Invariant Hopf 2-cocycles for affine algebraic groups
aa r X i v : . [ m a t h . QA ] O c t INVARIANT HOPF -COCYCLES FOR AFFINEALGEBRAIC GROUPS PAVEL ETINGOF AND SHLOMO GELAKI
Abstract.
We generalize the theory of the second invariant coho-mology group H ( G ) for finite groups G , developed in [Da2, Da3,GK], to the case of affine algebraic groups G , using the methods of[EG1, EG2, G1]. In particular, we show that for connected affinealgebraic groups G over an algebraically closed field of character-istic 0, the map Θ from [GK] is bijective (unlike for some finitegroups, as shown in [GK]). This allows us to compute H ( G ) inthis case, and in particular show that this group is commutative(while for finite groups it can be noncommutative, as shown in[GK]). Introduction
An interesting invariant of a tensor category C is the group of tensorstructures on the identity functor of C (i.e., the group of isomorphismclasses of tensor autoequivalences of C which act trivially on the un-derlying abelian category) up to an isomorphism [Da1, Da4, BC, GK,PSV, Sc]. This group is called the second invariant (or lazy) cohomol-ogy group of C and denoted by H ( C ). In particular, if C := Corep( H )is the category of finite dimensional comodules over a Hopf algebra H ,then H ( H ) := H ( C ) is the group of invariant Hopf 2-cocycles for H modulo the subgroup of coboundaries of invertible central elementsof H ∗ .In [Da2, GK] the authors study H ( C ) for C := Corep k ( O ( G )) =Rep k ( G ), where G is a finite group, and O ( G ) is the algebra of functionson G with values in an algebraically closed field k . Namely, they definethe set B ( G ) of pairs ( A, R ), where A is a normal commutative sub-group in G and R is a non-degenerate G -invariant class in H ( b A, k × ),and a map Θ : H ( G ) := H ( C ) → B ( G ). This allows one to com-pute H ( G ) in many examples, although not always, as in general themap Θ is neither surjective nor injective. For example, it was proved Date : October 12, 2017.
Key words and phrases. affine algebraic groups; tensor categories; second invari-ant cohomology group. in [Da2] that if the order of G is coprime to 6 then H ( G ) is the di-rect product of B ( G ) and the group Out cl ( G ) of class-preserving outerautomorphisms of G .The goal of this paper is to generalize the theory of [Da2, GK] to thecase of affine algebraic groups, using the methods of [EG1, EG2, G1].In particular, we show that for connected affine algebraic groups overa field k of characteristic zero, the map Θ is, in fact, bijective (so thesituation is simpler than for finite groups). This allows us to computethe second invariant cohomology group in this case.More specifically, we show in Theorem 5.1 that if k has character-istic 0 and U is a unipotent algebraic group with u := Lie( U ), then H ( U ) ∼ = ( ∧ u ) u .Furthermore, let G be any connected affine algebraic group over k of characteristic 0. Let G u be the unipotent radical of G , and let G r := G/G u . Let Z be the center of G , and let Z u , Z r be the unipotentand reductive parts of Z . Let g u , z u , z r be the Lie algebras of G u , Z u , Z r ,respectively. Our main result Theorem 7.8 is that one has H ( G ) ∼ = Hom( ∧ b Z r , k × ) × ( z r ⊗ z u ) × ( ∧ g u ) G , where b Z r is the character group of Z r . In particular, this group iscommutative (while for finite groups it can be noncommutative, asshown in [GK]).The organization of the paper is as follows. Section 2 is devotedto preliminaries. In particular, we recall the definition of the secondinvariant cohomology group of a Hopf algebra and its basic properties,and study the space ( ∧ g ) g for a Lie algebra g .In Section 3 we use [G1] to describe the structure of H ( G ), where G is a commutative affine algebraic group over k of characteristic 0 (seeTheorem 3.4).In Section 4 we use [G1] to study the support group A of an invariantHopf 2-cocycle J for the function Hopf algebra O ( G ) of an arbitraryaffine algebraic group over k . In particular, we show that A is a closednormal commutative subgroup of G (see Theorem 4.2).In Section 5 we extend the definitions of the set B ( G ) and mapΘ : H ( G ) → B ( G ) from [GK] to affine algebraic groups G over k ,and study the basic properties of Θ in Theorem 5.1.In Section 6 we focus on unipotent algebraic groups over k of char-acteristic 0, and prove Theorem 6.1.Finally, Section 7 is devoted to the proof of our main result Theorem7.8, which is a generalization of Theorem 6.1 to arbitrary connected affine algebraic groups G over k of characteristic 0. NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 3
Acknowledgements.
We are grateful to Vladimir Popov for com-municating to us Lemma 7.2. We thank A. Davydov and M. Yakimovfor suggesting the current formulation of Lemma 2.8, and A. Davydovfor references. The work of P.E. was partially supported by the NSFgrant DMS-1502244. S.G. thanks the University of Michigan and MITfor their hospitality. 2.
Preliminaries
Throughout the paper, unless otherwise specified, we shall work overan algebraically closed field k of arbitrary characteristic.2.1. Hopf -cocycles. Let H be a Hopf algebra over k with multipli-cation map m . An invertible element J ∈ ( H ⊗ H ) ∗ is called a (right) Hopf -cocycle for H if it satisfies the two conditions X J ( a b , c ) J ( a , b ) = X J ( a, b c ) J ( b , c ) , (1) J ( a,
1) = ε ( a ) = J (1 , a )for all a, b, c ∈ H (see, e.g., [Do]).We have a natural action of the group of invertible elements ( H ∗ ) × on Hopf 2-cocycles for H . Namely, if J is a Hopf 2-cocycle for H and x ∈ ( H ∗ ) × , then the linear map J x : H ⊗ H → k defined by J x ( a, b ) = X x ( a b ) J ( a , b ) x − ( a ) x − ( b ) , a, b ∈ H, is also a Hopf 2-cocycle for H . We say that two Hopf 2-cocycles for H are gauge equivalent if they belong to the same ( H ∗ ) × -orbit, andthat they are strongly gauge equivalent if they belong to the same orbitunder the action of the group of invertible central elements of H ∗ .Given a Hopf 2-cocycle J for H , one can construct a new Hopf algebra H J as follows. As a coalgebra H J = H , and the new multiplication m J is given by m J = J − ∗ m ∗ J , i.e.,(2) m J ( a ⊗ b ) = X J − ( a , b ) a b J ( a , b ) , a, b ∈ H. Equivalently, every Hopf 2-cocycle J for H defines a tensor structureon the forgetful functor Corep k ( H ) → Vec, where Corep k ( H ) is thetensor category of finite dimensional comodules over H . Recall that if J and J ′ are gauge equivalent then the Hopf algebras H J and H J ′ areisomorphic.Note that if K is a Hopf subalgebra of H and J is a Hopf 2-cocyclefor H then the restriction res( J ) of J to K defines a Hopf 2-cocycle for K . Also, if K is a Hopf algebra quotient of H and J is a Hopf 2-cocyclefor K then the lifting lif( J ) of J to H defines a Hopf 2-cocycle for H . PAVEL ETINGOF AND SHLOMO GELAKI
Invariant Hopf -cocycles. Let H be a Hopf algebra over k withmultiplication map m . A Hopf 2-cocycle J for H is called invariant (or lazy in, e.g., [GK, Section 1.4]) if m J = m (equivalently, if J ∗ m = m ∗ J ). In particular, when H is cocommutative, every Hopf 2-cocycleis invariant.Note that if K is a Hopf subalgebra of H and J is an invariant Hopf2-cocycle for H then res( J ) defines an invariant Hopf 2-cocycle for K .However, if K is a Hopf algebra quotient of H and J is an invariantHopf 2-cocycle for K then lif( J ) is not necessarily an invariant Hopf2-cocycle for H .It is clear that invariant Hopf 2-cocycles for H form a group undermultiplication, which is denoted by Z ( H ) (= Z ℓ ( H ) in [GK]). Infact, we make the following observation. Lemma 2.1.
Let J be an invariant Hopf -cocycle for H , and let F be a (not necessarily invariant) Hopf -cocycle for H . Then J ∗ F is aHopf -cocycle for H .Proof. By Eq. (1) for J and F , and the invariance of J , we have X ( J ∗ F )( a b , c )( J ∗ F )( a , b )= X J ( a b , c ) F ( a b , c ) J ( a , b ) F ( a , b )= X ( J ( a b , c ) J ( a , b )) ( F ( a b , c ) F ( a , b ))= X ( J ( a , b c ) J ( b , c )) ( F ( a , b c ) F ( b , c ))= X J ( a , b c ) F ( a , b c ) J ( b , c ) F ( b , c )= X ( J ∗ F )( a, b c )( J ∗ F )( b , c )for all a, b, c ∈ H , as desired. (cid:3) Furthermore, the group Z ( H ) contains a central subgroup B ( H )(= B ℓ ( H ) in [GK]) consisting of all the invariant Hopf 2-cocycles ∂ ( x ),where x ∈ ( H ∗ ) × is central in H ∗ and ∂ ( x )( a, b ) = X x ( a b ) x − ( a ) x − ( b ) , a, b ∈ H. Following [GK], we define the quotient group(3) H ( H ) := Z ( H ) /B ( H )(= H ℓ ( H ) in [GK]). For J ∈ Z ( H ), let [ J ] denote its class in H ( H ).By definition, [ J ] = [ ¯ J ] if and only if J and ¯ J are strongly gaugeequivalent (see Subsection 2.1). NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 5
It is straightforward to verify that a Hopf 2-cocycle J for H is invari-ant if and only if it defines a tensor structure on the identity functor on C := Corep k ( H ). Hence, H ( H ) is isomorphic to the group Aut ( C )of isomorphism classes of tensor structures on the identity functor of C , i.e., of tensor autoequivalences of C which act trivially on the under-lying abelian category. The second invariant cohomology Aut ( C ) foran arbitrary tensor category C was introduced first by Davydov [Da1]and studied also in [PSV]. Proposition 2.2. [DEN, Corollary 3.16] H inv ( H ) is an affine proal-gebraic group, and if H is finitely presented, it is an affine algebraicgroup. (cid:3) Example 2.3.
As pointed out in [Da1, Section 8] (see also [Sc]), when H is cocommutative , the group H ( H ) coincides with Sweedler’s sec-ond cohomology group H ( H, k ) of H with coefficients in the algebra k [Sw]. In particular, if G is a group then H ( k [ G ]) ∼ = H ( G, k × ) [Sw,Theorem 3.1], and if g is a Lie algebra then H ( U ( g )) ∼ = H ( g , k ) [Sw,Theorem 4.3].Recall that a Hopf algebra automorphism of H of the formAd( x ) := x ∗ id ∗ x − for some x ∈ ( H ∗ ) × is called cointernal (see, e.g., [BC]). For example,if x : H → k is an algebra homomorphism (i.e., x ∈ H ∗ fin is a grouplikeelement) then Ad( x ) is cointernal and it is called coinner . The set of allcointernal Hopf algebra automorphisms of H is denoted by CoInt( H ),and the set of all coinner Hopf algebra automorphisms of H is denotedby CoInn( H ). It is easy to see that CoInn( H ) ⊆ CoInt( H ) are normalsubgroups of Aut Hopf ( H ) (see, e.g., [BC, Lemma 1.12]).The following lemma is dual to [GK, Proposition 1.7(a)]. Lemma 2.4.
Let J ∈ Z inv ( H ) and let x ∈ ( H ∗ ) × . Then J x ∈ Z inv ( H ) if and only if Ad ( x ) ∈ CoInt ( H ) .Proof. By definition, J x ∈ Z ( H ) if and only if J x ∗ m = m ∗ J x , i.e.,if and only if( x ◦ m ) ∗ J ∗ ( x − ⊗ x − ) ∗ m = m ∗ ( x ◦ m ) ∗ J ∗ ( x − ⊗ x − ) . Hence, since J is invariant, it follows that J x ∈ Z ( H ) if and only if( x ◦ m ) ∗ ( x − ⊗ x − ) ∗ m = m ∗ ( x ◦ m ) ∗ ( x − ⊗ x − ) , which is equivalent to( x − ⊗ x − ) ∗ m ∗ ( x ⊗ x ) = ( x − ◦ m ) ∗ m ∗ ( x ◦ m ) . PAVEL ETINGOF AND SHLOMO GELAKI
But by definition, the latter is equivalent to Ad( x ) being a Hopf auto-morphism of H . We are done. (cid:3) Following [GK, Proposition 1.7], we have the following result.
Proposition 2.5.
The quotient group CoInt ( H ) / CoInn ( H ) acts freelyon H inv ( H ) , and the associated map (4) CoInt ( H ) / CoInn ( H ) → H inv ( H ) , Ad ( x ) ( ε ⊗ ε ) x , defines an embedding of CoInt ( H ) / CoInn ( H ) as a subgroup of H inv ( H ) .Proof. By Lemma 2.4, we have an action of the group CoInt( H ) on H ( H ), given by Ad( x ) · [ J ] = [ J x ] for every Ad( x ) ∈ CoInt( H ) and[ J ] ∈ H ( H ). This is a well-defined action since if Ad( x ) = Ad( y )then x ∗ y − is central in H ∗ , and hence [ J x ] = [ J y ].Now every invariant Hopf 2-cocycle J for H is fixed by every coinnerHopf automorphism Ad( x ) with x ∈ G ( H ∗ ), since J x = ( x ◦ m ) ∗ J ∗ ( x − ⊗ x − ) = J ∗ ( x ◦ m ) ∗ ( x − ⊗ x − ) = J. Hence, we get an action of the quotient group CoInt( H ) / CoInn( H ) on H ( H ).Fix [ J ] ∈ H ( H ), and suppose Ad( x ) ∈ CoInt( H ) is such thatAd( x ) · [ J ] = [ J ]. Then [ J ] = [ J x ], hence J and J x are strongly gaugeequivalent, i.e., J = ( z ◦ m ) ∗ ( x ◦ m ) ∗ J ∗ ( x − ⊗ x − ) ∗ ( z − ⊗ z − )for some central invertible element z in H ∗ . Since J is invariant,( z ◦ m ) ∗ ( x ◦ m ) ∗ J = J ∗ ( z ◦ m ) ∗ ( x ◦ m ) , hence we have ε ⊗ ε = (( z ∗ x ) ◦ m ) ∗ (( z ∗ x ) − ⊗ ( z ∗ x ) − ). In other words,we have z ∗ x ∈ G ( H ∗ ). But z is central, hence Ad( x ) = Ad( z ∗ x ) isin Inn( H ), which implies the freeness of the action.Finally, it follows from the above that the map (4) is injective, andsince ( ε ⊗ ε ) x is invariant for every x ∈ ( H ∗ ) × such that Ad( x ) ∈ CoInt( H ), it is straightforward to verify that ( ε ⊗ ε ) xy = ( ε ⊗ ε ) x ∗ ( ε ⊗ ε ) y for every x, y ∈ ( H ∗ ) × such that Ad( x ) , Ad( y ) ∈ CoInt( H ), hence (4)is a group homomorphism. (cid:3) Remark 2.6.
Let C := Corep k ( H ). It is straightforward to verify thatCoInt( H ) / CoInn( H ) is the stabilizer of the standard fiber functor of C in Aut ( C ). NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 7
Cotriangular Hopf algebras.
Recall (see, e.g., [EGNO, Defini-tion 8.3.19]) that (
H, R ) is a cotriangular
Hopf algebra if R ∈ ( H ⊗ H ) ∗ is an invertible element such that • P R ( h , g ) R ( g , h ) = ε ( g ) ε ( h ) (i.e., R − = R ), • R ( h, gl ) = P R ( h , g ) R ( h , l ), • R ( hg, l ) = P R ( g, l ) R ( h, l ), and • P R ( h , g ) g h = P h g R ( h , g )for every h, g, l ∈ H .Recall that the Drinfeld element of (
H, R ) is the grouplike element u in H ∗ , given by u ( h ) = P R ( S ( h ) , h ) for every h ∈ H .Given a Hopf 2-cocycle J for H , ( H J , R J ) is also cotriangular, where R J := J − ∗ R ∗ J . Recall that if J and ¯ J := J x are gauge equivalentthen Ad( x ) : ( H J , R J ) → ( H ¯ J , R ¯ J ) is an isomorphism of cotriangularHopf algebras. Proposition 2.7.
Let ( H, R ) be a cotriangular Hopf algebra, and let I := { a ∈ H | R ( b, a ) = 0 , b ∈ H } be the right radical of R . Then I coincides with the left radical { b ∈ H | R ( b, a ) = 0 , a ∈ H } of R , andis a Hopf ideal of H .Proof. See [G1, Proposition 2.1]. (cid:3)
A cotriangular Hopf algebra (
H, R ) such that R is non-degenerate(i.e., I = 0) is called minimal . By Proposition 2.7, any cotriangularHopf algebra ( H, R ) has a unique minimal cotriangular Hopf algebraquotient
H/I , which we will denote by ( H min , R ).Recall that the finite dual Hopf algebra H ∗ fin of H consists of all theelements in H ∗ that vanish on some finite codimensional ideal of H .Note that in the minimal case, the non-degenerate form R defines twoinjective Hopf algebra maps R + , R − : H ֒ → H ∗ fin , given by R + ( h )( a ) = R ( h, a ) and R − ( h )( a ) = R ( S ( a ) , h )for every a, h ∈ H .2.4. Invariant solutions to the classical Yang-Baxter equation.
Recall that a Lie algebra h over k is called quasi-Frobenius with symplec-tic form ω if ω ∈ ∧ h ∗ is a non-degenerate ω : h × h → k is a non-degenerate skew-symmetric bilinear form satisfying(5) ω ([ x, y ] , z ) + ω ([ z, x ] , y ) + ω ([ y, z ] , x ) = 0for every x, y, z ∈ h .Let g be a Lie algebra over k . Recall that an element r ∈ ∧ g is asolution to the classical Yang-Baxter equation if(6) CYB( r ) := [ r , r ] + [ r , r ] + [ r , r ] = 0 . PAVEL ETINGOF AND SHLOMO GELAKI
By Drinfeld [Dr], solutions r to the classical Yang-Baxter equation in ∧ g are classified by pairs ( h , ω ), via r = ω − ∈ ∧ h , where h ⊆ g is aquasi-Frobenius Lie subalgebra with symplectic form ω . We will call h the support of r .The following result appear in [Da3, Lemma 5.5.2]. Lemma 2.8.
Assume k has characteristic = 2 . Then the componentsof any element r ∈ ( ∧ g ) g commute. Hence, any element in ( ∧ g ) g isa solution to the classical Yang-Baxter equation.Proof. Let h be the span of the components of r . Since r is g -invariant, h is a Lie ideal of g . Moreover, h carries a g -invariant symplecticform ω = r − . Thus, ω ([ x, y ] , z ) = − ω ( z, [ x, y ]) = − ω ([ z, x ] , y ) forevery x, y, z ∈ h . I.e., ω ([ x, y ] , z ) is anti-invariant under the cyclicpermutation of x, y, z . Since this permutation has order 3, we get ω ([ x, y ] , z ) = 0 for every x, y, z ∈ h . But since ω is non-degenerate,[ x, y ] = 0 for every x, y ∈ h , as desired. (cid:3) Proposition 2.9.
Assume k has characteristic = 2 , . If r, s ∈ ( ∧ g ) g then [ r, s ] = ∆( z ) − z ⊗ − ⊗ z for some central element z ∈ U ( g ) of degree . Moreover, we have [ r, [ r, s ]] = [ s, [ r, s ]] = 0 .Proof. Let a and b be the supports of r and s , respectively. By Lemma2.8, a and b are abelian Lie ideals in g . Therefore, a + b is a Liesubalgebra (in fact, a Lie ideal) of g , which is nilpotent of index ≤ h := a ∩ b , and let a and b be some complements of h in a and b , respectively (as vector spaces). Then a + b = a ⊕ b ⊕ h . Since a , b are abelian Lie ideals in a + b , we have that h is central in a + b ,and the only nontrivial component of the bracket is [ , ] : a × b → h .We have r = r ′ + r ′′ , where r ′ ∈ ( a ⊗ h ) ⊕ ∧ h and r ′′ ∈ ∧ a . Let r ′′ = P i x i ⊗ x ′ i be a shortest presentation of r ′′ in ∧ a . Let b ∈ b .Since [∆( b ) , r ] = 0, we have P i ([ b, x i ] ⊗ x ′ i + x i ⊗ [ b, x ′ i ]) = 0. Hence,[ b, x i ] = [ b, x ′ i ] = 0 for all i . Thus, x i , x ′ i are central in a + b . So[ r ′′ , s ] = 0, and it suffices to show that [ r ′ , s ] = ∆( z ) − z ⊗ − ⊗ z forsome central element z ∈ U ( g ) of degree 3.Similarly, we have a decomposition s = s ′ + s ′′ , where the componentsof s ′′ are central in a + b . Therefore, [ r ′ , s ′′ ] = 0, and it suffices to showthat [ r ′ , s ′ ] = ∆( z ) − z ⊗ − ⊗ z .Let { h i } be a basis of h . Then r ′ = P i a i ∧ h i and s ′ = P i b i ∧ h i ,modulo ∧ h , where a i ∈ a and b i ∈ b . Thus,[ r ′ , s ′ ] = X i,j ([ a i , b j ] ⊗ h i h j + h i h j ⊗ [ a i , b j ]) . NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 9
Now write [ a i , b j ] = P k c ijk h k . Since r ′ is ( a + b )-invariant, it followsthat P i [ a i , b j ] ∧ h i = 0 for every j , i.e., P i,k c ijk h k ∧ h i = 0 for every j .Thus, c ijk is symmetric in i, k . Similarly, since s ′ is ( a + b )-invariant, c ijk is symmetric in j, k . Thus c ijk is symmetric in all three indices.Finally, we have[ r, s ] = [ r ′ , s ′ ] = X i,j,k c ijk ( h k ⊗ h i h j + h i h j ⊗ h k ) . Thus we may take z := P i,j,k c ijk h i h j h k = mult([ r, s ]), where multdenotes multiplication of components, and obtain that[ r, s ] = [ r ′ , s ′ ] = ∆( z ) − z ⊗ − ⊗ z, as desired. Note that z is central in U ( g ), since r and s are g -invariant,and [ r, [ r, s ]] = [ s, [ r, s ]] = 0 since [ r, h i ] = [ s, h i ] = 0 for every i . (cid:3) Example 2.10.
Let g be the Heisenberg Lie algebra with basis a, b, c such that [ a, b ] = c , [ a, c ] = [ b, c ] = 0, r := a ∧ c and s := b ∧ c . Then[ r, s ] = c ⊗ c + c ⊗ c , so we may take z := c /
3. (See [Da3, Example5.5.9] for more details.)Finally, we let B ( g ) denote the set of pairs ( h , ω ) such that h ⊆ g is anabelian Lie ideal with a g -invariant symplectic form ω . As mentionedabove, we have a bijection B ( g ) ∼ = −→ ( ∧ g ) g , given by ( h , ω ) ω − .3. Hopf -cocycles for commutative affine algebraicgroups In this section we will assume that k has characteristic 0.3.1. Invariant Hopf -cocycles for affine algebraic groups. Let G be a (possibly disconnected) affine algebraic group over k , and let G u be the unipotent radical of G . Recall that we have a split projection G → G/G u , i.e., G = G r ⋉ G u is a semidirect product for a (unique, upto conjugation) closed subgroup G r ∼ = G/G u (a “Levi subfactor”) suchthat G r is reductive.Equivalently, we have a split exact sequence of Hopf algebras O ( G r ) ι −→ O ( G ) p −→ O ( G u )with the Hopf algebra section O ( G ) q −→ O ( G r ) corresponding to theinclusion G r ֒ → G .Set Z ( G ) := Z ( O ( G )) and H ( G ) := H ( O ( G )) . Let us say that J is a Hopf 2-cocycle for G if J is a Hopf 2-cocyclefor O ( G ), and that J is minimal if the cotriangular Hopf algebra( O ( G ) J , J − ∗ J ) is minimal (see Subsection 2.3).For J ∈ Z ( G ), we will call J r := lif(res( J )) (with respect to ι and q ; see Subsection 2.1) the reductive part of J . It is clear that J r is aHopf 2-cocycle for G . Lemma 3.1.
Let J ∈ Z inv ( G ) be an invariant Hopf -cocycle for G .Then J − is a Hopf -cocycle for G .Proof. Since J is invariant, J − is also a Hopf 2-cocycle for G . Hence, J − = lif(res( J − )) is a Hopf 2-cocycle for G too. (cid:3) By Lemmas 2.1 and 3.1, if J ∈ Z ( G ) then J u := J ∗ J − is a Hopf2-cocycle for G . Clearly the restriction of J u to O ( G r ) is trivial (i.e., itdefines a fiber functor on Rep( G ), which coincides with the standardone on the semisimple tensor subcategory Rep( G/G u )). We will call J u the unipotent part of J . We thus obtain Proposition 3.2.
For every invariant Hopf -cocycle J for G , we have J = J u ∗ J r . (cid:3) Hopf -cocycles for commutative affine algebraic groups. Let A be a commutative affine algebraic group over k . Recall that A ∼ = A r × A u is a direct product, and A r ∼ = T × A f is a direct productof an algebraic torus T ∼ = G nm and a finite abelian group A f , and A u ∼ = G ma is an additive group. The diagonalizable closed subgroup A r of A is the reductive part of A , and the closed subgroup A u is theunipotent radical of A .Let b A := Hom( A, k × ) denote the character group of A . Recall that b T ∼ = Z n . We have O ( A r ) ∼ = k [ c A r ], the group algebra of c A r ∼ = b T × c A f with its standard Hopf algebra structure, and b A = c A r .Recall also that O ( A u ) is a polynomial algebra on m variables, withits standard Hopf algebra structure, and we have O ( A ) ∼ = O ( A r ) ⊗ O ( A u ) ∼ = k [ c A r ] ⊗ O ( A u ) , as Hopf algebras. Corollary 3.3.
Let A ∼ = A r × A u be a commutative affine algebraicgroup as above, and let a , a r and a u be the Lie algebras of A , A r and A u , respectively. Let J be a Hopf -cocycle for A . Then J r and J u are (invariant) Hopf -cocycles for A , and J = J r ∗ J u = J u ∗ J r .Furthermore, we have [ J u ] = [exp( r/ for a unique element r in a ∧ a u = a r ⊗ a u ⊕ ∧ a u . In particular, the Drinfeld element of thecotriangular Hopf algebra ( O ( A ) , J − ∗ J ) is trivial. NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 11
Proof.
The first assertion follows from Proposition 3.2. The secondassertion is a special case of [EG2, Theorem 3.2], but here is a directproof. Consider ρ := 2 log( J u ) (it is well defined since J u is unipotent).Then ρ is a Hochschild 2-cocycle of O ( A ) with trivial coefficients. Sinceby the Hochschild-Kostant-Rosenberg theorem, HH i ( O ( A ) , k ) ∼ = ∧ i a ,this isomorphism maps the class of ρ to an element r ∈ ∧ a , which ismoreover in a ∧ a u since ρ is nilpotent. The claim follows. (cid:3) Theorem 3.4.
Let A ∼ = A r × A u be a commutative affine algebraicgroup as above, and let a and a u be the Lie algebras of A and A u ,respectively. Then the following hold: (1) We have group isomorphisms H inv ( A r ) ∼ = H ( c A r , k × ) ∼ = Hom ( ∧ c A r , k × ) . Furthermore, minimal Hopf -cocycles for A r correspond underthis isomorphism to non-degenerate alternating bicharacters on c A r . (2) We have a vector space isomorphism ∧ a u ∼ = −→ H inv ( A u ) , r [exp( r/ . Furthermore, minimal Hopf -cocycles for A u correspond underthis isomorphism to non-degenerate elements of ∧ a u . In par-ticular, O ( A u ) has a minimal Hopf -cocycle if and only if m iseven. (3) We have group isomorphisms H inv ( A ) ∼ = H ( c A r , k × ) × ( a ∧ a u ) ∼ = Hom ( ∧ c A r , k × ) × ( a r ⊗ a u ⊕ ∧ a u ) . Proof. (1) and (2) follow from the definitions in a straightforward man-ner (see [G1, Theorem 5.3 & Proposition 5.4]).(3) follows from (1) and Corollary 3.3. (cid:3)
Remark 3.5.
The group H ( A ) plays the role of the group of skew-symmetric bicharacters of A when A u is non-trivial. Remark 3.6.
The group isomorphism H ( c A r , k × ) ∼ = Hom( ∧ c A r , k × )is given by J R J := J − J . Remark 3.7.
The dimension of an algebraic torus having a minimalHopf 2-cocycle need not be even (see [G1, Remark 4.2]).Moreover, if A ∼ = T × A f has a minimal Hopf 2-cocycle then theorder of A f does not have to be a perfect square (contrary to the caseof finite groups). For example, let A = G m × Z /n Z , let q ∈ k × be a non-root of unity, ζ ∈ k a primitive n -th root of unity, and let R bethe bicharacter of b A given by R (( x, y, a ) , ( x ′ , y ′ , a ′ )) = q xy ′ − yx ′ ζ xa ′ − ax ′ .Then R is non-degenerate, hence defines a minimal cotriangular struc-ture on O ( A ). It corresponds to a minimal Hopf 2-cocycle J for A given by J (( x, y, a ) , ( x ′ , y ′ , a ′ )) = q xy ′ ζ xa ′ .This raises a question which finite groups can arise as componentgroups H/H of affine algebraic groups H such that O ( H ) has a min-imal Hopf 2-cocycle. Clearly, this can be the product of any group Γof central type with any finite abelian group, by taking H to be theproduct of Γ with a number of copies of the example above for variousvalues of n . Example 3.8.
Let A r := G m and A u := G a , with Lie algebra bases x and y , respectively. Then r := x ∧ y defines a minimal Hopf 2-cocycleexp( r/
2) for A r × A u . Example 3.9.
Let A r := G m and A u := G a , with Lie algebra bases { x , x } and y . Then for every irrational a , r a := ( x + ax ) ∧ y definesa minimal Hopf 2-cocycle exp( r a /
2) for A r × A u , with a 3-dimensionalsupport. (See [G1, Example 4.13].) Remark 3.10.
Since a commutative affine algebraic group A over k has only 1-dimensional irreducible representations, all fiber functorson Rep k ( A ) = Corep k ( O ( A )) preserve dimensions, so all of them arisefrom Hopf 2-cocycles for A .Let us now consider commutative affine algebraic groups over a field k of characteristic p > Proposition 3.11. (1) If A is a finite p -group then H inv ( A ) is thetrivial group. (2) If A is a vector group, we have a vector space isomorphism H inv ( A ) ∼ = ∧ a , where a is the Lie algebra of A .Proof. (1) Follows from [G2, Corollary 6.9].(2) Since in this case O ( A ) ∼ = U ( a ∗ ), we have H ( A ) = H ( U ( a ∗ )) = H ( a ∗ , k ) = ∧ a (see Example 2.3). (cid:3) The support of an invariant Hopf -cocycle for anaffine algebraic group Let G be an affine algebraic group over k and let O ( G ) be its func-tion algebra. Then ( O ( G ) , ε ⊗ ε ) is a cotriangular commutative Hopfalgebra. NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 13
Theorem 4.1.
Let J be a Hopf -cocycle for G . Then there exist aclosed subgroup H of G (called the support of J ), determined uniquelyup to conjugation, and a minimal Hopf -cocycle ¯ J for H such that J is gauge equivalent to ¯ J (viewed as a Hopf -cocycle for G ).Proof. For the existence of H and ¯ J , see [G1, Theorem 3.1].Now let H, ¯ H ⊆ G be two closed normal subgroups of G , and let J ∈ ( O ( H ) ⊗O ( H )) ∗ and L ∈ ( O ( ¯ H ) ⊗O ( ¯ H )) ∗ be two minimal Hopf 2-cocycles for H and ¯ H , respectively. Suppose J, L are gauge equivalentas Hopf 2-cocycles for G . Then our job is to show that H, ¯ H areconjugate in G .By definition, L = J x = ( x ◦ m ) ∗ J ∗ ( x − ⊗ x − ) for some element x ∈ ( O ( G ) ∗ ) × . Hence, we have R L = ( x ⊗ x ) ∗ R J ∗ ( x − ⊗ x − ). Thenby the minimality of R L and R J , we have(7) ( x ⊗ x ) ∗ ( O ( H ) ⊗ O ( H )) ∗ ∗ ( x − ⊗ x − ) = ( O ( ¯ H ) ⊗ O ( ¯ H )) ∗ inside the algebra ( O ( G ) ⊗ O ( G )) ∗ . It follows that( x ◦ m ) ∗ ( x − ⊗ x − ) = L ∗ ( x ⊗ x ) ∗ J − ∗ ( x − ⊗ x − ) ∈ ( O ( ¯ H ) ⊗ O ( ¯ H )) ∗ is a symmetric Hopf 2-cocycle for ¯ H . Therefore,( x ◦ m ) ∗ ( x − ⊗ x − ) = ( y ◦ m ) ∗ ( y − ⊗ y − )for some y ∈ O (( ¯ H ) ∗ ) × . Equivalently, x ∗ y − ∈ O ( G ) ∗ is a grouplikeelement, so x ∗ y − = g for some g ∈ G . Hence, we have( g ⊗ g ) ∗ ( O ( H ) ⊗ O ( H )) ∗ ∗ ( g − ⊗ g − ) = ( O ( ¯ H ) ⊗ O ( ¯ H )) ∗ inside ( O ( G ) ⊗ O ( G )) ∗ , which is equivalent to gHg − = ¯ H . (cid:3) We will denote the conjugacy class of H by Supp( J ), and sometimeswrite H = Supp( J ) by abuse of notation.The following result was proved for finite groups in [Da2, Theorem2.4 ] (see also [GK, Lemma 4.4(a)]). Theorem 4.2.
Let J ∈ Z inv ( G ) be an invariant Hopf -cocycle withsupport A := Supp ( J ) . Then A is a closed normal commutative sub-group of G . Note that since A is normal, it is well defined as a closed subgroupof G , not just up to conjugation. Proof.
Since J is invariant, it follows that ( O ( G ) , J − ∗ J ) is a com-mutative cotriangular Hopf algebra and ( O ( A ) , J − ∗ J ) is a minimal commutative cotriangular Hopf algebra. Hence, O ( A ) is isomorphicto a Hopf subalgebra of its finite dual Hopf algebra O ( A ) ∗ fin (see Sub-section 2.3). Since O ( A ) ∗ fin is cocommutative , it follows that O ( A ) iscocommutative, which is equivalent to A being commutative. Take arbitrary g ∈ G . By Lemma 2.4, J g is also invariant, thus( O ( G ) , ( J g ) − ∗ J g ) is a commutative cotriangular Hopf algebra too.Now since J is invariant, we have( J g ) − ∗ J g = (cid:0) ( g ⊗ g ) ∗ J − ∗ ( g − ◦ m ) (cid:1) ∗ (cid:0) ( g ◦ m ) ∗ J ∗ ( g − ⊗ g − ) (cid:1) = ( g ⊗ g ) ∗ J − ∗ J ∗ ( g − ⊗ g − )= J − ∗ J, which implies that gAg − = A , as desired. (cid:3) Corollary 4.3.
The connected component of the identity of the supportof an invariant Hopf -cocycle J for a reductive algebraic group G over k is a torus. In particular, if G is semisimple then the support of J isa finite group. (cid:3) The set B ( G )Let G be an affine algebraic group over k . Following [Da2] (seealso [GK]), we let B ( G ) denote the set of pairs ( A, R ) such that A isa commutative closed normal subgroup of G , ( O ( A ) , R ) is a minimalcotriangular Hopf algebra with trivial Drinfeld element, and R is G -invariant (see Subsection 2.3). We set e := ( { } , ε ⊗ ε ).By Theorem 4.2, every invariant Hopf 2-cocycle J for G gives rise toan element (Supp( J ) , R J ) ∈ B ( G ), where R J := J − ∗ J .Set Int( G ) := CoInt( O ( G )), and observe that since G is the group ofgrouplike elements of O ( G ) ∗ , we have CoInn( O ( G )) = Inn( G ). Recallthat by Proposition 2.5, Int( G ) / Inn( G ) is a subgroup of H ( G ).Following [Da2, Section 6] (see also [GK, Lemma 4.4(b) & Theorem4.5]), we have the following result. Theorem 5.1.
The following hold: (1)
The assignment
Θ : H inv ( G ) → B ( G ) , [ J ] ( Supp ( J ) , R J ) , is a well-defined map of sets. (2) The fibers of Θ are the left cosets of Int ( G ) / Inn ( G ) inside H inv ( G ) .In particular, Θ is injective if and only if Int ( G ) = Inn ( G ) . (3) Θ − ( e ) ∼ = Int ( G ) / Inn ( G ) as groups.Proof. (1) If [ J ] = [ ¯ J ] then by Theorem 4.1 and Theorem 4.2, we haveSupp( J ) = Supp( ¯ J ) and R J = R ¯ J . Thus, Θ([ J ]) is independent of thechoice of a representative from [ J ].(2) Suppose Θ([ J ]) = Θ([ ¯ J ]). Then J, ¯ J are gauge equivalent as Hopf2-cocycles for G , supported on the same commutative closed normal NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 15 subgroup A of G , and R J = R ¯ J . Since J, ¯ J are invariant, the equal-ity between the R -matrices implies that ¯ J J − is a symmetric Hopf2-cocycle for G , hence ¯ J = J x for some element x ∈ O ( G ) ∗ such thatAd( x ) ∈ Int( G ), which defines a unique element in Int( G ) / Inn( G ).(3) Follows from (2). (cid:3) Corollary 5.2. If G is a commutative affine algebraic group then Θ isbijective and therefore induces a group structure on B ( G ) .Proof. Follows from Theorems 3.4, 5.1 and the fact Int( G ) = Inn( G ) istrivial. (cid:3) Corollary 5.2 implies that for a fixed closed normal commutativesubgroup A of G , the set of elements R such that ( A, R ) ∈ B ( G ) is inbijection with the set of non-degenerate elements in H ( A ) G , where H ( A ) is described in Theorem 3.4.6. Unipotent algebraic groups
In this section we will assume that k has characteristic 0. Theorem 6.1. If U is a unipotent algebraic group over k then Θ isbijective. Moreover, H inv ( U ) ∼ = ( ∧ u ) u as affine algebraic groups, where u := Lie ( U ) , so in particular H inv ( U ) is commutative.Proof. We first show that Θ is surjective. Let (
A, R ) ∈ B ( U ). ByTheorem 3.4, we have R = exp( r ) for some r ∈ ( ∧ u ) u (as A is com-mutative unipotent, i.e., a vector group). Let J := exp( r/ J is invariant since R is, and we have Θ([ J ]) = ( A, R ).Next we show that Θ is injective. By Theorem 5.1, we have to showthat Int( U ) = Inn( U ).Let m be the augmentation ideal of O ( U ) ∗ . Then O ( U ) ∗ is a completelocal ring with unique maximal ideal m . Thus, if α is a cointernalautomorphism of U then α is given by conjugation by an element a in 1 + m . Hence the element log( a ) defines a derivation of u , given by d ( x ) = [log( a ) , x ], and it suffices to show that this derivation is inner: d ( x ) = [ u, x ] for some u ∈ u . Then we may take a = exp( u ) ∈ U , andget that α is inner.The derivation d defines a class [ d ] ∈ H ( u , u ), and its image in H ( u , O ( U ) ∗ ) is zero (as d ( x ) = [log( a ) , x ], where log( a ) ∈ O ( U ) ∗ ).Thus, it suffices to show that u is a direct summand in O ( U ) ∗ as a u -module under the adjoint action. In other words, we need to show thatthe surjective morphism b : O ( U ) → u ∗ , given by β dβ (1), splits as amorphism of u -modules, i.e., there exists a u -morphism φ : u ∗ → O ( U )such that b ◦ φ = id. The morphism φ can be viewed as an element of O ( U ) ⊗ u , i.e., as a regular function φ : U → u . Thus it sufficesto show that there exists a regular u -invariant function φ : U → u such that dφ (1) = id. But in the unipotent case we can simply take φ ( X ) = log( X ). The proof that Θ is injective is complete.Finally, let us show that the bijectionΘ − ◦ exp : ( ∧ u ) u → H ( U ) , r [ J r ] , where J r := exp( r/ , is a group homomorphism. By Proposition 2.9, [ r, [ r, s ]] = [ s, [ r, s ]] = 0for any r, s ∈ ( ∧ u ) u . So we have J r ∗ J s = J r + s exp([ r, s ] / J r ∗ J s is gauge equivalent to J r + s by the gauge transformation given by theelement exp( z/ z is as in Proposition 2.9.The proof of the theorem is complete. (cid:3) Remark 6.2.
Theorem 6.1 is a special case of Theorem 7.8, but itsproof is simpler and we decided to give it separately.
Example 6.3.
Let U be the 3 − dimensional Heisenberg group. Then u has a basis a, b, c such that c is central and [ a, b ] = c . It is straight-forward to verify that ( ∧ u ) u = { ( αa + βb ) ∧ c | α, β ∈ k } , and byCorollary 6.1, H ( U ) ∼ = { exp(( αa + βb ) ∧ c ) | α, β ∈ k } ∼ = G a . Connected affine algebraic groups
In this section we will assume that k has characteristic 0.7.1. The bijectivity of Θ .Proposition 7.1. Let G be a connected affine algebraic group over k with Lie algebra g . Then g is a direct summand of O ( G ) ∗ as a g -module(under the adjoint action).Proof. We will show that the surjective morphism b : O ( G ) → g ∗ , givenby β dβ (1), splits as a morphism of g -modules, i.e., there exists a g -morphism φ : g ∗ → O ( G ) such that b ◦ φ = id. The morphism φ can beviewed as an element of O ( G ) ⊗ g , i.e., as a regular function φ : G → g .Thus it suffices to prove the following lemma, communicated to us byVladimir Popov. Lemma 7.2.
There exists a regular G -invariant function φ : G → g such that dφ (1) = id.Proof. This almost follows from [LPR, Theorem 10.2 and Proposition10.5]. More precisely, the map φ := γ c ◦ θ ◦ γ − C in [LPR, 2nd line ofp. 961] actually is defined at 1 and moreover is ´etale at 1 (i.e., itsdifferential is an isomorphism). Indeed, θ is ´etale at [1 ,
1] (because ǫ is NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 17 ´etale at 1 by [LPR, Lemma 10.3], and τ is an isomorphism [LPR, 4.3]),and by [LPR, (3.1)], γ c and γ C are ´etale at [1 ,
0] and [1 , φ by the inverse of dφ (1), we can make dφ (1) = id.The remaining problem is that φ is only a rational map. To make itregular, we use [LPR, Proposition 10.6 and Lemma 10.7] to constructa regular function f on G such that f (1) = 1, and f ◦ φ is regular.Then d ( f ◦ φ )(1) = dφ (1) = id, so we may replace φ with f ◦ φ . We aredone. (cid:3) This completes the proof of the proposition. (cid:3)
Proposition 7.3.
Let G be a connected affine algebraic group over k .Then Int ( G ) = Inn ( G ) .Proof. Assume first that G is reductive. Then it follows from the wellknown description of the automorphism group of G that an automor-phism of G is inner if and only if it does not permute the irreduciblerepresentations of G . Hence, the claim follows in this case.Consider now the general case. Suppose x ∈ ( O ( G ) ∗ ) × is such that α := Ad( x ) is an automorphism of G . Let G u be the unipotent radicalof G , let G r := G/G u , and let ψ : O ( G ) ∗ → O ( G r ) ∗ be the correspond-ing algebra surjection. Then ψ ( x ) implements an automorphism of G r ,which is inner by the above argument. Hence, multiplying x by anelement of G , we are reduced to the situation when ψ ( x ) = 1. Let I be the kernel of ψ , then x ∈ I .Thus we can define the element log( x ) := P ( − n − ( x − n /n ∈ I (as any series P n a n , where a n ∈ I n , converges in the topology of O ( G ) ∗ ). We have [log( x ) , y ] = d ( y ), where d := log( α ) is a derivationof g := Lie( G ). Thus log( x ) defines a class in H ( g , g ) which becomestrivial in H ( g , O ( G ) ∗ ) (where g acts on O ( G ) ∗ by the adjoint action).Therefore, since by Proposition 7.1, g is a direct summand of O ( G ) ∗ as a g -module, the class of log( x ) in H ( g , g ) is zero, i.e., [log( x ) , y ] = [ b, y ]for all y ∈ g and some element b ∈ g . Then it is easy to see thatAd( x ) y = Ad(exp( b )) y for all y ∈ g , hence xgx − = exp( b ) g exp( b ) − for all g ∈ G . Thus, Ad( x ) is an inner automorphism, as desired. (cid:3) Theorem 7.4.
Let G be a connected affine algebraic group over k .Then Θ is bijective.Proof. By Theorem 5.1 and Proposition 7.3, Θ is injective.We now show that Θ is surjective. Let (
A, R ) ∈ B ( G ). We have A = A r × A u , the product of the reductive and unipotent parts. SinceAut( A r ) is discrete, G acts on A r trivially, i.e., A r is central in G . Let a := Lie( A ), a u := Lie( A u ). By Theorem 3.4, we have R = R r ∗ R u , where R u = exp( s ) for some s ∈ a ∧ a u . Let J u := exp( s/ J r be any Hopf 2-cocycle for A r such that ( J r ) − ∗ J r = R r . Then J r isinvariant since A r is central in G . So setting J := J r ∗ J u , we see that J is G -invariant and Θ([ J ]) = ( A, R ). (cid:3) Remark 7.5. If G is a finite group of odd order, it follows from [Da2,Lemma 6.1] that Θ is surjective. (See also [GK, Corollary 4.6].) How-ever, there exist finite groups of even order for which Θ is not surjective(see [GK, 7.3])7.2. The structure of H inv ( G ) for nilpotent G with a commuta-tive reductive part. Let G := G r × G u be a (possibly disconnected) nilpotent affine algebraic group over k , with a commutative reductivepart . Let g u be the Lie algebra of G u , and let z u be the center of g u .Let g r be the Lie algebra of G r . Then g := g r ⊕ g u is the Lie algebra of G . Note that g r is central in g . Lemma 7.6.
Let r ∈ g ∧ g u be a g -invariant element. Write r = r ′ + r ′′ ,where r ′ ∈ g r ⊗ g u and r ′′ ∈ ∧ g u . Then r ′ ∈ g r ⊗ z u and r ′′ ∈ ( ∧ g u ) g u .Proof. Let b ∈ g u . We have [∆( b ) , r ′ + r ′′ ] = 0, [∆( b ) , r ′′ ] ∈ ∧ g u , and[∆( b ) , r ′ ] = [1 ⊗ b, r ′ ] ∈ g r ⊗ g u . Thus [∆( b ) , r ′′ ] = [∆( b ) , r ′ ] = 0, and itfollows that r ′ ∈ g r ⊗ z u . (cid:3) Lemma 7.7.
Let G = G r × G u be a nilpotent affine algebraic groupwith a commutative reductive part as above. The following hold: (1) Int ( G ) = Inn ( G ) , so Θ : H inv ( G ) → B ( G ) is injective. (2) The group H inv ( G ) is commutative and naturally isomorphic toHom ( ∧ c G r , k × ) × ( g r ⊗ z u ) × ( ∧ g u ) g u .Proof. (1) Every γ ∈ Int( G ) acts trivially on G r (since G r is central in G ), and defines an internal automorphism of G u by restriction. Hence,by Theorem 6.1, γ is inner.(2) We have a natural group homomorphism ψ : Hom( ∧ c G r , k × ) × ( g r ⊗ z u ) × ( ∧ g u ) g u → H ( G ) , given by ψ ( K, r ′ , r ′′ ) = [ K ∗ exp( r/ r := r ′ + r ′′ and K isviewed as an element in H ( G r ) = Hom( ∧ c G r , k × ). Thus, we have(Θ ◦ ψ )( K, r ′ , r ′′ ) = ( A, R ), where A is the corresponding commutativenormal subgroup to R := K − ∗ K ∗ exp( r ). It is clear that Θ ◦ ψ isinjective, hence ψ is injective as well.Now let us show that ψ is surjective. By Part (1), for this it sufficesto prove that Θ ◦ ψ is surjective. Let ( A, R ) ∈ B ( G ). Write A = A r × A u ,and let a r , a u be the Lie algebras of A r , A u . Then Theorem 3.4 impliesthat R = R r ∗ R u , where R r = K − ∗ K and R u = exp( r ) for some NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 19 r ∈ a r ⊗ a u ⊕ ∧ a u . Since A r ⊂ G r is central in G , we have that K and r are G -invariant. Hence, by Lemma 7.6, r = r ′ + r ′′ , where r ′ ∈ g r ⊗ z u and r ′′ ∈ ( ∧ g u ) g u . Thus, ( A, R ) = (Θ ◦ ψ )( K, r ′ , r ′′ ), i.e., ψ is anisomorphism, as desired. (cid:3) The structure of H inv ( G ) for connected G . Let G be a con-nected affine algebraic group over k . Let G u be the unipotent radicalof G , g u its Lie algebra, and G r := G/G u . Let Z be the center of G , Z r its reductive part and Z u its unipotent part (so, Z = Z r Z u ). Let z , z r and z u be the Lie algebras of Z , Z r and Z u , respectively. Theorem 7.8.
Let G be a connected affine algebraic group over k .Then the group H inv ( G ) is commutative and is naturally isomorphic toHom ( ∧ b Z r , k × ) × ( z r ⊗ z u ) × ( ∧ g u ) G . Proof.
We have a natural group homomorphism ψ : Hom( ∧ b Z r , k × ) × ( z r ⊗ z u ) × ( ∧ g u ) G → H ( G ) , given by ψ ( K, r ′ , r ′′ ) = [ K ∗ exp( r/ r := r ′ + r ′′ and K isviewed as an element in H ( Z r ) = Hom( ∧ b Z r , k × ). Thus, we have(Θ ◦ ψ )( K, r ′ , r ′′ ) = ( A, R ), where A is the corresponding commutativenormal subgroup to R := K − ∗ K ∗ exp( r ). It is clear that Θ ◦ ψ isinjective, hence ψ is injective as well.Let us now prove that ψ is surjective. By Theorem 7.4, Θ is bijective,so it suffices to show that Θ ◦ ψ is surjective. Let ( A, R ) ∈ B ( G ). Write A = A r × A u . Then A , A r and A u are normal in G . Hence, A r iscentral in G (since the group of automorphisms of A r is discrete, while G is connected). Hence, A ⊂ G ′ := Z r G u , and ( A, R ) is an elementof B ( G ′ ). Since G ′ is nilpotent with a commutative reductive part, itfollows from Lemma 7.7 that ( A, R ) = (Θ ◦ ψ )( K, r ′ , r ′′ ), where K is aHopf 2-cocycle for Z r , r ′ ∈ z r ⊗ g u , and r ′′ ∈ ( ∧ g u ) g u . Moreover, R is G -invariant, hence so are r ′ , r ′′ . Therefore, r ′ ∈ z r ⊗ z u and r ′′ ∈ ( ∧ g u ) G ,i.e., ψ is an isomorphism, as desired. (cid:3) Remark 7.9. If G is a reductive algebraic group over k , Theorem 7.8says that H inv ( G ) = H ( b Z, k × ) = Hom( ∧ b Z, k × ), which also followsfor k = C from [NT, Theorem 7]. In particular, if G is semisimple then H ( G ) is a finite group, and if the center of G is cyclic then H ( G )is the trivial group. Remark 7.10.
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NVARIANT HOPF 2-COCYCLES FOR AFFINE ALGEBRAIC GROUPS 21
Department of Mathematics, Massachusetts Institute of Technol-ogy, Cambridge, MA 02139, USA
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