Universal associative envelopes of (n+1)-dimensional n-Lie algebras

TL;DR

This paper proves the existence of universal associative envelopes for even-dimensional simple n-Lie algebras and constructs these envelopes explicitly, highlighting the complexity differences for odd n.

Contribution

It establishes the existence and explicit construction of universal associative envelopes for even n-Lie algebras, confirming Pozhidaev's conjecture and providing computational methods.

Findings

Universal envelopes exist for even n-Lie algebras.

Constructed U(L) as a quotient of free associative algebra.

Construction is more complex for odd n.

Abstract

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie algebras. More generally, for n even and any (n+1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map from L to U(L) is injective. We use noncommutative Grobner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.