Splitting of operations for alternative and Malcev structures

Abstract

Abstract. In this paper we define pre-Malcev algebras and alternative quadrialgebras and prove that they generalize pre-Lie algebras and quadri-algebras respectively to the alternative setting. We use the results and techniques from [4,14] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota-Baxter operators.

Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra

“…for all x, y, z, t ∈ A. The following result establishes the connection between Kupershmidt operators and pre-Malcev algebras which generalize the construction with Rota-Baxter operators (see in [22] for more details). A Kupershmidt operator on a Malcev algebra (A, [ , ]) with respect to a representation (V ; ϱ) is a linear map T :…”

“…Definition 4.1. [22] A pre-Malcev algebra is a vector space A endowed with a bilinear product " • "A × A → A satisfying the identity…”

“…which implies that S is a Nijenhuis operator on the pre-Malcev algebra (V, •) ( [22]). Thus S is a Nijenhuis operator on the sub-adjacent Malcev algebra (V, [ , ] T ).…”

Hom-pre-Malcev and Hom-M-Dendriform algebras

“…Just as the tangent algebra of a Lie group is a Lie algebra, the tangent algebra of a locally analytic Moufang loop is a Malcev algebra [26-28, 46, 53, 58], see also [17,52,54] for discussions about connections with physics. The notion of pre-Malcev algebra as a Malcev algebraic analogue of a pre-Lie algebra was introduced in [41]. A pre-Malcev algebra is a vector space A with a bilinear multiplication ′′ • ′′ such that the product [x, y] = x • y − y • x endows A with the structure of a Malcev algebra, and the left multiplication operator L • (x) : y → x • y define a representation of this Malcev algebra on A.…”

Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras

“…After the common operadic foundation for all of these varieties appeared [30], it was logical to try to unify all of these objects of the same "world". In light of the names of pre-Lie [24] and pre-Poisson [7] algebras given by M. Gerstenhaber and M. Aguiar respectively, the names of pre-Jordan, pre-alternative, pre-associative, and pre-Malcev algebras [39][40][41][42] appeared. Similar occurred for post-algebras after the name of post-Lie algebras was given by B. Vallette in [29]; for example, see [43].…”