Degenerations of nilpotent associative commutative algebras

Degenerations of nilpotent Lie algebras

In this paper we study degenerations of nilpotent Lie algebras. If λ, μ are two points in the variety of nilpotent Lie algebras, then λ is said to degenerate to μ , λ→deg μ , if μ lies in the Zariski closure of the orbit of λ . It is known that all degenerations of nilpotent Lie algebras of dimension n < 7 can be realized via a one-parameter subgroup. We construct degenerations between characte...

Degenerations of 7-dimensional Nilpotent Lie Algebras

We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of complex Lie algebra laws in low dimen...

A Class of Locally Nilpotent Commutative Algebras

This paper deals with the variety of commutative nonassociative algebras satisfying the identity Lx + γLx3 = 0, γ ∈ K. In [2] it is proved that if γ = 0, 1 then any finitely generated algebra is nilpotent. Here we generalize this result by proving that if γ 6= −1, then any such algebra is locally nilpotent. Our results require characteristic 6= 2, 3.

Dimension Computations in Non-Commutative Associative Algebras

ON m-STABLE COMMUTATIVE POWER- ASSOCIATIVE ALGEBRAS

A commutative power-associative algebra A of characteristic >5 with an idempotent u may be written1 as the supplementary sum ^=^4„(l)+4u(l/2)+^4u(0) where 4U(X) is the set of all xx in A with the property xx« =Xxx. The subspaces Au(l) and .4K(0) are orthogonal subalgebras, [AU(1/2)]2QAU(1)+AU(0) andAu(K)Au(l/2) C4„(l/2)+^4u(l—X) forX=0, 1. The algebra A is called w-stable if 4u(X)-4„(l/2)C.4u(l...