Let A be an algebra over a field K of characteristic zero, let δ1, . . . , δs ∈ DerK(A) be commuting locally nilpotent K-derivations such that δi(xj) = δij , the Kronecker delta, for some elements x1, . . . , xs ∈ A. A set of algebra generators for the algebra A := ∩i=1ker(δi) is found explicitly and a set of defining relations for the algebra A is described. Similarly, given a set σ1, . . . , ...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G| + 1, where |G| is the order of the commutator subgroup. Previously we determined the groups G for which the upper/lower nilpotency index is maximal or the upper nilpotency index is ‘almost maximal’ (that is, ...

If X is the complement of a hypersurface in P, then Kohno showed in [9] that the nilpotent completion of π1(X) is isomorphic to the nilpotent completion of the holonomy Lie algebra of X. When X is the complement of a hyperplane arrangement A, the ranks φk of the lower central series quotients of π1(X) are known in only two very special cases: if X is hypersolvable (in which case the quadratic c...

Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper an investigation is initiated into which zero-nonzero patterns are potentially nilpotent over F with a special emphasis on the case that F = Zp is a finite field. A necessary condition on F is observed for...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.

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...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G|+ 1, where |G| is the order of the commutator subgroup. The authors have previously determined the groups G for which this index is maximal and here they determine the G for which it is ‘almost maximal’, that ...

In this paper, we show that every finite-dimensional Zinbiel algebra over an arbitrary field is nilpotent, extending a previous result they are solvable by other authors.