alexa On Contractions of Three-Dimensional Complex Associative Algebras

ISSN: 1736-4337

Journal of Generalized Lie Theory and Applications

  • Commentary   
  • J Generalized Lie Theory Appl, Vol 11(3): 282
  • DOI: 10.4172/1736-4337.1000282

On Contractions of Three-Dimensional Complex Associative Algebras

Mohammed NF1,3, Rakhimov IS1,2* and Sh Said Husain K1,2
1Department of Mathematics, Faculty of Science, Institute for Mathematical Research, UPM, Malaysia
2Laboratory of Cryptography, Analysis and Structure, Institute for Mathematical Research (INSPEM), UPM, Malaysia
3Department of Mathematics, College of Education for Pure Sciences/Ibn-AL-Haithem, University of Baghdad, Iraq
*Corresponding Author: Rakhimov IS, Laboratory of Cryptography, Analysis and Structure, Institute for Mathematical Research (INSPEM), UPM, Malaysia, Tel: 03- 8946 6831, Email: [email protected]

Received Date: Jul 06, 2017 / Accepted Date: Sep 29, 2017 / Published Date: Oct 10, 2017

Abstract

Contraction is one of the most important concepts that motivated by numerous applications in different fields of physics and mathematics. In this work, the contractions of complex associative algebras are considered. We focus on the variety A3(Equation) of all complex associative algebras of dimension three (including nonunital). Various contractions criteria are collected and new criteria are proposed to test the possible existence of contraction for each pair of associative algebras. One of the main tools is the use of the low-dimensional cohomology groups of these algebras. As a result, we prove that the variety A3(Equation) has seven irreducible components, two of dimension 5, four of dimension 7 and one of dimension 9.

Introduction

The notion of contractions was first introduced by Segal [1] and Inonu [2] for Lie algebras. According to references, the contractions can be divided into two major categories. The first one is more physical that deals with the applications of contractions. Another one is pure algebraic that is mainly oriented to the abstract algebraic structure and mathematical background. For associative algebras, Gabriel [3] studied the irreducible components of the algebraic variety of 4-dimensional unital associative algebras. Mazzola’s paper [4] concerns unital associative algebras of dimension five. Classification of lowdimensional nilpotent rigid associative algebras and the description of the irreducible components have been treated by Makhlouf [5,6].

The main purpose of this work is to study the variety of all 3-dimensional complex associative algebras. In the paper, we deal with the algebraic point of view of the contractions. We study orbit closures of the variety of complex associative algebras of dimension three. The paper is organized as follows. Some notations on associative algebras, degeneration, rigidity and contractions of associative algebras are given in Section 2. In Section 3, we list some important invariance arguments for contractions. Calculation and collection of invariance arguments are adduced to conclude the possible existence of contractions for an arbitrary pair of associative algebras in Section 4.

Preliminaries

In this section, we recall some terminology that are used in the paper. Let A=(V,λ) be an algebra of dimension n with an underlying vector space V over a field Equation and product λ:V×VV. Let g:(0,1]→GL(V) be a continuous function. More precisely for any t∈(0,1], a nonsingular linear operator gt on V is assigned. A parameterized family of new isomorphic to A=(V,λ) algebra structures on V is determined as follows:

Equation

Definition 2.1

If for any x,y∈ , the limit Equation exists then algebraic structure λ0 is called a contraction of λ.

A contraction B of A to algebra B is called trivial if B is abelian and improper if B is isomorphic to A. Consider an n3-dimensional vector space Hom(VV,V) formed by bilinear maps V×VV, where V is an n-dimensional vector space over an algebraically closed field Equation denoted byEquation An algebra A=(V,λ) is given as an element λ(A) of Equation through the bilinear mapping λ:VVV. The linear reductive group Equation acts on Equation by

Equation

Under this action, two algebras A and B belong to the same orbit if and only if they are isomorphic. Moreover, we say that algebra A degenerates to algebra B, if B lies in Zariski closure of the orbit of A. This is denoted by Equation.

Definition 2.2

Let A be an algebra over a field Equation. We call A an associative algebra if its bilinear mapping λ satisfies the following condition

Equation

Let Equation be the set of all associative algebra structures on n-dimensional space over a field Equation. The set Equation is an algebraic subset of the affine variety Equation. For a fixed basis {e1,e2,…,en} of the vector space V, the multiplication table of A on this basis is given as a point Equation as follows

Equation

Let A be an associative Equation, D be an A-bimodule and Φ:ApD be a multilinear mapping. The set of all multilinear mappings from Ap to D is called p-dimensional cochain of A and denoted by Cp(A,D). The coboundary homomorphism is a mapping δ(p) from C p(A,D) to C p+1(A,D) given by

Equation

The kernel of the coboundary operator denoted by Z p(A,D) whose elements are called p-cocycles with values in D. The elements of the image of δ(p-1) denoted by B p(A,D) are called p-coboundaries with values in D. The quotient space:

H p (A,D) = Z p (A,D) / Bp (A,D) is called the cohomology space (group) of A of degree p. In this paper, we will consider a particular case that D=A as A-bimodule.

Definition 2.3

An associative algebra A is called geometrically rigid whenever its orbit is Zariski open in Equation and called algebraically rigid if the second cohomology group H2(A,A) is trivial [7].

Invariance Arguments

In this section, we list some invariance arguments which are helpful for studying the variety of a given class of algebras. Let A be an associative algebras over a field Equation. We define:

Ak = λ (Ak −1, A) - the k-th degree of A, where Equation;

R(A) = {xA | λ (A, x) = 0} - the right annihilator of A;

L(A) = {xA | λ (x, A) = 0} - the left annihilator of A;

Z(A) = {xA | λ (x, y) = λ ( y, x),∀yA} - the center of A;

Aut(A) = {d : AA | d(λ (x, y)) = λ (d(x),d( y)),∀x, yA} - the group of automorphisms of A;

SA(A) - the maximal abelian subalgebra of A;

Com(A) - the maximal commutative subalgebra of A;

O(A) - the orbit of A;

Der(A) = {dEnd(A) | d(λ (x, y)) = λ (d(x), y) +λ (x,d( y)),∀x, yA} - the algebra of derivations of A;

rn(A) -the nilpotency rank of associative algebra A;

Hi (A, A) - the ith cohomology group of A;

Z 2 (A, A) - the 2-cocycles of associative algebra A;

Equation is the space of (α,β,γ)-derivations of A, for fixed Equation.

The following theorem is very useful to study the irreducible components of the subvariety Equation of Equation [8,9].

Theorem 3.1

The the following subsets of Equation are closed relative to the Zariski topology for any Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

The proof of 1-4 is the same to that of Lie algebras [10,11]. For the parts 5 and 6, the proof is obtained by the following significant fact: let N be a Zariski closed subset of Equation and A1,A2 in Equation. If A1 lies in N and A1A2 then A2 also lies in A. More precisely, the subset N is not Equation-setwise stabilizer. However, it is B-setwise stabilizer, where B is the Borel subgroup Equation- made up of upper triangular matrices. The statements 7,8 and 9 are equivalent because of the following relation between the dimensions of Equation-- orbits, automorphism’s groups and derivation algebras.

dimO(A) = n2 − dimAut(A) = n2 − dimDer(A)

The proof of 10 is directly coming behind the following fact: let A be an associative algebra. We define the lower central series:

Equation

If A is a nilpotent associative algebra then it has nilpotency rank denoted by rn(A), i.e., it is a minimal positive integer l such that Al=0. It is not hard to see that if A1 degenerates to A2 then dim Equation. The proof of 11 and 12 are the same of Lie algebras [12].

The next corollary is used to reject existence of degenerations for each pair of associative algebras A and B.

Corollary 3.1

If an algebra A degenerates to an algebra B. Then the following conditions are valid:

0.60tw0.60tw

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

Equation

In the sequel, all algebras suppose to be over the field of complex numbers Equation.

The Variety of Complex Associative Algebras of Dimension Three

In this section, we recall the complete list of non-isomorphic classes of three-dimensional complex associative algebras, which was obtained in Rikhsiboev et al. [13] to study the subvariety Equation.

Theorem 4.1

Any 3-dimensional complex associative algebra A is isomorphic to one of the following pairwise non-isomorphic algebras.

Equation

In Table 1, we assemble some contraction invariants of threedimensional associative algebras that were obtained in refs. [14,15].

Table

In Table 2 below we present the values of more contraction invariants.

Table

Table 3 contains the values of contraction invariants applying the algorithm, which was stated in ref. [16] to describe the (α,,γ)- derivations of three- dimensional complex associative algebras.

Table

By using all the criteria presented above, we give in the following table all possibilities of degenerations for 3-dimensional associative algebras. The checkmark denotes that there is a degeneration Equation. The other symbols stand for the reason why such a degeneration is impossible. Indeed, there is more than just one reason for a non-degeneration. However, we have written down only one in the table (Table 4).

Table

According to Table 4, the algebras Equation and are geometrically rigid. More precisely, they are not degeneration of other 3-dimensional associative algebras structures.

Theorem 4.2

The rigid irreducible components of the variety Equation are generated by the algebras Equation and with the dimensions:

As a consequence, Equation = 9.

References

Citation: Mohammed NF, Rakhimov IS, Said Husain SK (2017) On Contractions of Three-Dimensional Complex Associative Algebras. J Generalized Lie Theory Appl 11: 282. Doi: 10.4172/1736-4337.1000282

Copyright: © 2017 Mohammed NF, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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