Rota-Baxter TD Algebra and Quinquedendriform Algebra

2017 ◽  
Vol 24 (01) ◽  
pp. 53-74 ◽  
Author(s):  
Shuyun Zhou ◽  
Li Guo
Keyword(s):  
Associative Algebra ◽  
Linear Operators ◽  
Associative Algebras ◽  
Defining Relations ◽  
Binary Operations ◽  
Dendriform Algebra ◽  
Baxter Operator

A dendriform algebra defined by Loday has two binary operations that give a two-part splitting of the associativity in the sense that their sum is associative. Similar dendriform type algebras with three-part and four-part splitting of the associativity were later obtained. These structures can also be derived from actions of suitable linear operators, such as a Rota-Baxter operator or TD operator, on an associative algebra. Motivated by finding a five-part splitting of the associativity, we consider the Rota-Baxter TD (RBTD) operator, an operator combining the Rota-Baxter operator and TD operator, and coming from a recent study of Rota’s problem concerning linear operators on associative algebras. Free RBTD algebras on rooted forests are constructed. We then introduce the concept of a quinquedendriform algebra and show that its defining relations are characterized by the action of an RBTD operator, similar to the cases of dendriform and tridendriform algebras.

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.

WEYL TYPE ALGEBRAS FROM QUANTUM TORI

2006 ◽  
Vol 08 (02) ◽  
pp. 135-165 ◽  
Author(s):  
KAIMING ZHAO
Keyword(s):  
Differential Operator ◽  
Lie Algebra ◽  
Associative Algebra ◽  
Operator Algebra ◽  
Laurent Polynomials ◽  
Associative Algebras ◽  
Defining Relations ◽  
Quantum Tori ◽  
Quantum Version ◽  
Algebra Over A Field

We introduce and study the quantum version of the differential operator algebra on Laurent polynomials and its associated Lie algebra over a field F of characteristic 0. The q-quantum torus Fq is the unital associative algebra over F generated by [Formula: see text] subject to the defining relations titj = qi,jtjti, where qi,i = 1, [Formula: see text]. Let D be a subspace of [Formula: see text] where ∂i is the derivation on Fq sending [Formula: see text] to [Formula: see text]. Then, the quantum differential operator algebra is the associative algebra Fq[D]. Assume that Fq[D] is simple as an associative algebra. We compute explicitly all 2-cocycles of Fq[D], viewed as a Lie algebra. More precisely, we show that the second cohomology group of Fq[D] has dimension n if D = 0, dimension 1 if dim D = 1, and dimension 0 if dim D > 1. We also determine all isomorphisms and anti-isomorphisms Fq[D] → Fq′[D′] of simple associative algebras, and all isomorphisms Fq[D]/F → Fq′[D′]/F of simple Lie algebras.

Quantum Homomorphisms of Associative Algebras

1997 ◽  
Vol 12 (38) ◽  
pp. 2963-2974
Author(s):  
A. E. F. Djemai
Keyword(s):  
Quantum Group ◽  
Associative Algebra ◽  
Associative Algebras ◽  
Type Formula ◽  
Algebra Of Functions ◽  
Inhomogeneous Quantum

Given an associative algebra A generated by {ek, k=1, 2,…} and with an internal law of type: [Formula: see text], we first show that it is possible to construct a quantum bi-algebra [Formula: see text] with unit and generated by (non-necessarily commutative) elements [Formula: see text] satisfying the relations: [Formula: see text]. This leads one to define a quantum homomorphism[Formula: see text]. We then treat the example of the algebra of functions on a set of N elements and we show, for the case N=2, that the resulting bihyphen;algebra is an inhomogeneous quantum group. We think that this method can be used to construct quantum inhomogeneous groups.

Cayley Symmetries in Associative Algebras

1963 ◽  
Vol 15 ◽  
pp. 285-290 ◽  
Author(s):  
Earl J. Taft
Keyword(s):  
Associative Algebra ◽  
Associative Algebras ◽  
Wedderburn Decomposition ◽  
Finite Dimensional ◽  
Principal Theorem ◽  
Algebra Over A Field

Let A be a finite-dimensional associative algebra over a field F. Let R denote the radical of A. Assume that A/R is separable. Then it is well known (the Wedderburn principal theorem) that A possesses a Wedderburn decomposition A = S + R (semi-direct), where S is a separable subalgebra isomorphic with A/R. We call S a Wedderburn factor of A.

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.

Cleft extensions and Galois extensions for Hom-associative algebras

2016 ◽  
Vol 27 (03) ◽  
pp. 1650025 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Associative Algebra ◽  
Galois Extension ◽  
Classical Case ◽  
Normal Basis ◽  
Associative Algebras ◽  
Galois Extensions ◽  
Cleft Extension ◽  
Cleft Extensions

In this paper, we consider Hom-(co)modules associated to a Hom-(co)associative algebra and define the notion of Hom-triple. We introduce the definitions of cleft extension and Galois extension with normal basis in this setting and we show that, as in the classical case, these notions are equivalent in the Hom setting.

Commutative non-power-associative algebras

2019 ◽  
Vol 29 (08) ◽  
pp. 1527-1539
Author(s):  
Manuel Arenas ◽  
Irvin Roy Hentzel ◽  
Alicia Labra
Keyword(s):  
Associative Algebra ◽  
Associative Algebras ◽  
Commutative Algebras ◽  
Dimension Formula

We study commutative algebras satisfying the identity [Formula: see text] It is known that for [Formula: see text] and for characteristic not [Formula: see text] or [Formula: see text], the algebra is a commutative power-associative algebra. These algebras have been widely studied by Albert, Gerstenhaber and Schafer. For [Formula: see text] Guzzo and Behn in 2014 proved that commutative algebras of dimension [Formula: see text] satisfying [Formula: see text] are solvable. We consider the remaining values of [Formula: see text] We prove that commutative algebras satisfying [Formula: see text] with [Formula: see text] and generated by one element are nilpotent of nilindex [Formula: see text] (we assume characteristic of the field [Formula: see text]).

Baer-invariants and extensions relative to a variety

1967 ◽  
Vol 63 (3) ◽  
pp. 569-578 ◽  
Author(s):  
Abraham S.-T. Lue
Keyword(s):  
Commutative Ring ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Jordan Algebras ◽  
Associative Algebras ◽  
Definition Of ◽  
The Relationship

This paper examines the relationship between extensions in a variety and general extensions in the category of associative algebras. Our associative algebras are all unitary, over some fixed commutative ring Λ with identity, but while our discussion will be restricted to this category, it is clear that obvious analogues exist for groups, Lie algebras and Jordan algebras. (We use the notion of a bimultiplication of an associative algebra. In (2), Knopfmacher gives the definition of a bimultiplication in any variety of linear algebras.)

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.

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