The Lie Algebra of the Structure Group of a Power-Associative Algebra

1972 ◽  
Vol 31 (2) ◽  
pp. 363
Author(s):  
D. R. Scribner
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Structure Group

scholarly journals The Lie algebra of the structure group of a power-associative algebra.

1972 ◽  
Vol 31 (2) ◽  
pp. 363-363
Author(s):  
D. R. Scribner
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Structure Group

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.

scholarly journals Decompositions of algebras and post-associative algebra structures

2019 ◽  
Vol 30 (03) ◽  
pp. 451-466
Author(s):  
Dietrich Burde ◽  
Vsevolod Gubarev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Algebra Structure ◽  
Associative Algebras ◽  
Simple Lie Algebra ◽  
Reductive Lie Algebra ◽  
Baxter Operator

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

On the Garsia Lie Idempotent

2005 ◽  
Vol 48 (3) ◽  
pp. 445-454 ◽  
Author(s):  
Frédéric Patras ◽  
Christophe Reutenauer ◽  
Manfred Schocker
Keyword(s):  
Symmetric Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Group Algebra ◽  
Orthogonal Projection ◽  
Gram Matrix ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Rational Group ◽  
Lie Idempotent

AbstractThe orthogonal projection of the free associative algebra onto the free Lie algebra is afforded by an idempotent in the rational group algebra of the symmetric group Sn, in each homogenous degree n. We give various characterizations of this Lie idempotent and show that it is uniquely determined by a certain unit in the group algebra of Sn−1. The inverse of this unit, or, equivalently, the Gram matrix of the orthogonal projection, is described explicitly. We also show that the Garsia Lie idempotent is not constant on descent classes (in fact, not even on coplactic classes) in Sn.

Weyl Type Non-Associative Algebras Using Additive Groups I

2007 ◽  
Vol 14 (03) ◽  
pp. 479-488 ◽  
Author(s):  
Seul Hee Choi ◽  
Ki-Bong Nam
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Simple Algebra ◽  
Associative Algebras

A Weyl type algebra is defined in the book [4]. A Weyl type non-associative algebra [Formula: see text] and its restricted subalgebra [Formula: see text] are defined in various papers (see [1, 3, 11, 12]). Several authors find all the derivations of an associative (a Lie, a non-associative) algebra (see [1, 2, 4, 6, 11, 12]). We define the non-associative simple algebra [Formula: see text] and the semi-Lie algebra [Formula: see text], where [Formula: see text]. We prove that the algebra is simple and find all its non-associative algebra derivations.

scholarly journals Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra

2013 ◽  
Vol 162 (5) ◽  
pp. 965-1002 ◽  
Author(s):  
Shigeyuki Morita ◽  
Takuya Sakasai ◽  
Masaaki Suzuki
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Free Associative Algebra ◽  
Free Lie Algebra

scholarly journals Carter subgroups in the group of units of an associative algebra

2005 ◽  
Vol 71 (3) ◽  
pp. 471-478 ◽  
Author(s):  
Thorsten Bauer ◽  
Salvatore Siciliano
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Group Of Units ◽  
Cartan Subalgebras

In this paper we examine some properties of the Carter subgroups in the group of units of certain associative algebras. A description of the Carter subgroups in the case of a solvable associative algebra is obtained. Moreover, given an associative algebra A, we study relationships between the Cartan subalgebras of the Lie algebra associated with A and the Carter subgroups of the group of units of A.

scholarly journals Anti-flexible bialgebras

2021 ◽  
pp. 2250212
Author(s):  
Mafoya Landry Dassoundo ◽  
Chengming Bai ◽  
Mahouton Norbert Hounkonnou
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Manin Triple ◽  
Unexpected Consequence ◽  
Special Case

We establish a bialgebra theory for anti-flexible algebras in this paper. We introduce the notion of an anti-flexible bialgebra which is equivalent to a Manin triple of anti-flexible algebras. The study of a special case of anti-flexible bialgebras leads to the introduction of anti-flexible Yang–Baxter equation in an anti-flexible algebra which is an analogue of the classical Yang–Baxter equation in a Lie algebra or the associative Yang–Baxter equation in an associative algebra. It is unexpected consequence that both the anti-flexible Yang–Baxter equation and the associative Yang–Baxter equation have the same form. A skew-symmetric solution of anti-flexible Yang–Baxter equation gives an anti-flexible bialgebra. Finally the notions of an [Formula: see text]-operator of an anti-flexible algebra and a pre-anti-flexible algebra are introduced to construct skew-symmetric solutions of anti-flexible Yang–Baxter equation.

On Reductive Lie Admissible Algebras

1971 ◽  
Vol 23 (2) ◽  
pp. 325-331 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Commutative Algebra ◽  
Jacobi Identity ◽  
Direct Sum ◽  
Derivation Algebra ◽  
Image Position

A Lie admissible algebra is a non-associative algebra A such that A− is a Lie algebra where A− denotes the anti-commutative algebra with vector space A and with commutation [X, Y] = XY – YX as multiplication; see [1; 2; 5]. Next let L−(X): A− → A−: Y → [X, Y] and H = {L−(X): X ∊ A−}; then, since A− is a Lie algebra, we see that H is contained in the derivation algebra of A− and consequently the direct sum g = A − ⊕ H can be naturally made into a Lie algebra with multiplication [PQ] given by: P = X + L−(U), Q = Y + L−(V) ∊ g, thenand note that for any P, [PP] = 0 so that [PQ] = −[QP] and the Jacobi identity for g follows from the fact that A− is Lie.

NILPOTENT LINEAR SEMIGROUPS

2006 ◽  
Vol 16 (01) ◽  
pp. 141-160 ◽  
Author(s):  
ERIC JESPERS ◽  
DAVID RILEY
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Adjoint Semigroup ◽  
Restricted Burnside Problem ◽  
Global And Local

We characterize the structure of linear semigroups satisfying certain global and local nilpotence conditions and deduce various Engel-type results. For example, using a form of Zel'manov's solution of the restricted Burnside problem we are able to show that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups we call WMN. Methods of linear semigroups then allow us to prove that a linear semigroup is Mal'cev nilpotent precisely when it satisfies WMN. As an application, we show that a finitely generated associative algebra is nilpotent when viewed as a Lie algebra if and only if its adjoint semigroup is WMN.

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