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Journal of Generalized Lie Theory and Applications
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Quantizations of Group Actions

Hilja L. Huru1* and Valentin V. Lychagin2

1Department of Sports and Science, Finnmark University College, 9509 Alta, Norway

2Department of Mathematics and Statistics, University of Tromsø, 9037 Tromsø, Norway

*Corresponding Author:
Hilja L. Huru
Department of Sports and Science,
Finnmark University College,
9509 Alta, Norway
E-mail:
[email protected]

Received date: 18 March 2012, Revised date: 06 June 2012 Accepted date: 11 June 2012

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Abstract

 We describe quantizations on monoidal categories of modules over finite groups. Those 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 give explicit forms for all quantizations.

Introduction

In [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 equation 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 qQ 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.

Let equation be the dual of G. We use the Fourier transform to reconsider these quantizers as sequences of operators equation in End equation where Eα,Eβ are irreducible representations corresponding to equation The coherence condition for the operators equation 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 qQ in C[G×G] realizing the non-trivial quantizations in the category.

To sum up, the procedure is as follows:

equation

We illustrate this method for abelian groups (see, e.g., [2]) and the permutation groups S3 and A4. For S3 there is a 1-parameter family of quantizations. For A4 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 equation and two natural isomorphisms: an associativity constraint assocCequation and a braiding equation 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 a functor. A quantization Q of the functor Φ is a natural transformation of the tensor bifunctor

equation

that satisfies the following coherence condition:

equation

for X,Y ∈ Obj(C) (see [4] for details).

Note that 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

equation

commutes.

The orbits of the above action we denote by Q(Φ).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 IdC : CC, we call Q a quantization of the category C and use the notations q(C) = q(IdC) and Q(C) = Q(IdC).

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 QΦ,QΨ be quantizations of Φ and Ψ. The following formula defines the quantization

equation

of the composition Ψ◦Φ.

We call QΦ◦Ψ composition of quantizations. Then the composition defines an associative multiplication

q(Φ)×q(Ψ) −→ q(Ψ◦Φ)

on the sets of quantizations.

In particular, the composition (2.3), where Φ = Ψ = IdC, defines a multiplication

q(C)×q(C) −→ q(C)

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

equation

commutes, where

equation

Then σQ is a braiding too.

Algebras. Let A be an associative algebra in the category C with multiplication equation Then we define a new multiplication μQ on A by requiring that the following diagram

equation

commutes.

Then (A,μQ) 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 C with multiplication equation Let Q ∈ q(C). We define a quantized multiplication νQ on E by requiring that the following diagram

equation

commutes.

Then (E,νQ) is a module over the quantized algebra (A,μQ). We call it the quantized module.

Right modules are quantized in a similar way.

Coalgebras. Let A* be a coalgebra in the category C with comultiplication μ* : A* →A* ⊗A*. Let Q ∈ q◦(C). We define a new comultiplication (μ*)Q on A* by requiring that the following diagram

equation

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 C with multiplication equation and comultiplication equation The quantized bialgebra is the same object BQ = B equipped with the quantized multiplication equation and quantized comultiplication equation 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 ModR(G) the monoidal category of finitely generated G-modules over R and let q(G) and Q(G) be the sets of quantizations and orbits of this category.

Let X and Y be G-modules. Recall that the tensor product X equation Y over R is a G-module with action

equation

for g ∈ G,x ∈ X,y ∈ Y .

Let R[G] be the group algebra of G over R.

There is an isomorphism between the categories of G-modules and R[G]-modules, ModR(G) = ModR(R[G]) (see, e.g., [2]).

Hence,

q(G) = q(ModR(R[G])), Q(G) = Q(ModR(R[G])).

Theorem 1. Any quantization Q ∈ q(G) of the category of G-modules has the form

equation

where

equation

are elements of the group algebra R[G×G], for x ∈ X, y ∈ Y , and X,Y ∈ Obj(ModR(G)).

Proof. We identify elements x ∈ X for X ∈ Obj(ModR(G)) with morphisms

equation

Then, for any elements x ∈ X,y ∈ Y , X,Y ∈ Obj(ModR(G)) and a quantization Q ∈ q(G), the following diagram

equation

commutes. Therefore,

equation

where equation

equation

We call the elements qQ quantizers and identify q(G) with the set {qQ ∈ R[G×G]}.

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:

(i) the coherence condition

equation

where Δ is the diagonal in R[G]×R[G];

(ii) the normalization condition

equation

(iii) the naturality condition

q ·Δ(g) = Δ(g) · q, (3.3)

for g ∈ G.

Proof. The coherence condition follows from (2.1) where

equation

is represented as follows:

equation

and similarly for QX,Y Z.

The two other conditions follow straightforwardly from the normalization and naturality conditions on the quantizations.

See also [4] for similar settings.

We will from now on use the notion quantizer instead of quantization.

Remark that the conditions (3.1), (3.2), and (3.3) are some kind of 2-cyclic condition on q(G) (see, e.g., Section 6 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):

equation

where l ∈ U(G).

Proof. Representing as above elements x ∈ X by morphisms Øx : R[G] → X, we get the following commutative diagram with λX : X →X:

equation

for any unit preserving natural isomorphism of the identity functor, λ : IdModR(R[G]) →IdModR(R[G]).

Therefore, λ is uniquely defined by elements l ∈ λR[G](1), and

equation

Let q ∈ q(G). Then the action (2.2) gives

equation

with l ∈ U(G).

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 will 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, e.g., [6]).

Denote by equation the dual of G. For each α ∈ equation we pick the corresponding irreducible representation on Eα, dimEα = dα, and an explicit realization of this representation by a dα ×dα-matrix equation for each g ∈ G.

The elements equation 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

equation

We will consider equation as a “function” on the dual group which at each point equation takes values in End(Eα):

equation

and

equation

The inverse of the Fourier transform has the following form:

equation

where the * denotes the adjoint.

As we have seen the quantizers are elements of C[G×G].

The dual of G×G is equation and

equation

for equation with the action

equation

In this case, the Fourier transform

equation

and its inverse have the following forms:

equation

equation

where

equation

and

equation

Let χα(g) = Tr(Dα(g)) be the character of the irreducible representation equation

Splitting of the tensor product of equation into a sum of irreducible representations, we get isomorphisms

equation

where equation are the Clebsch-Gordan integers.

These integers can be computed as follows:

equation

Projections pα of G-modules equationonto its irreducible components cαEα =E(α) are the following:

equation

where equation They satisfy orthogonality conditions

equation

For the tensor products (4.3), the projectors take the form

equation

The matrix representation is equation where

equation

The Fourier transform on quantizers

We now rewrite the coherence condition (2.1) for quantizers in terms of their Fourier transforms.

equation

equation

The operators equation are G-morphisms.

Therefore, due to isomorphisms να,β, each equation is a direct sum

equation

of operators equation

Note that the operators equation are given byequation -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 ModC(G). Then the coherence condition diagram (2.1) under the Fourier transform takes the following form:

equation

where equation ∈ End equation and equation ∈ End equation

Assuming that our category is strict, we get the following conditions for the quantizers.

Theorem 5. The set of operators equation

equation

if and only if these operators are solutions of the following system of quadratic equations:

equation

for all α,β,γ ∈equation

Proof. The first condition follows from Theorem 4.

The second condition is the normalization condition, where equation

equation

Let equation be the Fourier transform of l ∈ U(G), where equation due to the Shur lemma, and l0 = 1.

Then action (3.4) can be rewritten as follows:

equation

where equation

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 will revisit this case by using the Fourier transform.

By theorem 1 the quantizations of G-modules have the form equation for elements x ∈ X,y ∈ Y in G-modules X and Y where equation

Let equation be the dual of G. All irreducible representations of G are 1-dimensional and identified with characters equation

The Fourier transform has the form

equation

for equation The inverse of this Fourier transform is

equation

Then

equation

is the operator

equation

where equation

Clearly equation

Corresponding to Theorem 5 we thus have the following conditions on equation

equation

for all equation where the first condition is given by the coherence condition and the second is the normalization condition.

Denote by equation the group of all functions satisfying these conditions. We see that they are 2-cocycles.

Hence equation is represented by the multiplicative 2-cocycles equation with coefficients in C *, where

equation

The C *-action (5.2) on the operators equation has the form

equation

where equation

Summing up, we get the following result.

Theorem 6. Let G be a finite abelian group. Then the group of regular quantizations equation is isomorphic to the 2nd multiplicative cohomology group equation

Moreover, any 2-cocycle equation defines a quantizer qz in the following way:

equation

Quantizations of S3-modules

We consider the symmetric group G = S3. Let the representatives of the orbits of the adjoint action be (), (1,2), and (1,2,3) and let χ0, χ1, and χ2 be the characters of the irreducible representations corresponding to these orbits. These irreducible representations are the trivial, sign, and standard representations on modules E0, E1, and E2 with matrix realizations D0, D1, and D2, respectively.

Theorem 7. For S3-modules over C the set of quantizers Q(S3) consists of the following:

(i) the trivial quantizer q = 1;

(ii) the 1-parameter family of quantizers

equation

where p ∈ C;

(iii) the discrete set of discrete quantizers

equation

The operators equation re the components of D2,2 corresponding to the decomposition of the tensor product equation

Proof. The multiplication table for the characters of S3 is

equation

and by (4.3) we get the multiplication table for irreducible representations

equation

where the irreducible representations E0, E1, and E2 are 1, 1, and 2 dimensional, respectively.

By (5.1) the quantizers equation in End equation are decomposed as follows:

equation

By normalization condition

equation

Theorem 5 for triple tensor products of all combinations of E0,E1,E2 gives the following relations (see the appendix for the details of the calculations):

equation

The action of the group U(S3) has the following form:

equation

where equation

If the quantizers all are nonzero, we may choose equation such that equation by (7.1), (7.2) then also equation and by (7.3) equation We then have the following sequence of quantizers depending on one parameter λ ∈ C:

equation

Equivalently, the representatives can be chosen as follows:

equation

If one or both of the quantizers equation are equal to zero, then the rest will either be equal to 0 or map to 1 by choosing l1, l2 appropriately.

By the conditions (7.1)–(7.3), the quantizers vary as follows:

equation

We now apply the inverse Fourier transform (4.2) to the quantizers (a)–(g) and get the corresponding element q in the group algebra,

equation

where equation Since Tr(trivial(g)*) is 1 for all g ∈ S3; Tr(sign(g)*) is 1 for (), (1,2,3), and (1,3,2) and −1 for (1,2), (1,3), and (2,3); and Tr(D2(g)*) is 2 for (), −1 for (1,2,3) and (1,3,2) and 0 for (1,2), (1,3), and (2,3), most of the sum cancels out.

Then for option (a) we let equation for p ∈ C and get the following class of quantizers:

equation

In addition to this case, the combinations (b)-(e) give the following quantizers:

equation

The combinations (f) and (g) give the trivial quantizer.

Remark that from (7.4) the quantizer q(a) has the following equivalent forms:

equation

Quantizations of A4

Let G=A4 be the alternating group. Elements (), (12)(34), (123), (132) represent the orbits of the adjoint G-action and we let χ0, χ1, χ2, and χ3 be the characters of the irreducible representations corresponding to these orbits.

These irreducible representations are the trivial representation, the first and second nontrivial one-dimensional representations, and the three-dimensional irreducible representation on modules E0, E1, E2, and E3 with matrix realizations D0, D1, D2, and D3, respectively.

Theorem 8. For A4-modules, the set of quantizers Q(A4) consists of the following:

(i) the trivial quantizer q = 1,

(ii) equation

where equation

equation

where P are 2×2-matrices of the form equation

The operators equation i = 0,1,2,3, are components of D3,3 corresponding to the decomposition equationequation

Proof. The multiplication table for the characters of A4 has the form

equation

and by (4.3) we get the multiplication table for irreducible representations

equation

Recall that the irreducible representations E0, e1, E2, E3 are 1, 1, 1, and 3 dimensional, respectively.

By (5.1), the quantizers equation in End equation split as follows:

equation

where we use the notation equation to keep in mind that this is a 2×2-matrix acting on 2E3.

Further, by normalization condition

equation

Theorem 5 for triple tensor products of all combinations of E0,E1,E2,E3 give the following relations (see the appendix for details of the calculations)

equation

Moreover, the action of the group U(A4) has the form

equation

where equation

Assume that the quantizers are non-zero. Then we may choose equation in such a way that all

equation

are equal to 1.

What remains is the 2×2-matrix equation which by choosing of basis can be transformed to one of the following forms:

equation

where λ,κ ∈ C.

Hence we have the sequence

equation

Assume now one or more of the quantizers equation may be equal to zero. The rest will then map to 1 by choosing equation properly.

By choosing equation we reduce the matrix M to one of the following matrices P:

equation

Finally by the conditions, (8.1) give the following possible sequences:

equation

Applying the inverse Fourier transform (4.2) to the sequences, we get the corresponding element in the group algebra,

equation

where, as mentioned, most of the sum cancels out.

Here equation are components of D3,3 operating on the decomposition equation

Writing M +I instead of M in the sequence (a) we get a shorter form for q:

equation

Further we get

equation

The combinations (f), (h), (j), and (k) give the same quantizer as (b), the combinations (g) and (c), (i) and (d) give the same quantizers.

We adjust the constants and have the result of the theorem.

Appendix

Let G = S3. By Theorem 5 the quantizers on tensor products of triples of irreducible representations satisfy

equation

Then by (A.1a)

equation

by (A.1d)

equation

and by for example (A.1e)

equation

Let G = A4. By Theorem 5 the quantizers on tensor products of triples of irreducible representations satisfy

equation

where equation is a 2×2-matrix on the 6-dimensional 2E3.

Then by (A.2b) and (A.2c),

equation

by (A.2l)

equation

which together with (A.2j) gives

equation

further by (A.2o)

equation

and the last conditions are given by (A.2k) and (A.2r) as follows:

equation

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