Polynomial Identities and Enveloping Algebras for n-ary Structures

Abstract

This thesis is devoted to studying the polynomial identities of alternating quaternary algebras structures, and the universal associative enveloping algebras of the (n+1)-dimensional n-Lie (Filippov) algebras, the 2-dimensional non-associative triple systems and the anti-Jordan triple system of n x n matrices. Firstly, we determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl2(C) as a direct summand of its fourth exterior power. The multiplicity is 1 (resp. 2) if and only if n = 4, 6 (resp. n = 8, 10). For these n we determine the multilinear polynomial Identities of degree <= 7 satisfied by the sl2(C)-invariant alternating quaternary algebra structures obtained from the projections of the fourth exterior power of V(n) to V(n). Secondly, we study the universal associative enveloping algebras of n-Lie algebras. For n even and any (n+1)-dimensional n-Lie algebra, we construct a universal associative envelope and establish a generalization of the Poincare-Birkhoff-Witt theorem for universal envelopes using noncommutative Grobner bases. We provide computational evidence that the construction is much more difficult for n odd. Thirdly, we construct universal associative envelopes for the non-associative triple systems arising from trilinear operations applied to the 2-dimensional simple associative triple system. We use noncommutative Grobner bases to determine the monomial bases, the structure constants, and the centers of the universal envelopes. For the finite dimensional envelopes, we determine the Wedderburn decompositions and classify the irreducible representations. Finally, we show that the universal associative envelope, of the simple anti-Jordan triple system of all n x n matrices (n > = 2) over an algebraically closed field of characteristic 0, is finite dimensional. We investigate the structure of the universal envelope and focus on the monomial basis, the structure constants, and the center. We explicitly determine the decomposition of the universal envelope into matrix algebras. We classify all finite dimensional irreducible representations of this simple anti-Jordan triple system, and show that the universal envelope is semisimple.