We shall say that an automorphism a is nilpotent or acts nilpotently on a group G if in the holomorph H= [G](a) of G with a, a is a bounded left Engel element, that is, [H, ¿a] = l for some natural number ¿. Here [H, ka] means [H, (k — l)a] with [H, Oa] denoting H. Let G' denote the commutator subgroup [G, G], and let $(G) denote the Frattini subgroup of G. If a is an automorphism of a nilpoten...

We state that all Rota-Baxter operators of nonzero weight on Grassmann algebra over a field characteristic zero are projections subalgebra along another one. show the one-to-one correspondence between solutions associative Yang-Baxter equation and matrix $M_n(F)$ (joint with P. Kolesnikov). prove unital (alternative, Jordan) algebraic nilpotent. For an $A$, we introduce its new invariant rb-ind...

Let G be a classical group and let g be its Lie algebra. For a nilpotent element X E g, the ring R(Ox) of regular functions on the nilpotent orbit Ox is a Gmodule. In this thesis, we will decompose it into irreducible representations of G for some spherical nilpotent orbits. Thesis Supervisor: David Alexander Vogan Title: Professor of Mathematics

We study the word length entropy of automorphisms of residually nilpotent groups, and how the entropy of such group automorphisms relates to the entropy of induced automorphisms on various nilpotent quotients. We show that much like the structure of a nilpotent group is dictated to a large degree by its abelianization, the entropy of an automorphism of a nilpotent group is dictated by its entro...

We prove that every ω-categorical, generically stable group is nilpotent-byfinite and that every ω-categorical, generically stable ring is nilpotent-by-finite.

‎Let $L$ be a Lie algebra‎, ‎$mathrm{Der}(L)$ be the set of all derivations of $L$ and $mathrm{Der}_c(L)$ denote the set of all derivations $alphainmathrm{Der}(L)$ for which $alpha(x)in [x,L]:={[x,y]vert yin L}$ for all $xin L$‎. ‎We obtain an upper bound for dimension of $mathrm{Der}_c(L)$ of the finite dimensional nilpotent Lie algebra $L$ over algebraically closed fields‎. ‎Also‎, ‎we classi...

An ideal I of a ring A is essentially nilpotent if I contains a nilpotent ideal N of A such that J 91N # 0 whenever J is a nonzero ideal of A contained in I. We show that each ring A has a unique largest essentially nilpotent ideal EN(A). We study the properties of EN(A) and, in particular, we investigate how this ideal behaves with respect to related rings.

We associate to each nilpotent orbit of a real semisimple Lie algebra go a weighted Vogan diagram, that is a Dynkin diagram with an involution of the diagram, a subset of painted nodes and a weight for each node. Every nilpotent element of go is noticed in some subalgebra of go. In this paper we characterize the weighted Vogan diagrams associated to orbits of noticed nilpotent elements.