scholarly journals The module structure of Solomon's descent algebra

2002 ◽  
Vol 72 (3) ◽  
pp. 317-334 ◽  
Author(s):  
Dieter Blessenohl ◽  
Hartmut Laue
Keyword(s):  
Associative Algebra ◽  
Lower Central Series ◽  
Close Connection ◽  
Free Associative Algebra ◽  
Module Structure ◽  
Central Series ◽  
Descent Algebra ◽  
Direct Sum ◽  
Precise Condition ◽  
Countable Rank

AbstractA close connection is uncovered between the lower central series of the free associative algebra of countable rank and the descending Loewy series of the direct sum of all Solomon descent algebras Δn, n ∈ ℕ0. Each irreducible Δn-module is shown to occur in at most one Loewy section of any principal indecomposable Δn-module.A precise condition for his occurence and formulae for the Cartan numbers are obtained.

scholarly journals Lower central series of a free associative algebra over the integers and finite fields

2012 ◽  
Vol 372 ◽  
pp. 251-274 ◽  
Author(s):  
Surya Bhupatiraju ◽  
Pavel Etingof ◽  
David Jordan ◽  
William Kuszmaul ◽  
Jason Li
Keyword(s):  
Associative Algebra ◽  
Finite Fields ◽  
Lower Central Series ◽  
Free Associative Algebra ◽  
Central Series

scholarly journals On the lower central series quotients of a graded associative algebra

2011 ◽  
Vol 328 (1) ◽  
pp. 287-300 ◽  
Author(s):  
Martina Balagović ◽  
Anirudha Balasubramanian
Keyword(s):  
Associative Algebra ◽  
Lower Central Series ◽  
Central Series

scholarly journals FREE CENTRE-BY-NILPOTENT-BY-ABELIAN LIE RINGS OF RANK 2

2015 ◽  
Vol 102 (1) ◽  
pp. 63-73 ◽  
Author(s):  
MARIA ALEXANDROU ◽  
RALPH STÖHR
Keyword(s):  
Free Abelian Group ◽  
Torsion Group ◽  
Lower Central Series ◽  
Torsion Subgroup ◽  
Central Series ◽  
Lie Ring ◽  
Direct Sum ◽  
Lie Rings ◽  
Rank 2 ◽  
Derived Series

We study the free Lie ring of rank $2$ in the variety of all centre-by-nilpotent-by-abelian Lie rings of derived length $3$. This is the quotient $L/([\unicode[STIX]{x1D6FE}_{c}(L^{\prime }),L]+L^{\prime \prime \prime })$ with $c\geqslant 2$ where $L$ is the free Lie ring of rank $2$, $\unicode[STIX]{x1D6FE}_{c}(L^{\prime })$ is the $c$th term of the lower central series of the derived ideal $L^{\prime }$ of $L$, and $L^{\prime \prime \prime }$ is the third term of the derived series of $L$. We show that the quotient $\unicode[STIX]{x1D6FE}_{c}(L^{\prime })+L^{\prime \prime \prime }/[\unicode[STIX]{x1D6FE}_{c}(L^{\prime }),L]+L^{\prime \prime \prime }$ is a direct sum of a free abelian group and a torsion group of exponent $c$. We exhibit an explicit generating set for the torsion subgroup.

scholarly journals On the lower central series of an associative algebra

2008 ◽  
Vol 320 (1) ◽  
pp. 213-237 ◽  
Author(s):  
Galyna Dobrovolska ◽  
John Kim ◽  
Xiaoguang Ma
Keyword(s):  
Associative Algebra ◽  
Lower Central Series ◽  
Central Series

scholarly journals THE DECOMPOSITION OF LIE POWERS

2006 ◽  
Vol 93 (1) ◽  
pp. 175-196 ◽  
Author(s):  
R. M. BRYANT ◽  
M. SCHOCKER
Keyword(s):  
Positive Integer ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Tensor Algebra ◽  
Prime Characteristic ◽  
Characteristic P ◽  
Descent Algebra ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Solomon Descent Algebra

Let $G$ be a group, $F$ a field of prime characteristic $p$ and $V$ a finite-dimensional $FG$-module. Let $L(V)$ denote the free Lie algebra on $V$ regarded as an $FG$-submodule of the free associative algebra (or tensor algebra) $T(V)$. For each positive integer $r$, let $L^r (V)$ and $T^r (V)$ be the $r$th homogeneous components of $L(V)$ and $T(V)$, respectively. Here $L^r (V)$ is called the $r$th Lie power of $V$. Our main result is that there are submodules $B_1$, $B_2$, ... of $L(V)$ such that, for all $r$, $B_r$ is a direct summand of $T^r(V)$ and, whenever $m \geqslant 0$ and $k$ is not divisible by $p$, the module $L^{p^mk} (V)$ is the direct sum of $L^{p^m} (B_k)$, $L^{p^{m - 1}} (B_{pk})$, ..., $L^1 (B_{p^mk})$. Thus every Lie power is a direct sum of Lie powers of $p$-power degree. The approach builds on an analysis of $T^r (V)$ as a bimodule for $G$ and the Solomon descent algebra.

scholarly journals CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

Sign in / Sign up

Export Citation Format

Share Document