Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On minimal Leibniz algebras with nilpotent commutator subalgebra


Author: S. M. Ratseev
Translated by: V. A. Vavilov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 1.
Journal: St. Petersburg Math. J. 27 (2016), 125-136
MSC (2010): Primary 17A32
DOI: https://doi.org/10.1090/spmj/1379
Published electronically: December 7, 2015
MathSciNet review: 3443269
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \{c_n({\mathbf V})\}_{n\geq 1}$ be the codimension sequence of a variety of Leibniz algebras $ {\mathbf V}$. The complexity function $ \mathcal {C}({\mathbf V},z)=\sum _{n=1}^{\infty }c_n({\mathbf V})z^n/n!$ is studied. This is an exponential generating function for the codimension sequence. Before, complexity functions were used to study Lie algebras and associative algebras. In this paper, an explicit formula is obtained for the complexity function of a variety of Leibniz algebras with nilpotent commutator subalgebra, specifically, the variety determined by the identity $ x_0(x_1x_2)(x_3x_4)\dots (x_{2s-1}x_{2s})=0$. By using this function, an explicit formula is derived for the codimensions of these algebras, which grow exponentially. Also, two series of varieties of Leibniz algebras with nilpotent commutator subalgebra of polynomial growth are constructed; they are minimal in a certain sense. Namely, the codimension sequence of any variety in the first of these series grows as a polynomial of a certain degree $ k$, but for all its proper subvarieties the codimension sequence grows as a polynomial of some degree strictly smaller than $ k$. The codimension sequence of any variety of the second series grows as a polynomial with some value of the leading coefficient $ q$, whereas for all its proper subvarieties the codimension sequence grows as a polynomial whose leading coefficient is strictly smaller than $ q$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 17A32

Retrieve articles in all journals with MSC (2010): 17A32


Additional Information

S. M. Ratseev
Affiliation: Ulyanovsk State University, 42 Lev Tolstoy str., 432017 Ulyanovsk, Russia
Email: ratseevsm@mail.ru

DOI: https://doi.org/10.1090/spmj/1379
Keywords: Leibniz algebras, Lie algebras, varieties of algebras, growth of varieties
Received by editor(s): June 2, 2014
Published electronically: December 7, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society