τ ◦ X + τ ◦ d ◦ X ◦ τ − ) , Ad( τ )(Θ) = ( I + τ ◦ d ) ◦ Θ ◦ ( I + d ◦ τ ) − , where l A ◦ X ◦ A − is defined by l A ◦ X ◦ A − ( x, y ) = A l X ( A − x, A − y ) − A X A − A ( A − x, A − y )+ A ( X A − x, A − y ) + A ( A − x, X A − y ) . (37)For any Lie group K with Lie algebra k , we have k · e X · k − = e Ad( k ) X , ∀ k ∈ K, X ∈ k . Depending on this, we can derive several interesting facts.
Corollary 6.7.
For all ( X, l X ) ∈ Der ( g ) , Θ ∈ Der − ( g ) , A ∈ Aut ( g ) and τ ∈ Aut − ( g ) , we have (i) A ⋄ e ( X,l X ) ⋄ A − = e ( A ◦ X ◦ A − ,l A ◦ X ◦ A − ) , where l A ◦ X ◦ A − is given by (37) ; (ii) τ ⋆ e Θ ⋆ τ − = e ( I + τ ◦ d ) ◦ Θ ◦ ( I + d ◦ τ ) − ;(iii) A ⊲ e Θ = e A ◦ Θ ◦ A − ;(iv) τ ⋆ ( e ( X,l X ) ⊲ τ − ) = e X ◦ τ − + τ ◦ X + τ ◦ d ◦ X ◦ τ − . Corollary 6.8.
For all x ∈ g , Θ ∈ Der − ( g ) and A ∈ Aut ( g ) , we have A ⋄ e ¯ d (Θ) ⋄ A − = e ¯ d ( A ◦ Θ ◦ A − ) , (38) A ⋄ e ad ( x ) ⋄ A − = e ad ( A x )+ ¯ d ( A ( x,A − · )) . (39) Proof.
Note that ¯ d (Θ) = ( δ (Θ) , l δ (Θ) ). Firstly, it is simple to check that A ◦ δ (Θ) ◦ A − = δ ( A ◦ Θ ◦ A − ) . Then, by (37) and the fact that
A, A − are Lie 2-algebra homomorphisms, we find that l δ ( A ◦ Θ ◦ A − ) ( y, z )= A Θ A − [ y, z ] − [ A Θ A − y, z ] − [ y, A Θ A − z ]= A Θ[ A − y, A − z ] − A Θ dA − A ( A − y, A − z ) − A [Θ A − y, A − z ] + A ( d Θ A − y, A − z ) − A [ A − y, Θ A − z ] + A ( A − y, d Θ A − z )= A l δ (Θ) ( A − y, A − z ) − A Θ dA − A ( A − y, A − z ) + A ( d Θ A − y, A − z ) + A ( A − y, d Θ A − z )= l A ◦ δ (Θ) ◦ A − ( y, z ) . ad ( x ) = (ad ( x ) , l ( x, · , · )) . Since A is a Lie 2-algebra homomorphism, it is simpleto verify A ◦ ad ( x ) ◦ A − = ad ( A x ) + δ ( A ( x, A − · )) . Set Λ := A ( x, A − · ) and ad x := ad ( x ) for simplicity. Using again the fact that A and A − areLie 2-algebra homomorphisms, we have( l ( A x, · , · ) + l δ (Λ) )( y, z ) = l ( A x, y, z ) + A ( x, [ A − y, A − z ] − dA − A ( A − y, A − z )) − [ A ( x, A − y ) , z ] − [ y, A ( x, A − z )]= A ( A − y, ad x A − z ) + A (ad x A − y, A − z ) − ad A x A ( A − y, A − z ) − A ( x, dA − A ( A − y, A − z )) + A l ( x, A − y, A − z )= − (ad A x + Λ d ) A ( A − y, A − z ) + A l ( x, A − y, A − z )+ A ( A − y, ad x A − z ) + A (ad x A − y, A − z )= l A ◦ ad ( x ) ◦ A − ( y, z ) , where the second equality follows from the condition (iii) of Definition 2.2. By (i) of Corollary 6.7,we obtain (39).Denote by Inn ( g ) and Inn − ( g ) the connected Lie groups e inn ( g ) and e inn − ( g ) respectively.Then we get a natural sub-crossed module of Aut( g ):Inn( g ) : Inn − ( g ) ∂ −−−−→ Inn ( g ) . Theorem 6.9.
With the notations above,
Inn( g ) is a normal sub-crossed module of Aut( g ) . Since Inn( g ) is the image of the inner derivation Lie 2-algebra under the exponential map, wecall it the inner automorphism -group of g .For a strict Lie 2-algebra g = ( g ⊕ g − , d, [ · , · ]), consider the sub-complex Sinn( g ) of SDer( g )given by Sinn ( g ) = Im(ad ) and Sinn − ( g ) = Im(ad ). It is straightforward to see that Sinn( g )is an ideal of SDer( g ). Denote by SInn ( g ) and SInn − ( g ) the connected Lie groups e Sinn ( g ) and e Sinn − ( g ) respectively. Then SInn( g ) is a normal sub-crossed module of SAut( g ), which isconsistent with the inner automorphism 2-group given in [13]. Remark 6.10.
In [15], an inner automorphism 3-group of a strict Lie 2-group is given from aslight different viewpoint. Their inner automorphism 3-group can remember the center, which isdifferent from ours.
References [1] J. C. Baez and A. S. Crans, Higher-dimensional algebra VI: Lie 2-algebras,
Theory Appl.Categ.
12 (2004), 492-538.[2] J. C. Baez and A. Lauda, Higher-dimensional algebra 5: 2-groups,
Theory Appl. Categ.
Comm. Math. Phys.
293 (3) (2010), 701-725.[4] J. C. Baez and C. L. Rogers, Categorified symplectic geometry and the string Lie 2-algebra.
Homology, Homotopy Appl.
12 (1) (2010), 221-236.215] S. Chen, Y. Sheng and Z. Zheng, Non-abelian extensions of Lie 2-algebras,
Sci. China Math.
55 (8) (2012), 1655-1668.[6] M. Crainic and R. L. Fernandes, Integrability of Lie brackets,
Ann. of Math. (2)
157 (2)(2003), 575-620.[7] M. Doubek and T. Lada, Homotopy derivations, arXiv:1409.1691.[8] E. Getzler. Lie theory for nilpotent L ∞ -algebras. Ann. of Math. (2)
170 (1) (2009), 271-301.[9] A. Henriques. Integrating L ∞ -algebras. Compos. Math.
144 (4) (2008), 1017-1045.[10] T. Lada and M. Markl, Strongly homotopy Lie algebras.
Comm. Alg.
23 (6) (1995), 2147-2161.[11] H. Lang and Z. Liu, Crossed modules for Lie 2-algebras, arXiv: 1402.7226. to appear in
Applied Categorical Structures.
DOI: 10.1007/s10485-015-9389-8.[12] D. Li-Bland and P. ˇSevera. Integration of exact Courant algebroids.
Electron. Res. Announc.Math.
Sci. 19 (2012), 58-76.[13] K. L. Norrie, Actions and automorphisms of crossed modules,
Bull. Soc. Math. France
11 (2)(1990), 129-146.[14] R. A. Mehta and X. Tang. From double Lie groupoids to local Lie 2-groupoids.
Bull. Braz.Math. Soc.
42 (4) (2011), 651-681.[15] D. Roberts and U. Schreiber, The inner automorphism 3-group of a strict 2-group.
J. Ho-motopy Relat. Struct.
J. Pure Appl. Alg.
38 (1985), 313-322.[17] Y. Sheng, C. Zhu, Integration of semidirect product Lie 2-algebras,
Inter. J. Geom. MethodsModern Phys.
Lett. Math. Phys.