Group algebras with almost maximal lie nilpotency index

LetK G be a non-commutative Lie nilpotent group algebra of a groupG over a fieldK. It is known that the Lie nilpotency index ofKG is at most |G′|+1, where |G′| is the order of the commutator subgroup ofG. In [4] the groupsG for which this index is maximal were determined. Here we list theG’s for which it assumes the next highest possible value. The present paper is a part of the PhD dissertation of the author.     

Finiteness of basis of identities of finite-dimensional representations of solvable lie algebras

Moscow. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 3, pp. 78–86, May–June, 1988.     

Quadratic linear Keller maps of nilpotency index three

Let be a homogeneous polynomial map of degree d2 and a power linear map such that f and F_A are a generalized Gorni-Zampieri pair. We discuss the relation between the nilpotency indices of JH and J(Ay)^(d) and we show that f is linearly triangularizable if and only if F_A is linearly triangularizable. As a consequence, we show that a quadratic linear Keller map F_A=y+(Ay)⁽²⁾ with nilpotency index three, i.e., (J(Ay)⁽²⁾)³=0, is linearly triangularizable.     

The nilpotency index of the radicals of blocks with abelian defect groups

LetF be an algebraically closed field of characteristicp > 0, letG be a finitep-solvable group and letB be a block of the group algebraFG with abelian defect groupD. In this note we provide a precise formula for the nilpotency indext(B) ofJ(B), whereJ(B) is the Jacobson radical ofB. Namely, we prove that if $$D \cong Z_{p^{n_1 } } \times Z_{p^{n_2 } } \times \ldots \times Z_{p^{n_k } }$$ then $$t(B) = 1 - k + \sum\limits_{i = 1}^k {p^{n_i } .}$$      

Group algebras of Lie nilpotency index 12 and 13

Let G be a group and let K be a field of characteristic p>0. Lie nilpotent group algebras of strong Lie nilpotency index up to 11 have already been classified. In this paper, our aim is to classify the group algebras KG which are strongly Lie nilpotent of index 12 or 13.     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

The right-normed basis for a free lie superalgebra and Lyndon-Shirshov words

We construct a basis for a free Lie superalgebra consisting of right-normed words , where are free generators. Supported By RFBR Grant No. 05-01-00230. __________ Translated from Algebra i Logika, Vol. 45, No. 4, pp. 458–483, July–August, 2006.     

Finiteness of basis of identities of a finitely generated alternative PI-algebra over a field of characteristic zero

Dedicated to the memory of Anatolii Illarionovich Shirshov.     

Essential and inessential elements of a standard basis

In this paper we introduce the concept of inessential element of a standard basis B(I), where I is any homogeneous ideal of a polynomial ring. An inessential element is, roughly speaking, a form of B(I) whose omission produces an ideal having the same saturation as I; it becomes useless in any dehomogenization of I with respect to a linear form. We study the properties of B(I) linked to the presence of inessential elements and give some examples.     

The residual nilpotency of the augmentation ideal and the residual nilpotency of some classes of groups

The following theorem is proved: LetG be any group. Then the augmentation ideal ofZG is residually nilpotent if and only ifG is approximated by nilpotent groups without torsion or discriminated by nilpotent p_i,-groups,i ∈I, of finite exponents. This theorem is applied to obtain conditions under which the groupsF/N′ are residually nilpotent whereF is a free non-cyclic group and N?F.     

On nilpotency of the barideal of a bernstein algebra

It is given a new bound for the index of nilpotency of the barideal of a finited generated Bernstein algebra. Also, it is found the index of solvability of the barideal of a Bernstein algebra and a bound of nilpotency for the square of the barideal.     

Finiteness of the quadratic primal simplex method when s-monotone index selection rules are applied

Central European Journal of Operations Research - This paper considers the primal quadratic simplex method for linearly constrained convex quadratic programming problems. Finiteness of the...