Quantizations of group actions

We describe quantizations on monoidal categories of modules over finite groups. They are given by quantizers which are elements of a group algebra. Over the complex numbers we find these explicitly. For modules over S3 and A4 we are given explicit forms for all quantizations.


Introduction
In the papers [1,3] we found the quantizations of the monoidal categories of modules graded by finite abelian groups. Quantizations are natural isomorphisms of the tensor bifunctor Q : ⊗ → ⊗ that satisfy the coherence condition. By this condition the quantizations are 2-cocycles, and under action by isomorphisms of the identity functor they are representatives of the second cohomology of the group.
With these explicit descriptions of quantizations we showed that classical non-commutative algebras like the quaternions and the octonions are obtained by quantizing a required number of copies of R. Moreover we obtained new classes of non-commutative algebras. The resulting non-commutativity is governed by a braiding which is the quantization of the twist.
In this paper we will investigate the situation for quantizations of modules with action of finite groups that are not necessarily abelian. Further the definition of quantizations is widened as they may be natural transformations, not only natural isomorphisms.
Quantizations Q of the monoidal category of modules over finite groups G are realized by elements q Q in the group algebra C[G × G] called quantizers. These satisfy a form of the coherence condition, normalization and invariance with respect to G-action.
LetĜ be the dual of G. We use the Fourier transform to reconsider these quantizers as sequences of operatorsq α,β in End(E α ⊗ E β ) where E α , E β are irreducible representations corresponding to α, β ∈Ĝ. The coherence condition for the operatorsq α,β gives a system of quadratic matrix equations.
There is an equivalence relation on these quantizers by the action by natural isomorphisms of the identity functor. Taking the orbits of this action we arrive at the final expressions of the quantizers.
We apply the inverse of the Fourier transform to move back to the group algebra where we now have the quantizers q Q in C[G × G] realizing the non-trivial quantizations in the category.
To sum up, the procedure is as follows Explicit sequences {q α,β ∈ End(E α ⊗ E β )} α,β∈Ĝ We illustrate this method for abelian groups, see e.g. [2], and the permutation groups S 3 and A 4 . For S 3 there is a 1-parameter family of quantizations. For A 4 we have a larger selection with 2 parameters producing quantizations.

Quantizations in braided monoidal categories
A braided monoidal category C is a category equipped with a tensor product ⊗ and two natural isomorphisms: an associativity constraint assoc C : X ⊗ (Y ⊗ Z) → (X ⊗ Y ) ⊗ Z and a braiding σ C : X ⊗ Y → Y ⊗ X, for objects X, Y, Z in C, that both satisfy MacLane coherence conditions, see [5].
Let C, D be braided monoidal categories and Φ : C → D be a functor. A quantization Q of the functor Φ is a natural transformation of the tensor bifunctor that satisfies the following coherence condition for X, Y ∈ Obj(C) (see [4], for details). Note that in this paper we do not require the quantizations to be natural isomorphisms, only natural transformations (cf. [4]).
We denote the set of quantizations of Φ by q(Φ).
Let λ : Φ → Φ be a unit preserving natural isomorphism of the functor Φ. Then we define an action of λ : Q → λ(Q) on the quantizations by requiring that the diagram commutes.
The orbits of the above action we denote by If a quantization is a natural isomorphism we will call it a regular quantization and use the notations q • (Φ) and Q • (Φ).
If the functor Φ is the identity Id C : C → C we call Q a quantization of the category C and use the notations q(C) = q(Id C ) and Q(C) = Q(Id C ).
For the following result, see also [4]. Let Φ : B → C, Ψ : C → D be functors between the monoidal categories B, C, D and let Q Φ , Q Ψ be quantizations of Φ and Ψ. The following formula defines the quantization of the composition Ψ • Φ.
We call Q Φ•Ψ composition of quantizations. Then the composition defines an associative multiplication on the sets of quantizations. In partucular, the composition (6), where Φ = Ψ = Id C , defines a multiplication and gives a monoid structure on the set of quantizations. Moreover, the regular quantizations form a group in this monoid. The monoid q(C) acts on a variety of objects, see e.g. [4]. We list some examples here.
Braidings. Let σ be a braiding in the category C. If Q ∈ q • (C) then we define an action of Q on braidings by requiring that the following diagram Algebras. Let A be an associative algebra in the category with multiplication µ : A⊗A → A. Let Q ∈ q(C).
Then we define a new multiplication µ Q on A by requiring that the following diagram commutes.
Then (A, µ Q ) is an algebra in the category C too. We call it the quantized algebra.
Modules. Let E be a left module over the algebra A in the category with multiplication ν : We define a quantized multiplication ν Q on E by requiring that the following diagram commutes.
Then (E, ν Q ) is a module over the quantized algebra (A, µ Q ). We call it the quantized module.
Right modules are quantized in a similar way.
Coalgebras. Let A * be a coalgebra in the category with comultiplication µ * : A * → A * ⊗ A * . We define a new comultiplication (µ * ) Q on A * by requiring that the following diagram commutes.
Then (A * , (µ * ) Q ) is a coalgebra in the category C too. We call it the quantized coalgebra.
Bialgebras. Let B be a bialgebra in the category with multiplication µ B : B ⊗ B → B and comultiplication µ * The quantized bialgebra is the same object B Q = B equipped with the quantized multiplication µ Q B and quantized comultiplication (µ * B ) Q , quantized as above. This is also a bialgebra in the category.

Quantizations of G-modules
Let R be a commutative ring with unit and let G be a finite group. Denote by M od R (G) the monoidal category of finitely generated G-modules over R and let q(G) and Q(G) be the sets of quantizations of this category.
Let X and Y be G-modules. Recall that the tensor product X ⊗ Y over R is a G-module with action be the group algebra of G over R.
Then there is an isomorphism between the categories of G-modules and R[G]-modules, M od R (G) = M od R (R[G]) (see e.g. [2]). Hence, Theorem 1 Any quantization Q ∈ q(G) of the category of G-modules has the form where are elements of the group algebra R[G × G], for x ∈ X, y ∈ Y and X, Y ∈ Obj(M od R (G)).
Proof. We identify elements x ∈ X for X ∈ Obj(M od R (G)) with morphisms Then, for any elements x ∈ X, y ∈ Y , X, Y ∈ Obj(M od R (G)) and a quantization Q ∈ q(G) the following diagram Therefore, We call the elements q Q quantizers and denote by the set of quantizers.
Theorem 2 An element q ∈ R[G × G] defines a quantization on the category of G-modules if and only if it satisfies the following conditions: • The naturality condition q · ∆(g) = ∆(g) · q (16) for g ∈ G.
Proof. The coherence condition follows from (3) where and similarly for Q X,Y Z . The two other conditions follow straight foreward from the normalization and naturality conditions on the quantizations.
See also [4] for similar settings. We shall from now on use the notion quantizer instead of quantization. Remark that the conditions (14), (15) and (16) are some kind of 2-cyclic condition on q(G) (see, for example, section 4 on finite abelian groups).
Let U (G) be the set of units of R[G].
Theorem 3 The set of quantizers of G-modules Q(G) is the orbit space of the following U (G)-action on q(G): where l ∈ U (G).
Proof. Representing as above elements x ∈ X by morphisms φ x : R[G] → X we get the following commutative diagram with λ X : X → X, for any unit preserving natural isomorphism of the identity functor, λ : Therefore λ is uniquely defined by elements l ∈ λ R[G] (1), and Let q ∈ q(G). Then the action (4) gives We say that two quantizers p, q ∈ q(G) are equivalent if p = l(q) for some l ∈ U (G).

The Fourier transform
In this section we'll use the Fourier transform to find the quantizers, under the assumption that R = C. Below we list necessary formulae from representation theory of groups (see for example [6]). Denote byĜ the dual of G which is the set of equivalence classes of the irreducible representations of G. For each α ∈Ĝ we pick the corresponding irreducible representation on E α , dimE α = d α , and an explicit realisation of this representation by a span the group algebra C[G] and C[G] is isomorphic as an algebra to a direct sum of matrix algebras by the Fourier transform We'll considerf as a "function" on the dual group which at each point α ∈Ĝ takes values in End(E α ): and The inverse of the Fourier transform has the following form As we have seen the quantizers are elements of In this case the Fourier transform and its inverse have the following forms and Let χ α (g) = T r(D α (g)) be the character of the irreducible representation E α , α ∈Ĝ.
Splitting of the tensor product of E α ⊗ E β into a sum of irreducible representations we get isomorphisms where c γ αβ ∈ N are called the Clebsch-Gordan integers. These integers can be computed as follows Projections p α of G-modules E = α∈Ĝ c α E α onto its irreducible components c α E α = E (α) are the following where E (α) ≃ C c α ⊗ E α . They satisfy orthogonality conditions For the tensor products (24) the projectors take the form The matrix representation is

The Fourier transform on quantizers
We now rewrite the coherence condition (3) for quantizers in terms of their Fourier transforms. Let q = g,h∈G q g,h (g, h) be a quantizer and The operatorsq α,β : Therefore, due to isomorphisms ν α,β , eachq α,β is a direct sum of operatorsq γ α,β : c γ αβ E γ → c γ αβ E γ . Note that the operatorsq γ α,β are given by c γ αβ × c γ αβ -matrices. Rewriting the coherence condition in terms of these operators we get the following result.
Theorem 4 Let q be a quantizer on the monoidal category M od C (G). Then the coherence condition diagram (3) under the Fourier transform take the following form Assuming that our category is strict we get the following conditions for the quantizers.
Theorem 5 The set of operatorsq γ α,β ∈ End(c γ αβ E γ , c γ αβ E γ ) defines a quantizer if and only if these operators are solutions of the following system of quadratic equations: for all α, β, γ ∈Ĝ.

Finite abelian groups
Let G be a finite abelian group and R = C.
In [1,2] we investigated regular quantizations of modules with action and coaction by finite abelian groups. In this section we shall revisit this case by using the Fourier transform.
By theorem 1 the quantizations of G-modules have the form x ⊗ y → q · (x ⊗ y) for elements x ∈ X, y ∈ Y in G-modules X and Y where q = g,h∈G q g,h (g, h) ∈ C[G × G].
LetĜ be the dual of G. All irreducible representations of G are 1-dimensional and identified with characters α ∈ Hom(G, C * ) =Ĝ.
The Fourier transform has the form . The inverse of this Fourier transform is Then is an operatorq Corresponding to theorem 5 we thus have the following conditions onq α,β q α·β,γqα,β =q α,β·γqβ,γ (37) q 0,α =q α,0 = 1 (38) for all α, β, γ ∈Ĝ where the first condition is given by the coherence condition and the second is the normalization condition. Denote by q(Ĝ) the group of all functions satisfying these conditions. We see that they are 2-cocycles. Hence q(Ĝ) is represented by the multiplicative 2-cocyclesq onĜ with coefficients in C * , wherê q(α, β) =q α,β .
Theorem 6 Let G be a finite abelian group. Then the group of regular quantizations Q • (Ĝ) is isomorphic to the 2nd multiplicative cohomology group H 2 (Ĝ, C * ). Moreover, any 2-cocycle z ∈ Z 2 (Ĝ, C * ) defines an quantizer q z in the following way (40)

Quantizations of S 3 -modules
We consider the symmetric group G = S 3 . Let the representatives of the orbits of the adjoint action be (), (1,2) and (1, 2, 3) and let χ 0 , χ 1 and χ 2 be the characters of the irreducible representations corresponding to these orbits. These irreducible representations are the trivial, sign and standard representations on modules E 0 , E 1 and E 2 with matrix realisations D 0 , D 1 and D 2 respectively.
The operators D 2,2 i : E i → E i , i = 0, 1, 2 are the components of D 2,2 corresponding to the decomposition of the tensor product Proof. The multiplication table for the characters of S 3 is and by (24) we get the multiplication table for irreducible representations where the irreducible representations E 0 , E 1 , E 2 are 1, 1 and 2 dimensional respectively. By (30) the quantizersq ij in End(E i ⊗ E j ) are decomposed as followŝ q 11 =q 0 11 , q 12 =q 2 12 , q 21 =q 2 21 , q 22 =q 0 22 ⊕q 1 22 ⊕q 2 22 .
Theorem 5 for triple tensor products of all combinations of E 0 , E 1 , E 2 gives the following relations (see the appendix for the details of the calculations) The action of the group U (S 3 ) have the following form: wherel 0 = 1 andl 1 ,l 2 ∈ C * .
If the quantizers all are nonzero we may chosel 1 ,l 2 such thatq 12 ,q 2 22 → 1, by (48-49) then alsoq 11 ,q 21 → 1 and by (50)q 0 22 =q 1 22 . We then have the following sequence of quantizers depending on one parameter λ ∈ Ĉ q 11q12q21q 0 22q 1 22q 2 22 Equivalently, the representatives can be chosen as follows: If one or both of the quantizersq 12 ,q 2 22 are equal zero then rest will either be equal to 0 or map to 1 by choosing l 1 , l 2 appropriately.
Remark that from (52) the quantizer q a has the following equivalent forms 6 Quantizations of A 4 Let G = A 4 be the alternating group. Elements (), (12)(34), (123), (132) represent the orbits of the adjoint G-action and let χ 0 , χ 1 , χ 2 and χ 3 be the characters of the irreducible representations corresponding to these orbits. These irreducible representations are the trivial representation, the first and second nontrivial onedimensional representations and the three-dimensional irreducible representation on modules E 0 , E 1 , E 2 and E 3 with matrix realisations D 0 , D 1 , D 2 and D 3 respectively.
By choosingl 3 we reduce the matrix M to one of the following matrices P :