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Department of Mathematics, Tromsø University, N-9037 Tromsø, Norway
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E-mails: **[email protected] and [email protected]

**Received** August 02, 2007 **Revised** September 26, 2007

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In this paper we study quantizations, associativity constraints and braidings in the monoidal category of monoid graded modules over a commutative ring. All of them can be described in terms of the cohomology of underlying monoid. The case when the monoid is a finite topology has the main interest for us. The cohomology classes which are invariant with respect to homeomorphism group produce remarkable algebraic constructions. We study in details the Sierpinski and discrete topology and show the relations with the Clifford algebras, the Cayley algebra and their quantizations. All of them are ®-associative and ¾-commutative for suitable associativity constraints ® and braidings ¾.

Throughout this paper, let *M* be a finite commutative monoid. Let *k* be a commutative ring
with unit. Denote by *k _{M}*-mod the strict monoidal category of M-graded k-modules. The objects
in the category are M-graded k-modules and the arrows are the M-graded morphisms.

The ring *k* is a unit object e as we define *k* to be indexed by 0 ∈ M and components indexed
by *m* ∈ M, *m* 6= 0, are all zeros. Thus we have isomorphisms ¸ and

An algebra in the monoidal category k_{M}-mod is called *M-graded k-algebra*. Here
*μ* and *η* are morphisms of multiplication such that μ is
associative and unit-preserving. Note that μ maps In the same manner one
defines an *M-graded A-module* as a module X with an action in the monoidal
category *k _{M}*-mod.

Let X be a M-graded k-module. Denote by a family of projectors where

if Any such family of projectors determines a *M*-grading on a k-module *X*.
Consider the algebra of all k-valued functions on *M, k (M).* Then functions

for *m,m' ∈ M*, constitute a basis of the algebra. We define a *k (M)*-module structure on the
*M*-graded k-module *X* by putting For any function
*f ∈ k (M)* we define the action of f on X as follows

If *X* is a k-module, then *X* is a *k (M)*-module if and only if it is a M-graded k-module. Hence,
we get an isomorphism between the monoidal category of M-graded k-modules and the category
of *k (M)*-modules.

**Group gradings and actions**

Let *G* be a finite abelian group and be the dual group of G consisting of all group homomorphisms is the 1-dimensional torus.

In this section we will show that -grading gives G-action, and the other way around, and that we have an isomorphism between the -graded modules and G-modules. This isomorphism will be useful in applications of the theory developed in this paper.

Denote by the the basis consisting of Dirac δ-functions in the dual of
the function algebra is the group algebra of *G*.
Note that G-module structure and -module structure induce each other.

The Fourier transform *F* is the algebra isomorphism between

which shows that if *X* is a G-module we get a -module structure on for and, conversely, if X is a -graded -module, then we get an action of This establishes an isomorphism between the
monoidal category of G-modules and -graded modules over , [3].

Recall that an associativity constraint α, see [7], in a monoidal category *C* is a natural isomorphism

that satisfies the Mac Lane coherence condition.

A braiding of a monoidal category *C* is a natural isomorphism

for *X, Y* ∈ *Ob (C)*, which preserves the unit and the associativity constraint, Let *A ∈ Ob (C)* be an algebra with
multiplication We say that A is α -commutative if μ = μ o α, see [5].

A quantization, see [6], of a monoidal category *C* is a natural isomorphism of the tensor
bifunctor

for *X, Y ∈ Ob (C),* which preserves the unit and the associativity,

Any braiding α can be quantized as follows

and αq is a braiding too.

We define a *quantization* *A _{q}* of an algebra

is an algebra, see [6]. If an algebra A is α-commutative, then A_{q} is α-
commutative. If *X* is a left A-module in the category with the action then by
*a quantization* X_{q} of the A-module X we mean the same object X equipped with a new action

is also a left A-module in C, see [6].

The following theorems give the complete description of the associativity constraints, quantizations and braidings in the category of graded modules as in [3].

**Theorem 1.** Any associativity constraint α in the category kM-mod of M-graded k-modules has
the form

*where is a normalized 3-cocycle, with values in the
group of units U (k). Furthermore, the orbits of all associativity constraints under the action
of natural isomorphisms of the tensor bifunctor are in one-to-one correspondence with the 3 ^{rd}
cohomology group H^{3} (M,U (k)).*

**Theorem 2.** *Any quantization q of the category of M-graded k-modules has the form*

*where is a normalized 2-cocycle. Moreover, the orbits of all quantizations
under the action of unit-preserving natural isomorphisms of the identity functor are in one-to-one
correspondence with the 2 ^{nd} cohomology group, H^{2} (M,U (k)).*

**Theorem 3.*** A braiding α in the category k _{M}-mod a normalized 2-cochain of M such that that is, any braiding in k_{M}-mod is a symmetry. Furthermore, the following
equations are satisfied*

(2.1)

(2.2)

*If the associativity constraint α is trivial we get the bihomomorphism conditions*

In this section we calculate the zero, first, second and third cohomology groups of the Sierpinski topology. We will use these groups to find possible quantizations and associativity constraints of the category of modules graded by the Sierpinski topology.

The Sierpinski topology τ is the topology of the Sierpinski set; the two-point set where the point a is open and the point b is closed. Hence, the topology τ consists of Denote by 0, {a} by *a* and Ω by 1, then the monoid structure on τ is given by

Note that, a τ-graded k-algebra can be viewed as algebra

where *A*_{0} is a *k*-algebra,* A*_{a} and *A*_{1} are *A*_{0}-algebras and, in addition, *A*_{1} is an *A*_{a}-algebra.
Straight forward calculations, see [2], show that cohomology groups of the Sierpinski topology
with coefficients in an abelian group *G* are the following:

Hence, for the category of τ-graded *k*-modules there are no non-trivial quantizations, but there
are non-trivial associativity constraints. They have the following description.

**Theorem 4.** *Any 3-cocycle on the Sierpinski topology ¿ has the following
form:*

*where* *are functions such that* *and* *for all other*

Let Ω be a set consisting of n elements Ω = {a1, . . . , an} and let denote the power algebra of Ω, that is, the algebra of all subsets of Ω. has the group structure with respect to symmetric difference The passing to characteristic functions gives us an isomorphism from this group to the additive group of characteristic functions on Ω with values in

From now on we distinguish the following two group structures; the multiplicative group, denoted by and the additive group, denoted by

**The Fourier-Hadamard transform**

Let be a k-module and be a representation. Note that Then the operators

are projectors such that is a -graded *k*-module.

Let be the basis of as a vector space over *k* and let the
basis of We have

for all If we define

then we get - module structure on X. This operator one can consider as a ”change of rings”, see [3], with respect to the algebra isomorphism

This is the Fourier transform for we define an algebra isomorphism

This is the Fourier-Hadamard transform and it allows us to establish an isomorphisms between the monoidal categories of - graded modules and of -modules.

**The cohomology group of with coefficients in**

Here we describe cohomology of the power algebra with coefficients in a form suitable for us. Let We only need to calculate the cohomology with coefficients in Now,

for all r ≥ 0. The trivial cohomology classes are represented by *f* = 1. The non-trivial
cohomology classes are represented by Hence,

where is the polynomial algebra with coefficients in

Let n ≥ 2. The correspondence establishes an
isomorphism between for any monoid *M*. Therefore

by the K¨unneth formula, we get

where each li is the ith coordinate function from to the commutative ring Hence, for the power algebra we get

which is the subspace of the polynomial algebra consisting of all homogeneous polynomials of
degree *r* with coefficients in has the form

is a homogeneous polynomial of degree r and It follows that a representative
*f* for the cohomology classes of degree 1 is represented by a vector in the vector space a representative of the cohomology of degree 2 is represented by a symmetric n × n matrix with entries inand similarly for higher dimensions.

In this section we investigate quantizations and associativity constraints on graded algebras invariant under permutations.

Denote by the group of all automorphisms of Ω. Any introduces an automorphism on the power algebra by

and *X*_{B} is the characteristic function of B on . We need cohomologies represented by
*Aut* (Ω)-invariant cohomology classes, i.e. that are independent on the labelling of the elements
of Ω. They correspond to where if *P* is
symmetric and therefore is a polynomial of the elementary symmetric polynomials *s _{1}*, . . . ,

**The Aut (Ω)-invariant second cohomology and quantizations**

Let a representative of a non-trivial second cohomology class be
Aut (Ω)-invariant. Then *P* is a linear combination of the symmetric polynomials

**Quantizations of the two-point algebra**

For the case *n* = 1, Ω has only one point, and we have only one symmetric
quantization,

Consider invariant quantizations of the - graded algebra(the *two-point
algebra*) with the basis {1, *e*}, with the property e^{2} = 1. We get the quantized algebra
by introducing a new multiplication,

has as the basis {1, *e*} with the property e^{2} = −1, hence If we now again quantize

by *q* we are back at the monoidal algebra

**Quantizations of the four-point algebra**

For n ≥ 2 we have the three non-trivial possibilities for *P* described above. For the case *n* = 2,
Ω consists of two points and has four elements.

Let be the four-point algebra of rank 4 over with the basis where e_{1} corresponds to the grading by (1, 0) and e_{2} to (0, 1) and has the properties We get the following quantizations of this algebra.

**The matrix algebra,** Take *P* = s_{2} = l_{1}l_{2}. Then the quantized algebra A_{s2} is the
algebra with properties

This algebra is isomorphic to

**The tensor algebra,** Then for the quantized algebra is the algebra with properties

**Quaternions,** Then in the quantized algebra is
the algebra with properties

which is isomorphic to the quaternion algebra

**Remark 1.** The 3 algebras we now have described, make out the complete
list of the semisimple -algebras of rank 4, and the two last ones are the Clifford algebras.

**Quantization of the n-point algebra**

For the case *n* ≥ 2 we get 3 different quantizations of the n-point algebra For
the cohomology class represented by *P* = s_{2} the quantized algebra has a basis consisting of
all combinations satisfying

is isomorphic to the Clifford algebra

For the algebra has the same basis but with the properties

Further, for we get the algebra with the basis as above with the properties

is isomorphic to the Clifford algebra *C _{n}*.

**The Aut (Ω)-invariant third cohomology and associativity constraints**

For Ω with n elements let be an Aut (Ω)-invariant representation of a third cohomology class of with coefficients in There are 7 possibilities for Aut (Ω)- invariant P:

The representatives α of cohomology groups of degree 3 are associativity constraints on graded such that

We require

Let Ω consist of one point. Then we have one possible symmetric associativity constraint but this does not satisfy *P (i, i, i)* = 0.

Let Ω consist of two points. We have 3 possibilities for P, that is,

Only does satisfy the condition *P (i, i, i)* = 0, but s_{1}s_{2} = 0 as a function over

so in this case we only have the trivial associativity constraint.

Let Ω consist of three points. For algebras graded by there are 7 possibilities for P, but only satisfies the property

is an alternating algebra of rank 8 over and is in classical point of view nonassociative and noncommutative is obviously graded by .

Let 1, e_{1}, e_{2} and e_{3} have the gradings (0, 0, 0), (1, 0, 0), (0, 1, 0) and (0, 0, 1) respectively
in We see that the Cayley algebra is α-associative with respect to the associativity
constraint

and is σ-commutative with respect to the symmetry

where *g* ((1, 1, 1)) = 1 and *g (x)* = 0 for

If we now quantize , the three non-trivial *Aut* (Ω)-invariant quantizations, the result is new α-associative algebras with commutativities
different from the commutativity of . Denote these byrespectively. All
are α-associative as quantizations preserve the associativity given by α. They have the basis of and all combinations satisfying

Assume is of dimension n. The 2n-dimensional Cayley algebra is in the category of -graded modules equipped with the associativity constraint and the symmetry

Note that has the basis consisting of all combinations satisfying

When is quantized we get the following 3 algebras with the same basis and all are α-associative with respect to such that

for all *i, j, l = 1, . . . , n*. The quantization produces the algebra such that

-commutative with respect to the symmetry

The quantization gives the algebra with the properties

which is σ-commutative with respect to the symmetry

Furthermore, the quantization produces with the properties

which is commutative, with respect to

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- Huru HL (2002) Quantizations and associativity constraints of the category of graded modules. Cand.Scient. Theses in Mathematics, University of Tromsø.
- Huru HL (2006) Associativity constraints, braidings and quantizations of modules with grading and action. Lobachevskii J Math 23: 5–27.
- Husemoller D (1994) Fibre Bundles.Grad. Texts in Math.20, Springer-Verlag, New York.
- Lychagin VV (1998) Calculus and quantizations over Hopf algebras. Acta Appl Math 51: 303–352.
- Lychagin V (2001) Quantizations of differential equations. Nonlinear Analysis 47: 2621–2632.
- Mac Lane S (1998) Categories for the Working Mathematician.Grad. Texts in Math.5, Springer-Verlag,New York.
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