k-Modules Over Linear Spaces by n-Linear Maps Admitting a Multiplicative Basis

Abstract

We study the structure of certain k-modules V over linear spaces W with restrictions neither on the dimensions of V and W nor on the base field F.. . , w jn ]σ ∈ Fvr σ for some rσ ∈ I. We show that if V admits a multiplicative basis then it decomposes as the direct sum V = α Vα of well described k-submodules Vα each one admitting a multiplicative basis. Also the minimality of V is characterized in terms of the multiplicative basis and it is shown that the above direct sum is by means of the family of its minimal k-submodules, admitting each one a multiplicative basis. Finally we study an application of k-modules with a multiplicative basis over an arbitrary n-ary algebra with multiplicative basis.

Non-associative algebraic structures: classification and structure

“…In our joint paper with Barreiro and Sánchez [24], we generalize the cited results to n-ary case. We study the structure of certain k-modules V over linear spaces W without restrictions neither on the dimensions of V and W nor on the base field F. A basis B = {v i } i∈I of V is called multiplicative with respect to another basis B ′ = {w j } j∈J of W if for any σ ∈ S n , i 1 , .…”

“…The present part is based on the papers written together with Alexandre Pozhidaev, Antonio Jesús Calderón, Elisabete Barreiro, José María Sánchez, Paulo Saraiva, and Yury Popov [22,24,50,168,172].…”

Decompositions of linear spaces induced by n-linear maps

“…At this point, a parenthesis is due to underline the considerable amount of recent works where the above mentioned and similar connection techniques are applied as a tool to obtain interesting results in the frameworks of several types of algebras. Without being exhaustive, these techniques were used, for instance, along with the notions of multiplicative basis and quasi-multiplicative basis not only related with algebras (see Caledrón and Navarro, [3,4]), but also with some n-ary generalizations (see, e.g., the works of Calderón, Barreiro, Kaygorodov and Sánchez in [1,2,7]). Further, connection techniques were also applied in the context of graded Lie algebras (see [5]) and to obtain structural results on graded Leibniz triple systems (see Cao and Chen (2016) [8]).…”