In this paper and its sequel we continue our study of nilpotent symplectic alternating algebras. In particular we give a full classification of such algebras of dimension 10 over any field. It is known that symplectic alternating algebras over GF(3) correspond to a special rich class C of 2-Engel 3-groups of exponent 27 and under this correspondance we will see that the nilpotent algebras corre...

We develop a method of creating skew congruences on subpowers of finite algebras using groups of twin polynomials, and apply it to the investigation of residually small varieties generated by nilpotent algebras. We prove that a residually small variety generated by a finite nilpotent (in particular, a solvable Eminimal) algebra is weakly abelian. Conversely, we show in two special cases that a ...

A matrix M is nilpotent of index 2 if M = 0. Let V be a space of nilpotent n × n matrices of index 2 over a field k where card k > n and suppose that r is the maximum rank of any matrix in V . The object of this paper is to give an elementary proof of the fact that dim V ≤ r(n−r). We show that the inequality is sharp and construct all such subspaces of maximum dimension. We use the result to fi...

We consider a reductive dual pair (G,G′) in the stable range with G′ the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent K ′ C-orbits, where K ′ is a maximal compact subgroup of G′ and we describe the precise KC-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair (G,K). As an application, we pro...

The integral representation algebra A(RG) is C ®z a(RG). When does a(RG) contain nontrivial nilpotent elements? Let | G\ = pn, where p\n, p prime. Denote by Zp the £-adic valuation ring in Q, and by Zp* its completion. Reiner has shown (i) If a = l , then A(ZPG) and A(Z*G) have no nonzero nilpotent elements (see [ l ] ) . (ii) If ce^2, and G has an element of order p, then both A(ZPG) and A{Z*G...

We study the orbits of G = GL(V ) in the enhanced nilpotent cone V ×N , where N is the variety of nilpotent endomorphisms of V . These orbits are parametrized by bipartitions of n = dimV , and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition a...

We exhibit counterexamples to a Conjecture of Nesin, since we build a connected solvable group with finite center and of finite Morley rank in which no normal nilpotent subgroup has a nilpotent complement. The main result says that each centerless connected solvable group G of finite Morley has a normal nilpotent subgroup U and an abelian subgroup T such that G = U o T , if and only if, for any...

If R is a binomial ring, then a nilpotent R-powered group G is termed power-commutative if for any α ∈ R, [gα, h] = 1 implies [g, h] = 1 whenever gα 6= 1. In this paper, we further contribute to the theory of nilpotent R-powered groups. In particular, we prove that if G is a nilpotent R-powered group of finite type which is not of finite π-type for any prime π ∈ R, then G is PC if and only if i...

We prove that any Novikov algebra over a field of characteristic [Formula: see text] is Lie-solvable if and only its commutator ideal right nilpotent. also construct examples infinite-dimensional algebras with non-nilpotent text].