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Twisted automorphisms of group algebras

We continue the study of twisted automorphisms of Hopf algebras started in"Twisted automorphisms of Hopf algebras". In this paper we concentrate on the group algebra case. We describe the group of twisted automorphisms of the group algebra of a group of order coprime to 6. The description turns out to be very similar to the one for the universal enveloping algebra given in"Twisted automorphisms of Hopf algebras".


Introduction
In [3], using the language of Galois algebras, monoidal (auto-)equivalences of categories of representations of finite groups were described in terms of some group-theoretic data. Composition of monoidal equivalences corresponding to tensor product of Galois algebras turns out to have a quite complicated form in terms of that data. In particular, the group structure on isomorphism classes of monoidal auto-equivalences (bi-Galois algebras) is not very easy to deal with.
Here we describe this structure using a different presentation (as twisted automorphisms) for monoidal (auto-)equivalences, which was developed in [4]. Since any Galois algebra over a finite group has a normal basis ( [11]) the results of [3,4] imply that any bi-Galois algebra corresponds to a twisted automorphism. In [4] general structure of the (Cat-)group of twisted automorphisms of a Hopf algebra was examined. As an example, the case of the universal enveloping algebra of a Lie algebra (over the ring of formal power series) was treated. In that situation any twisted automorphism is a bialgebra automorphism together with an invariant twist (the so-called separated case). The gauge classes of invariant twists form an abelian group isomorphic to the invariant elements of the exterior square of the Lie algebra. The group of gauge classes of twisted automorphisms is a crossed product of the group of automorphisms of the Lie algebra and the group of invariant twists.
For the group algebra case the situation is more complicated yet resembles closely the case of a universal enveloping algebra. For simplicity we restrict ourselves to groups of odd order. We still have the separation property but separation into a composition of a group automorphism and an invariant twist is not unique: there are invariant twists (symmetric twists), which are gauge isomorphic to group automorphisms (class-preserving automorphisms). However we can make it unique by using only anti-symmetric twists. The gauge classes of anti-symmetric twists form an abelian group Aut anti Tw (k[G]) with the operation closely related to the one on twists of a universal enveloping algebra. If the order of the group is coprime to 6, the group of anti-symmetric twists is a subgroup of the group of all invariant twists Aut 1 Tw (k[G]). Moreover, the group of invariant twists is the direct product of the group of anti-symmetric twists and the group of symmetric twists (the group of class-preserving outer automorphisms): We can summarise the main results of the paper in the form of a commutative diagram with exact rows and columns:

Twists on group algebras
Here we revise Movshev's classification of twists on the group algebra k[G] of a finite group in terms of Galois algebras [11]. An important ingredient of that classification is a group with a non-degenerate 2-cocycle. Recall that a group 2-cocycle α ∈ Z 2 (S, k · ) is non-degenerate if, for any s ∈ S, the homomorphism C G (s) → k · , x → α(x, s)α(s, x) −1 is nontrivial. Note that a cocycle cohomologous to a non-degenerate cocycle is also non-degenerate. There is a correspondence between groups with nondegenerate 2-cocycles and the so-called groups of central type. A group is of central type if it has an irreducible representation of dimension equal to the index of its centre (the largest possible). The quotient by the centre of a group of central type is a group with non-degenerate 2-cocycle (thus the alternative name central type factor group). Conversely a group with non-degenerate 2cocycle has a central extension, which is a group of central type. Using the classification of finite simple groups, Howlett and Isaacs proved that groups of central type are solvable (the conjecture of Iwahori and Matsumoto) [9].
For the proof of this slightly modified version of Movshev's result see [3]. A Galois G-algebra corresponding to the pair (S, α) can be realized explicitly as the algebra of functions: R(G, S, α) = {r : G → k[S, α] : r(sg) = s(r(g)) ∀s ∈ S, g ∈ G} with the G-action given by (f φ)(g) = φ(gf ).
It was proved in [11] (see also [3]) that all Galois G algebras possess normal bases thus (according to section 2.2. of [4]) establishing the equivalence of the groupoids Aut Tw (k[G]) → Aut Gal (k[G]). Although a quasi-inverse to this equivalence is very hard to construct in general, in some cases twists corresponding to Galois algebras can be written explicitly. Note that, for an abelian A, the alternation map identifies the cohomology group H 2 (A, k · ) with the group Hom(Λ 2 A, k · ) of alternative bi-multiplicative forms Moreover a bi-multiplicative form (not necessarily alternative) is always a 2cocycle. It is not hard to see that any alternative form on an abelian group is the alternation of some bi-multiplicative form. Thus any 2-cocycle on an abelian A is cohomologous to some bi-multiplicative form. It follows from the definition that a bi-multiplicative form is non-degenerate as a 2-cocycle if and only if it is non-degenerate in the ordinary sense (establishes an isomorphism A →Â). For a non-degenerate bi-multiplicative form β on abelian A denote by b the adjoint bi-multiplicative form onÂ. Note that b is also non-degenerate. Define corresponding to the character χ. In the next lemma we will verify that F (A,b) (as an element of k[G] ⊗2 ) is the twist corresponding to the Galois G-algebra R(G, A, β).
) be a twist corresponding to an abelian subgroup A ⊂ G and a bi-multiplicative form β. Then the function algebra (k(G), * F ) with the F -twisted multiplication is isomorphic to the Galois G-algebra R(G, A, β).
can be established explicitly by assigning e χ → l χ = a∈A χ(a)p a . Indeed, since p χ (l ψ ) = δ χ,ψ l ψ and l χ i ψ = l χψ , the twisted product has the form Automorphisms of Galois G-algebras were studied in [3]. In view of theorem 2.1, they are automorphisms of objects of the groupoid Ctf (G) of central type factor subgroups of G. Theorem 2.3. The group of G-automorphisms Aut G (R) of the Galois Galgebra R = R(G, S, α) fits into an exact sequence: HereŜ = Hom(S, k · ) is the group of characters of S, N G (S) is the normaliser of S in G, and A ∈ H 2 (S, k · ) is the class of α. In particular, the automorphism group Aut G (R) has the same order as G only if S is normal abelian and the class A is G-invariant. Now following [2,3] we will prove that (up to gauge equivalence) twists for twisted isomorphisms are supported by normal abelian subgroups. Very similar results were proved in [5].
is gauge isomorphic to a twisted isomorphism of the form (f ′ , F (S,β) ) for some normal abelian S ⊂ G and a non-degenerate bi-multiplicative form β with Ginvariant alternation.
Proof. For a twisted automorphism (f, F ) of k[G] the f -twist F must correspond to a Galois G-algebra R with the automorphism group isomorphic to G. Thus R is isomorphic (as a Galois G-algebra) to R(G, S, α) for a normal abelian S with a non-degenerate 2-cocycle with G-invariant cohomology class. Finding a bilinear form b with the same class, we can replace R(G, S, α) with (the isomorphic Galois G-algebra) R(G, S, b). Finally the Galois G-algebra R(G, S, b) corresponds to the twist F (S,β) . Thus the twist F is gauge isomorphic to the twist F (S,β) : , which can be viewed as a gauge transformation

Separation of twisted isomorphisms
The fact that twists of twisted isomorphisms are supported by abelian normal subgroups allows us to prove separation for twisted automorphisms of the group algebra of a group of odd order. Proof. For odd order S the alternation map is bijective on Hom(Λ 2 S, k · ). Thus we can strengthen the statement of the theorem 2.4 assuming that β is a nondegenerate alternative G-invariant form. This assumption makes the twist F (S,β) G-invariant. Hence the twisted automorphism (f ′ , F (S,β) ) is separated and f ′ is a bialgebra automorphism of k[G], which has to be induced by a group automorphism of G.
Below we show that group algebras of groups of even order can have nonseparated twisted isomorphisms.
Let β :Â ×Â → k · be a non-degenerate bi-multiplicative form on the dual group of an abelian normal subgroup This means that for any g ∈ G the form β g (χ, ξ) = β(g(χ), g(ξ))β(χ, ξ) −1 is symmetric. Definẽ It is a group with the product so that (f, F ) is a twisted homomorphism. The subgroup , form a normal subgroup inG(G) isomorphic to A. Moreover, the quotient group G/A is isomorphic to the quotient group Q = G/A and the Q-action on A, coming from G, coincides with the one coming from G. The class of the extension A → G → Q in H 2 (Q, A) was calculated in [3]. It is equal to the class which is the differential of the second term of the Hochschild-Serre spectral sequence associated with the split extension of A by Q. The map D is trivial in the absence of 2-torsion. Thus for odd order G the twisted homomorphism (f, F ) is a twisted automorphism and, in particular, is separable. In the case when G is not isomorphic to G the twisted homomorphism (f, F ) can not be separable. Of course non-triviality of D(b) is not enough to guarantee that G is not isomorphic to G as abstract groups, but in the next example (taken from [3], see also [5]) it can be checked directly.

Example 3.2. Affine symplectic and metaplectic groups.
Let A be an elementary abelian 2-group with a symplectic form ( , ) : Let Q be the group of automorphisms of A preserving ( , ), i.e. the symplectic group Sp(n, 2), where n is the rank of A. Let G be the semi-direct product of A and Q, i.e. the affine symplectic group ASp(n, 2). Let By the definition the alternation of β is G-invariant. The group G corresponding to the abelian normal subgroup A with the form β coincides with the quotient of the group of pairs which is called the metaplectic group M p(n, 2) [14]. It is known that for n > 2 the metaplectic group M p(n, 2) is not isomorphic to the affine symplectic group ASp(n, 2). Thus the twisted homomorphism is not separable.

Class-preserving automorphisms
Here we describe invariant twists, which stabilise the unital inclusion k → k[G] (as in the section 4.3 of [4]) and link them with class-preserving automorphisms of G. Recall that an automorphism f : G → G is class-preserving if f preserves conjugacy classes of G: for any g ∈ G, Clearly, class-preserving automorphisms are closed under composition, forming a normal subgroup Aut cl (G) of the automorphism group Aut(G). Inner automorphisms are obviously class-preserving. Moreover, the homomorphism sending g ∈ G to the inner automorphism ( ) g defines a crossed module of groups Aut cl (G). In particular, π 0 (Aut cl (G)) = Out cl (G) is the group of classpreserving automorphisms modulo inner. The study of class-preserving automorphisms was initiated in [1], where the first examples of groups with nontrivial Out cl (G) were constructed. Since then many more examples were produced and some triviality results were proved. In particular, using the classification of finite simple groups, Sah proved solvability of Out cl (G) [12] and Feit and Seitz verified triviality of Out cl (G) for simple G [6]. Recently, class-preserving automorphisms were used to produce a counterexample to the isomorphism problem of integer group rings [8].
Here we characterize class-preserving automorphisms as invariant twists on the group algebra stabilizing the unital inclusion k → k[G].
Proof. Since k is algebraically closed (of characteristic zero), k[G] is a sum of matrix algebras and the restriction map is spanned by class sums (sums over conjugacy classes of G). Thus the kernel of ) and is Aut cl (G).
We will say that a class preserving automorphism φ is supported by a normal Below we give a cohomological description of class-preserving automorphisms supported by abelian normal subgroups. Here By definition, the class of ψ lies in the kernel of the homomorphism induced by the natural inclusion A → k[A] · . Since the field k is algebraically closed, the algebra k[A] is isomorphic to the function algebra k(Â). Moreover, this isomorphism preserves G-actions. Thus k[A] · is a permutation G-module and where the sum is taken over G-orbits inÂ and St G (O) is the stabilizer of an orbit O. So the class of a cocycle ψ is in the kernel of (3) iff which proves the proposition. Let V be a vector spaces over F = F 2 (the two element field). Let b : V ⊗ V → F be a non-degenerate symmetric bilinear form on V and q : V → F an associated quadratic form: Note that the existence of q implies that b is alternative: b(v, v) = 0 for any v ∈ V . Let Q = Aut(V, b) be the group of automorphisms of b (a symplectic group) and G = Q ⋉ V * be a semi-direct product (an affine symplectic group). Define an element in the group ring k[V * ] of the dual space V * by Since g preserves the linearisation of q, ψ(g) is linear in v. Now it is straightforward to see that [x, g] is given by the formula (4): Obviously the cocycle (4) satisfies the condition (2). To see that it is non-trivial we will follow the arguments of [7]. Recall that for a vector v ∈ V the map . Define q by q(u, l) = l(u). For a vector (u, l) ∈ V the corresponding symplectic transvection has the form: Now the value of the 1-cocycle ψ on the transvection is q(τ (u,l) (v, m)) − q(v, m) = (m(u) + l(v))(l(u) + 1), which coincides with b((u, l), (v, m))(q(u, l) + 1). Thus we have the following formula The cocycle ψ is a coboundary if there is a linear function l ∈ V * such that  Let V be a vector spaces over F = F 3 (the three element field). Let τ : V ⊗ V ⊗ V → F be a non-degenerate symmetric tri-linear form on V and c : V → F an associated cubic form: Note that c is a depolarization of 2τ : 2τ (u, v, w) and the form τ satisfies: τ (v, v, v) = 0 for any v ∈ V . Let Q = Aut(V, τ ) be the group of automorphisms of the form τ and G = Q ⋉ V * be a semi-direct product. As in the previous example define an element in the group ring k Here ω ∈ k is a primitive cubic root of unity and the p v = Since g preserves the linearisation of q (which is τ In contrast with the previous example it is much more difficult to find a tri-linear form with a non-trivial 1-cocycle ψ. Following [13], define on F 3 5 (the field of 243 elements), considered as a vectors space V over F 3 , a symmetric tri-linear form τ (x, y, z) = T r(xyz 9 + xy 9 z + x 9 yz).
Here T r : F 3 5 → F 3 is the trace of the field extension F 3 ⊂ F 3 5 . Let ǫ ∈ F 3 5 be a primitive root of unity of degree 11 such that T r(ǫ) = −1. It can be checked directly that the following linear operators r, s, t on V preserve τ : To define r note that the powers ǫ, ǫ 3 , ǫ 4 , ǫ 5 , ǫ 9 span the space V . In this basis r has the form r(ǫ) = −ǫ, r(ǫ 3 ) = −ǫ 9 , r(ǫ 4 ) = −ǫ 5 , It was proved in [13] that r, s, t generate the group Aut(V, τ ) and that the group Aut(V, τ ) is isomorphic to the Mathieu group M 11 . Now define c by c(x) = T r(x 11 ). It can be checked that r 2 , s, t stabilize c (it was also proved in [13] that r 2 , s, t generate the subgroup of M 11 isomorphic to P SL 2 (11)). Now if we assume that the 1-cocycle (5) is a coboundary we should have a non-zero linear function l ∈ V * invariant under s, which is not possible. Indeed, writing l as l(v) = T r(λv) for some λ ∈ F 3 5 we would have T r(λǫv) = T r(λv) for all v ∈ V , which implies λ = 0.

Invariant anti-symmetric twists and triangular structures on group algebras
It follows from the proposition (3.1) that up to gauge transformations invariant anti-symmetric twists on the group algebra k[G] of a group G of odd order correspond to normal abelian subgroups A ⊂ G with non-degenerate alternative bi-multiplicative G-invariant forms α : A × A → k · : A bijective correspondence between non-degenerate bi-multiplicative forms β : A × A → k · on an abelian group A and non-degenerate bi-multiplicative forms b :Â ×Â → k · on its group of charactersÂ can be defined explicitly by the rule: for any x ∈ A there is a unique χ ∈Â (and vice versa) such that This correspondence allows us to give a different presentation for anti-symmetric twists. Denote by p χ = 1 |A| x∈A χ −1 (x)x the minimal idempotent of the group algebra k[A] corresponding to the character χ ∈Â: can be written as Proof. Indeed, Since b(χ, ψ)χ(a 1 )ψ(a 2 )) = ψ(x)b(x, a 1 )ψ(a 2 ) = b(x, a 1 )ψ(xa 2 ) and 1 |A| ψ∈Â ψ(xa) = δ x,a −1 the expression (6) can be rewritten as It remains to notice that β(a −1 2 , a 1 ) = β(a 1 , a 2 ).
For a subgroup B of an abelian group A with a non-degenerate alternative bi-multiplicative form β : A × A → k · denote by with the last morphism being induced by the form β: a → (x ∈ B → β(a, b)).

Suppose that there exists a multiplicative splittingB → A. Then we have a Lagrangian decomposition
A ≃ B ⊕B where elements of A can be written as pairs (x, χ) where x ∈ B and χ ∈B.

Lemma 5.2. An abelian group with an alternative bi-multiplicative non-degenerate form has a Lagrangian decomposition.
Proof. We prove this by induction on the order of the group. Let a ∈ A be an element with the property: for any x ∈ A such that x m = a there is n so that a n = x. Then the inclusion a → A of the cyclic subgroup generated by a splits as well as the surjection A → a induced by the form β on A. So we can write A ≃ A ′ ⊕ a ⊕ a where A ′ ≃ a ⊥ / a with the induced form. By the induction it follows that, A ′ has a Lagrangian decomposition, and hence so has A.
With any Lagrangian decomposition A ≃ B ⊕B there are associated two more presentations for the twist F (A,β) , which, in a way, are mixtures of the previous two. Define

Lemma 5.3. The anti-symmetric twist
can be written as Proof. Indeed, Analogously (using the symmetry between B andB) we can write As the next example shows, not any Lagrangian subgroup fits into a Lagrangian decomposition.

Example 5.4.
Let A be Z/Z p 2 ⊕Z/Z p 2 with the standard alternative form β((1, 0), (0, 1)) = ε for a primitive root ε of degree p 2 . The subgroup B = pA is Lagrangian but the extension B → A →B does not split.
Of course we can always split the short exact sequence B → A →B settheoretically. Choosing a section The bi-multiplicative form β on A transports to B ×B: where β(χ, ψ) = β(s(χ), s(ψ)).
Repeating the calculations for the twist on Lagrangian decomposition, we will have the following form for the twist F (A,β) on B ×B: We finish this section with a well-known remark on triangular structures on group algebras. We include the proof of the proposition below because we will use its argument later on.

Proposition 5.5. The set T r (k[G]) of triangular structures on the group algebra is isomorphic to the set of pairs (A, α) consisting of a normal abelian subgroup A ⊂ G and a non-degenerate skew-symmetric bi-multiplicative
Proof. By the proposition (4.4.1) of [4] a support sub-bialgebra (minimal triangular sub-bialgebra) of a triangular structure R is a commutative and cocommutative normal sub-bialgebra H R . Thus it is k[A] for an abelian subgroup A = G(H R ) ⊂ G, which must be normal since H R is normal. The isomorphism b : H * R → H R induces (and is induced by) the homomorphism of groups of group-like elements G(b) : is the group of characters of A. Thus the isomorphism G(b) : Hom(A, k · ) → A induces (and is induced by) a non-degenerate bimultiplicative form α : A × A → k · . The G-invariance of α follows from the H-invariance of b while skew-symmetricity of α is equivalent to self-duality of b.

Group structure on anti-symmetric twists
We start with the following abstract situation. Let T be a group and t : T → T be an automorphism of order 2. Define a left T -action on itself via In particular, the stabiliser of the identity element is T t = {s ∈ T | t(s) = s} the subgroup of τ -invariant elements. Note also that this action preserves the subset and that X is closed under power maps x → x m for m ∈ Z.
Lemma 6.1. Suppose that ( ) 2 is bijective on X. Then In particular, for any x, y ∈ X, there is a unique factorisation: The operations s, • satisfy Here [x, y] = xyx −1 y −1 .
Proof. First we notice that in the assumption of the lemma (the bijectivity of ( ) 2 ) X coincides with the orbit {t(g) −1 g, g ∈ T } of the identity element. Indeed, solving y 2 = x for x ∈ X we will be able to write x as t(y) −1 y (since y is also in X). The next thing to observe is that the intersection X ∩ T t is trivial: for x ∈ X ∩ T t , we have x = t(x) = x −1 or x 2 = 1, which implies x = 1. Now we write T as the product T t X. For g ∈ T , solve y 2 = t(g) −1 g in X. Then for s = gy −1 , we have Since T t normalizes X (the action of T t on X via (8) is the conjugation), we can write xyz as an element of T t X in two ways: x(yz) = xs(y, z)(y•z) = s(y, z)x s(y,z) (y•z) = s(y, z)s(x s(y,z) , (y•z))(x s(y,z) •(y•z)) which gives the equations of the lemma. For Now let T be the group Aut 1 Tw (k[G]) of gauge classes of invariant twists with t being the automorphism of transposition of tensor functors of a twist. Then by the section 4, the stabiliser T t is the group Out cl (G) of outer class-preserving automorphisms of G and X is (isomorphic to) the set of abelian normal subgroups of G equipped with G-invariant non-degenerate bimultiplicative form. If G is odd then the order of any element from X is also odd so we are in the situation of the lemma 6.1. This defines a binary operation • on X: α1) . Note that the right hand side belongs to the group algebra of the subgroup First we describe the support group A. Let be the kernel of the bimultiplicative form on A 1 ∩ A 2 , which is the product of the restrictions of α 1 and α 2 .
Proof. The definition of π does not depend on the factorisation a 1 a 2 . Then for d ∈ A 1 ∩ A 2 : To see that A is abelian note first that the commutant [A 1 , A 2 ] lies in K. To check that the commutant [A, A] is trivial, it is enough to verify that b 1 ([u 1 u 2 , v 1 v 2 ], y) = 0 for any u 1 u 2 , v 1 v 2 ∈ A and any y ∈ A 1 . Writing y). Indeed, by G-invariance of b i and the defining relations for u 1 u 2 , v 1 v 2 (together with [A 1 , A 2 ] ⊂ K), we have the chain of equalities: We need to verify that the value of the form α does not depend on factorisations of its arguments. To see that, we give another presentation of the group A. Denote by A 1 ⊲⊳ A 2 the group of pairs (x 1 , x 2 ), x i ∈ A i with the product: is a group homomorphism with the image A 1 A 2 . The group A fits into a commutative diagram with exact rows and columns: The homomorphism Π is given by the same formula as the map π. We can define a skew-symmetric pairing on A 1 ⊲⊳ A 2 using the formula It follows from the non-degeneracy of α i that the orthogonal to ker(Π) in A 1 ⊲⊳ A 2 is the anti-diagonal image of A 1 ∩A 2 . Hence that the kernel of the restriction of the pairing from A 1 ⊲⊳ A 2 to ker(Π) coincides with the anti-diagonal image of K. Thus the pairing restricts from ker(Π) to a non-degenerate pairing on ker(Π)/K = A. Proof. First we prove that A contains the support subgroup. Note that the supports of F (A1,α1) • F (A2,α2) and R = F (A1,α1) F 2 (A2,α2) F (A1,α1) coincide. To see that R belongs to k[A]⊗k [A] it is enough to check that (I⊗π)(R) = 1. Indeed, as an R-matrix, R defines a homomorphism of Hopf algebras l → (l ⊗ I)(R) whose image is the group algebra of the support subgroup of R. The condition (I ⊗ π)(R) = 1 will mean that the support subgroup is in the kernel of π. We check this condition by showing that (I ⊗ π)(F (A1,α1) ) = (I ⊗ π)(F (A2,α2) ) −1 or, more precisely, by showing that for any c ∈ K the evaluation (I ⊗ π)(F (A1,α1) )(I ⊗ c) is the inverse of (I ⊗ π)(F (A2,α2) )(I ⊗ c) (here we think of k[K] as the function algebra k[K] * ). Since for y ∈ A 1 π(y)(c) = α 1 (y, c) The inner sum 1 |A1| y∈A1 α 1 (xc −1 , y) is the δ-function δ x,c . Hence (I⊗π)(F (A1,α1) )(I⊗ c) = c. Similarly, using that π(y)(c) = α 2 (y, c) −1 for y ∈ A 2 , we get that (I ⊗ π)(F (A2,α2) )(I ⊗ c) = c −1 .
Let χ u1u2 be the character on A corresponding to an element u 1 u 2 ∈ A via the form α: To show that α 2 is the form corresponding to R ∈ k[A] ⊗2 we need to check that which by multiplicativity and invariance of the forms α i equals This expression can be simplified since 1 Now, writing y2y −1 1 u 1 = y 2 y −1 1 u1 (y 2 y −1 1 ) −1 u 1 and using that u −1 1 v 1 = v 1 by commutativity of A, we have This allows us to simplify the expression (10) further: Finally 1 |A2| y1∈A2 α 2 (y 1 , y 2 2 u −1 2 ) is the δ-function δ y2,u2 , which leaves us with Clearly the operation • on anti-symmetric twists is commutative. Thus the commutator of any two anti-symmetric twists F 1 = F (A1,b1) and F 2 = F (A2,b2) can be written as . An explicit form of such u is closely related to a certain invariant of the pair (A 1 , b 1 ), (A 2 , b 2 ), which we are going to describe now. The construction is very similar to the one for twists on universal enveloping algebras from [4]. Denote by B = [A 1 , A 2 ] the commutant of A 1 and A 2 in G. To a pair of characters (χ 1 , χ 2 ) on B we assign an element [x 1 , The elements x i are defined up to subgroups Note that these subgroups have the following commutation property: Indeed, for arbitrary x ∈ B ⊥ b1 , y ∈ A 2 and for any z by c(χ 1 , χ 2 , . This map is multiplicative in each variable. It's also G-invariant by the construction. Moreover this map is symmetric. Indeed, let y i ∈ a i be such that Then which means that c(χ 1 , χ 2 , χ 3 ) = c(χ 3 , χ 2 , χ 1 ). Similarly which means that c(χ 1 , χ 2 , χ 3 ) = c(χ 1 , χ 3 , χ 2 ). c(χ, χ, ψ)c(χ, ψ, ψ)p ψ ⊗ p χ .
Here c :B ×B ×B → k · is the map defined above.
Since c is symmetric we can always find a solution. It is much more subtle to check that this solution will be G-invariant, which will guarantee that u is in the centre of k[G]. However if 3 is coprime to |G| then we can always do that by solving u(χ) 3 = c(χ, χ, χ). Thus we have the following result. x i p −j ⊗ x j p i for some non-central x ∈ E. Here p i = 1 p p−1 t=0 ǫ −it c t are central idempotents. Since F −1 x = F x −1 , the commutator of two invariant twists can be expressed as follows: [F x , F y ] = x i1 p j1 y k1 p l1 x −i2 p j2 y −k2 p l2 ⊗ x j1 p −i1 y l1 p −k1 x j2 p i2 y l2 p k2 = ǫ (i 2 j+j 2 i)b(x,y) p j ⊗p i .
Here, following the Lie algebra case, we construct a group with two abelian normal subgroups equipped with invariant bi-multiplicative forms with given invariant (11). Let C be an abelian group and c : C ⊕ C ⊕ C → k · be a symmetric tri-multiplicative map. Define on M (C, c) = C ⊕ C ⊕Ĉ the structure of a meta-abelian group: (x 1 , y 1 , χ 1 )(x 2 , y 2 , χ 2 ) = (x 1 + x 2 , y 1 + y 2 , χ 1 χ 2 c(x 1 , y 2 , −)). corresponding to the tri-linear form τ from the example (4.4) with Q = Aut(τ ), we get an example of two invariant anti-symmetric twists whose commutant corresponds to a non-trivial class-preserving automorphism.