scholarly journals An Upper Bound for the Lower Central Series Quotients of a Free Associative Algebra

2008 ◽  
Vol 2008 ◽  
Author(s):  
Galyna Dobrovolska ◽  
Pavel Etingof
Keyword(s):  
Associative Algebra ◽  
Upper Bound ◽  
Lower Central Series ◽  
Free Associative Algebra ◽  
Central Series

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 The class of a nilpotent wreath product

1977 ◽  
Vol 17 (1) ◽  
pp. 53-89 ◽  
Author(s):  
David Shield
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Group Ring ◽  
Upper Bound ◽  
Lower Central Series ◽  
Nilpotency Class ◽  
Central Series ◽  
Augmentation Ideal ◽  
Cambridge Philos ◽  
Base Group

Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.

On the triple tensor product of prime-power groups

2020 ◽  
Vol 23 (5) ◽  
pp. 879-892
Author(s):  
S. Hadi Jafari ◽  
Halimeh Hadizadeh
Keyword(s):  
Tensor Product ◽  
Nilpotent Group ◽  
Upper Bound ◽  
Prime Power ◽  
Tensor Products ◽  
Lower Central Series ◽  
Central Series ◽  
Derived Subgroup

AbstractLet G be a finite p-group, and let {\otimes^{3}G} be its triple tensor product. In this paper, we obtain an upper bound for the order of {\otimes^{3}G}, which sharpens the bound given by G. Ellis and A. McDermott, [Tensor products of prime-power groups, J. Pure Appl. Algebra 132 1998, 2, 119–128]. In particular, when G has a derived subgroup of order at most p, we classify those groups G for which the bound is attained. Furthermore, by improvement of a result about the exponent of {\otimes^{3}G} determined by G. Ellis [On the relation between upper central quotients and lower central series of a group, Trans. Amer. Math. Soc. 353 2001, 10, 4219–4234], we show that, when G is a nilpotent group of class at most 4, {\exp(\otimes^{3}G)} divides {\exp(G)}.

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 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.

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