Medical, Pharma, Engineering, Science, Technology and Business

**Hilja L. Huru ^{1*} and Valentin V. Lychagin^{2}**

^{1}Department of Sports and Science, Finnmark University *C*ollege, 9509 Alta, Norway

^{2}Department of Mathematics and Statistics, University of Tromsø, 9037 Tromsø, Norway

- *Corresponding Author:
- Hilja L. Huru

Department of Sports and Science,

Finnmark University*C*ollege,

9509 Alta, Norway

E-mail:

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

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

In [1,3], we found the quantizations of the monoidal categories of modules graded by finite abelian groups. *Q*uantizations
are natural isomorphisms of the tensor bifunctor 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.

*Q*uantizations *Q* of the monoidal category of modules over finite groups G are realized by elements *q _{Q}* in the
group algebra

Let be the dual of G. We use the Fourier transform to reconsider these quantizers as sequences of operators in End where *E _{α}*,Eβ are irreducible representations corresponding to The coherence
condition for the operators 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

To sum up, the procedure is as follows:

We illustrate this method for abelian groups (see, e.g., [2]) and the permutation groups *S _{3}* and

A braided monoidal category *C* is a category equipped with a tensor product and two natural isomorphisms: an
associativity constraint assoc*C* and a braiding for objects
X,Y,Z in *C*, that both satisfy MacLane coherence conditions; see [5].

Let * C,D *be braided monoidal categories and

that satisfies the following coherence condition:

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

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 *I _{dC}* :

For the following result, see also [4].

Let Φ :B→*C*,Ψ: * C→D* be functors between the monoidal categories

of the composition *Ψ◦Φ.*

We call * Q^{Φ◦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 = *I _{dC}*, 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

commutes, where

Then *σ _{Q}* is a braiding too.

**Algebras.** Let A be an associative algebra in the category *C* with multiplication 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

**Modules.** Let E be a left module over the algebra A in the category *C* with multiplication Let
*Q* ∈ q(*C*). 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

Right modules are quantized in a similar way.

** Coalgebras.** Let A* be a coalgebra in the category

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 and comultiplication The quantized bialgebra is the same object *B _{Q}* = B equipped with the quantized
multiplication and quantized comultiplication quantized as above. This is also a bialgebra in the category.

Let R be a commutative ring with unit and let G be a finite group. Denote by Mod* _{R}*(G) the monoidal category of
finitely generated

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

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, Mod* _{R}*(G) = Mod

Hence,

q(G) = q(Mod* _{R}*(R[G])),

**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(Mod* _{R}*(G)).

*Proof. *We identify elements x ∈ X for X ∈ Obj(Mod* _{R}*(G)) with morphisms

Then, for any elements x ∈ X,y ∈ Y , X,Y ∈ Obj(Mod* _{R}*(G)) and a quantization

commutes. Therefore,

where

We call the elements *q _{Q}* quantizers and identify q(G) with the set {

**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*

*where Δ is the diagonal in R[G]×R[G];*

*(ii) the normalization condition*

*(iii) the naturality condition*

* q ·Δ(g) = Δ(g) · q, (3.3)*

*for g ∈ G.*

*Proof. *The coherence condition follows from (2.1) where

is represented as follows:

and similarly for *Q _{X,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):*

where l ∈ U(G).

*Proof. *Representing as above elements x ∈ X by morphisms* Ø _{x}* : R[G] → X, we get the following commutative
diagram with λ

for any unit preserving natural isomorphism of the identity functor, λ : IdMod* _{R}*(R[G])
→IdMod

Therefore, λ is uniquely defined by elements l ∈ λ_{R[G]}(1), and

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

with l ∈ U(G).

We say that two quantizers p, q ∈ q(G) are equivalent if p = l(q) for some l ∈ U(G).

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 the dual of G. For each α ∈ we pick the corresponding irreducible representation on *E _{α}*,
dim

The elements 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 will consider as a “function” on the dual group which at each point takes values in End(*E _{α}*):

and

The inverse of the Fourier transform has the following form:

where the * denotes the adjoint.

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

The dual of G×G is and

for with the action

In this case, the Fourier transform

and its inverse have the following forms:

where

and

Let *χ _{α}*(g) = Tr(D

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

where are the *C*lebsch-Gordan integers.

These integers can be computed as follows:

Projections *p _{α}* of

where They satisfy orthogonality conditions

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

The matrix representation is where

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

The operators are G-morphisms.

Therefore, due to isomorphisms* ν _{α,β}*, each is a direct sum

of operators

Note that the operators are given by -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:*

*where ∈ End and ∈ End *

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

Theorem 5. *The set of operators *

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

*for all α,β,γ ∈*

*Proof. *The first condition follows from Theorem 4.

The second condition is the normalization condition, where

Let be the Fourier transform of l ∈ U(G), where due to the Shur lemma, and *l _{0}* = 1.

Then action (3.4) can be rewritten as follows:

where

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 for elements x ∈ X,y ∈ Y in
*G*-modules X and Y where

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

The Fourier transform has the form

for The inverse of this Fourier transform is

Then

is the operator

where

*C*learly

*C*orresponding to Theorem 5 we thus have the following conditions on

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

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

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

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

where

Summing up, we get the following result.

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

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

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

**Theorem 7. ***For S _{3}-modules over C the set of quantizers Q(S_{3}) consists of the following:*

*(i) the trivial quantizer q = 1;*

*(ii) the 1-parameter family of quantizers*

*where p ∈ C;*

*(iii) the discrete set of discrete quantizers*

*The operators re 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 (4.3) we get the multiplication table for irreducible representations

where the irreducible representations E_{0}, E_{1}, and E_{2} are 1, 1, and 2 dimensional, respectively.

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

By normalization condition

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}*) has the following form:

where

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

Equivalently, the representatives can be chosen as follows:

If one or both of the quantizers are equal to zero, then the rest will either be equal to 0 or map to 1 by
choosing *l _{1}*,

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

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

where Since Tr(trivial*(g)**) is 1 for all
*g ∈ S _{3};* Tr(sign

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

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

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

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

Let G=*A _{4}* be the alternating group. Elements (), (12)(34), (123), (132) represent the orbits of the adjoint G-action
and we let χ

These irreducible representations are the trivial representation, the first and second nontrivial one-dimensional
representations, and the three-dimensional irreducible representation on modules E_{0}, E_{1}, E_{2}, and E_{3} with matrix
realizations D^{0}, D^{1}, D^{2}, and D^{3}, respectively.

**Theorem 8.** *For A _{4}-modules, the set of quantizers Q(A_{4}) consists of the following:*

*(i) the trivial quantizer q = 1,*

*(ii) *

*where *

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

*The operators i = 0,1,2,3, are components of D ^{3,3} corresponding to the decomposition *

*Proof. The multiplication table for the characters of A _{4} has the form*

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

Recall that the irreducible representations E_{0}, e1, E_{2}, E_{3} are 1, 1, 1, and 3 dimensional, respectively.

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

where we use the notation to keep in mind that this is a 2×2-matrix acting on 2E_{3}.

Further, by normalization condition

Theorem 5 for triple tensor products of all combinations of E_{0},E_{1},E_{2},E_{3} give the following relations (see the
appendix for details of the calculations)

Moreover, the action of the group U(*A _{4}*) has the form

where

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

are equal to 1.

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

where* λ,κ ∈ C.*

Hence we have the sequence

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

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

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

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

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

Here are components of D^{3,3} operating on the decomposition

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

Further we get

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.

*Let G = S _{3}. By Theorem 5 the quantizers on tensor products of triples of irreducible representations satisfy*

Then by (A.1a)

by (A.1d)

and by for example (A.1e)

Let G = *A _{4}*. By Theorem 5 the quantizers on tensor products of triples of irreducible representations satisfy

where is a 2×2-matrix on the 6-dimensional 2E_{3}.

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

by (A.2l)

which together with (A.2j) gives

further by (A.2o)

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

- Huru HL (2007) Quantizations of braided derivations. II. Graded modules Lobachevskii. J Math 25: 131–160.
- Huru HL (2007) Quantizations of braided derivations. III. Modules with action by a group Lobachevskii. J Math 25: 161–185.
- Huru HL, Lychagin V (2008) Quantizations and classical non-commutative non-associative algebras. J Gen Lie Theory Appl 2: 35–44.
- Lychagin V (2001) Quantizations of differential equations. Proceedings of the Third World Congress of Nonlinear Analysts, Part 4 (Catania, 2000), Nonlinear Anal 47: 2621–2632.
- Mac Lane S (1998) Categories for the Working Mathematician, vol. 5 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2nd ed.
- Simon B (1996) Representations of Finite and Compact Groups, vol. 10 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI.

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