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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Quantizations and classical non-commutative non-associative algebras

Hilja Lisa HURU and Valentin LYCHAGIN

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

Received August 02, 2007 Revised September 26, 2007

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Abstract

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.

The monoidal category of graded modules

Throughout this paper, let M be a finite commutative monoid. Let k be a commutative ring with unit. Denote by kM-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. kM-mod has the strict monoidal structure where the tensor product of two objects image andimage is the k-module image with grading

image

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 image

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

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

image

if image 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 image

image

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 image For any function f ∈ k (M) we define the action of f on X as follows

image

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 image be the dual group of G consisting of all group homomorphisms image is the 1-dimensional torus.

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

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

The Fourier transform F is the algebra isomorphism between image

image

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

Associativity constraints, braidings and quantizations

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

image

that satisfies the Mac Lane coherence condition.

A braiding of a monoidal category C is a natural isomorphism

image

for X, YOb (C), which preserves the unit and the associativity constraint, imageimage Let A ∈ Ob (C) be an algebra with multiplication image 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

image

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

Any braiding α can be quantized as follows

image

and αq is a braiding too.

We define a quantization Aq of an algebra A given by a quantization q in the category C to be the same object A equipped with a new multiplication

image

image is an algebra, see [6]. If an algebra A is α-commutative, then Aq is α- commutative. If X is a left A-module in the category with the action image then by a quantization Xq of the A-module X we mean the same object X equipped with a new action

image

image 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

image

where image is a normalized 3-cocycle,image 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 3rd cohomology group H3 (M,U (k)).

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

image

where image 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 2nd cohomology group, H2 (M,U (k)).

Theorem 3. A braiding α in the category kM-mod a normalized 2-cochain of M such that image that is, any braiding in kM-mod is a symmetry. Furthermore, the following equations are satisfied

image (2.1)

image (2.2)

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

image

Cohomology of the Sierpinski topology

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 image where the point a is open and the point b is closed. Hence, the topology τ consists of image Denoteimage by 0, {a} by a and Ω by 1, then the monoid structure on τ is given by

image

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

image

where A0 is a k-algebra, Aa and A1 are A0-algebras and, in addition, A1 is an Aa-algebra. Straight forward calculations, see [2], show that cohomology groups of the Sierpinski topology with coefficients in an abelian group G are the following:

image

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 image on the Sierpinski topology ¿ has the following form:

image

where image are functions such that image and imagefor all other image

The power algebra image

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

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

The Fourier-Hadamard transform

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

image

are projectors such that image is aimage -graded k-module.

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

image

for all image If we define

image

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

image

This is the Fourier transform forimage we define an algebra isomorphism

image

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

The cohomology group of image with coefficients inimage

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

image

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

image

where image is the polynomial algebra with coefficients inimage

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

image

by the K¨unneth formula, we get

image

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

image

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

image

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

Aut (Ω)-invariant cohomology groups

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

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

image

image and XB 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 image where if P is symmetric and therefore is a polynomial of the elementary symmetric polynomials s1, . . . , sn in n variables l1, . . . , ln.

The Aut (Ω)-invariant second cohomology and quantizations

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

image

Quantizations of the two-point algebra

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

Consider invariant quantizations of the image- graded algebraimage(the two-point algebra) with the basis {1, e}, with the property e2 = 1. We get the quantized image algebra image by introducing a new multiplication, image

image

image has as the basis {1, e} with the property e2 = −1, hence If we now again quantize

image by q we are back at the monoidal algebra image

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 imagehas four elements.

Let imagebe the four-point algebra of rank 4 over image with the basis image where e1 corresponds to the grading by (1, 0) and e2 to (0, 1) and has the properties imageimageWe get the following quantizations of this algebra.

The matrix algebra,image Take P = s2 = l1l2. Then the quantized algebra As2 is the algebra with properties

image

This algebra is isomorphic to image

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

image

Quaternions,image Then in the quantized algebraimage is the algebra with properties

image

which is isomorphic to the quaternion algebra image

Remark 1. The 3 algebras we now have described, image make out the complete list of the semisimple image-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 image For the cohomology class represented by P = s2 the quantized algebra imagehas a basis consisting of all combinations imagesatisfying

image

imageis isomorphic to the Clifford algebra image

For imagethe algebra imagehas the same basis but with the properties

image

Further, for imagewe get the algebra image with the basis as above with the properties

image

image is isomorphic to the Clifford algebra Cn.

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

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

image

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

image

We require image

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

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

Only image does satisfy the condition P (i, i, i) = 0, but s1s2 = 0 as a function over image

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

The Cayley algebra

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

image

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

Let 1, e1, e2 and e3 have the gradings (0, 0, 0), (1, 0, 0), (0, 1, 0) and (0, 0, 1) respectively in imageWe see that the Cayley algebra is α-associative with respect to the associativity constraint

image

and is σ-commutative with respect to the symmetry

image

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

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

image

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

image

Note that image has the basis consisting of all combinations image satisfying

image

image

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

image

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

image

image -commutative with respect to the symmetry

image

The quantization image gives the algebraimage with the properties

image

which is σ-commutative with respect to the symmetry

image

Furthermore, the quantization image producesimage with the properties

image

which is imagecommutative, with respect to

image

References

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