Search results for "Mathematics::Rings and Algebras"
showing 10 items of 79 documents
Two-step nilpotent Leibniz algebras
2022
In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. In particular, we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical $(2n+1)-$dimensional Heisenberg Lie algebra $\mathfrak{h}_{2n+1}$. Then we use the Leibniz algebras - Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. We also show that every Lie quandle integr…
Multialternating Jordan polynomials and codimension growth of matrix algebras
2007
Abstract Let R be the Jordan algebra of k × k matrices over a field of characteristic zero. We exhibit a noncommutative Jordan polynomial f multialternating on disjoint sets of variables of order k 2 and we prove that f is not a polynomial identity of R . We then study the growth of the polynomial identities of the Jordan algebra R through an analysis of its sequence of Jordan codimensions. By exploiting the basic properties of the polynomial f , we are able to prove that the exponential rate of growth of the sequence of Jordan codimensions of R in precisely k 2 .
Gradings on the algebra of upper triangular matrices of size three
2013
Abstract Let UT 3 ( F ) be the algebra of 3 × 3 upper triangular matrices over a field F . On UT 3 ( F ) , up to isomorphism, there are at most five non-trivial elementary gradings and we study the graded polynomial identities for such gradings. In case F is of characteristic zero we give a complete description of the space of multilinear graded identities in the language of Young diagrams through the representation theory of a Young subgroup of S n . We finally compute the multiplicities in the graded cocharacter sequence for every elementary G -grading on UT 3 ( F ) .
On the Leibniz bracket, the Schouten bracket and the Laplacian
2003
International audience; The Leibniz bracket of an operator on a (graded) algebra is defined and some of its properties are studied. A basic theorem relating the Leibniz bracket of the commutator of two operators to the Leibniz bracket of them is obtained. Under some natural conditions, the Leibniz bracket gives rise to a (graded) Lie algebra structure. In particular, those algebras generated by the Leibniz bracket of the divergence and the Laplacian operators on the exterior algebra are considered, and the expression of the Laplacian for the product of two functions is generalized for arbitrary exterior forms.
Local spectral theory for Drazin invertible operators
2016
Abstract In this paper we investigate the transmission of some local spectral properties from a bounded linear operator R, as SVEP, Dunford property (C), and property (β), to its Drazin inverse S, when this does exist.
Cocharacters of group graded algebras and multiplicities bounded by one
2017
Let G be a finite group and A a G-graded algebra over a field F of characteristic zero. We characterize the (Formula presented.)-ideals (Formula presented.) of graded identities of A such that the multiplicities (Formula presented.) in the graded cocharacter of A are bounded by one. We do so by exhibiting a set of identities of the (Formula presented.)-ideal. As a consequence we characterize the varieties of G-graded algebras whose lattice of subvarieties is distributive.
Singular quadratic Lie superalgebras
2012
In this paper, we give a generalization of results in \cite{PU07} and \cite{DPU10} by applying the tools of graded Lie algebras to quadratic Lie superalgebras. In this way, we obtain a numerical invariant of quadratic Lie superalgebras and a classification of singular quadratic Lie superalgebras, i.e. those with a nonzero invariant. Finally, we study a class of quadratic Lie superalgebras obtained by the method of generalized double extensions.
Lie properties of symmetric elements in group rings
2009
Abstract Let ∗ be an involution of a group G extended linearly to the group algebra KG . We prove that if G contains no 2-elements and K is a field of characteristic p ≠ 2 , then the ∗-symmetric elements of KG are Lie nilpotent (Lie n -Engel) if and only if KG is Lie nilpotent (Lie n -Engel).
THE AMITSUR–LEVITZKI THEOREM FOR THE ORTHOSYMPLECTIC LIE SUPERALGEBRA osp(1, 2n)
2006
http://www.worldscinet.com/jaa/05/0503/S0219498806001740.html; International audience; Based on Kostant's cohomological interpretation of the Amitsur–Levitzki theorem, we prove a super version of this theorem for the Lie superalgebras osp(1, 2n). We conjecture that no other classical Lie superalgebra can satisfy an Amitsur–Levitzki type super identity. We show several (super) identities for the standard super polynomials. Finally, a combinatorial conjecture on the standard skew supersymmetric polynomials is stated.
Charakterisierung von Kettengeometrien �ber semilokalen Algebren
1991
Chain geometries over semilocal rings are characterized as special touching structures possessing a distinguished group of automorphisms fixing two distant points.