Page 147 - Textos de Matemática Vol. 39

QUASIASSOCIATIVE ALGEBRAS 137
2. Quasirepresentations and quasilinear algebra
We start this section with the definition of representation of a G-graded quasiassociative algebra which leads us to the study of what we might call ‘Quasilinear’ Algebra.
Definition 2.1. [3] Let G be a group and A be a G-graded quasiassociative algebra with structure given by a cocycle φ. A representation or ‘action’ of the G-graded quasiassociative algebra A is a G-graded vector space V and a degree- preserving map ◦ : A⊗V → V such that (ab)◦v = φ(a¯,¯b,v¯)a◦(b◦v), 1◦v = v, for homogeneous elements a ∈ Aa¯, b ∈ A¯b, v ∈ Vv¯.
Let G be a group of order n with a cocycle φ. An action of the G-graded quasiassociative algebra A can be expressed as an algebra map ρ : A → Mn,φ(K), where Mn,φ(K) is a ‘deformation’ of the associative matrix algebra Mn(K) of n × n matrices with elements in K [3].
To define Mn,φ(K), we consider a grading function |i| ∈ G for i = 1, · · · , n. Let Eij be usual basis element of Mn(K) with 1 in (ij)-entry and 0 elsewhere. The vector space Mn(K) becomes a G-graded quasiassocia- tive algebra by establishing the degree of Eij as |i||j|−1 ∈ G and defining the multiplication by
n φ(|i|, |p|−1, |p||j|−1)
(X · Y )ij = xipypj φ(|p|−1, |p|, |j|−1) , ∀ X, Y ∈ Mn,φ(K).
p=1
This quasiassociative algebra is denoted by Mn,φ(K).
Based on this approach it becomes natural to consider a ‘Quasilinear’ Algebra setting. In case φ(x,y,z) = 1, for all x,y,z ∈ G, we are back in the traditional associative Linear Algebra setting. As an example we will study a ‘quasi-LU’ decomposition for elements in Mn,φ(K). This generalizes the well- known LU decomposition of a matrix in the associative algebra of square n × n matrices.
For simplicity, from now on we will write XY for X · Y, and φ(i, j, k) for φ(|i|, |j|, |k|), for i, j, k = 1, ..., n.
Lemma 2.2. The algebra Mn,φ(K) has identity Iφ=[ekj ], where, for i, j = 1,...,n, eij = φ(i−1, i, i−1) if i = j, and 0 otherwise.