CMMSE: Combinatorial structures and finite-dimensional alternative algebras

In this paper, the link between combinatorial structures and alternative algebras is studied, determining which configurations are associated with those algebras. Moreover, the isomorphism classes of each 2-dimensional configuration associated with these algebras is analyzed, providing a new method to classify them. In order to complement the theoretical study, two algorithmic methods are implemented: the first one constructs and draws the (pseudo)digraph associated with a given alternative algebra and the second one tests if a given combinatorial structure is associated with some alternative algebra.

A New Approach to Slice Analysis Via Slice Topology

In this paper we summarize some known facts on slice topology in the quaternionic case, and we deepen some of them by proving new results and discussing some examples. We then show, following Dou et al. (A representation formula for slice regular functions over slice-cones in several variables, arXiv:2011.13770, 2020), how this setting allows us to generalize slice analysis to the general case of functions with values in a real left alternative algebra, which includes the case of slice monogenic functions with values in Clifford algebra. Moreover, we further extend slice analysis, in one and several variables, to functions with values in a Euclidean space of even dimension. In this framework, we study the domains of slice regularity, we prove some extension properties and the validity of a Taylor expansion for a slice regular function.

Central Extension of Left Alternative Algebras and the Second Cohomology Group

The purpose of this paper is to prove that the second cohomology group H 2 A , F of a left alternative algebra A over an algebraically closed field F of characteristic 0 can be interpreted as the set of equivalent classes of one-dimensional central extensions of A .

Kupershmidt-(dual-)Nijenhuis structure on the alternative algebra with a representation

The aim of this paper is to study Kupershmidt-(dual-)Nijenhuis structures on alternative algebras with representations. The notion of a (dual-)Nijenhuis pair is introduced and it can generate a trivial deformation of an alternative algebra with a representation. We introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on an alternative algebra with a representation. Furthermore, we verify that Kupershmidt operators and Kupershmidt-(dual-)Nijenhuis structures can give rise to each other under some conditions. Finally, we study the notions of Rota–Baxter–Nijenhuis structures and alternative [Formula: see text]-matrix-Nijenhuis structures. Meanwhile, we investigate the relation between them.

Some finitely generated associative algebras with a Lie nilpotency identity

It is proved that the algebra of multiplications of the free commutative alternative algebra of finite rank [Formula: see text] is strongly Lie nilpotent of class [Formula: see text]. It is found the class of nilpotency of the ideal, generated by commutators in the free [Formula: see text]-generated associative algebra with identity of Lie nilpotency of degree [Formula: see text] under the condition that [Formula: see text] or [Formula: see text].

Action theory of alternative algebras

We present the category of alternative algebras as a category of interest. This kind of approach enables us to describe derived actions in this category, study their properties and construct a universal strict general actor of any alternative algebra. We apply the results obtained in this direction to investigate the problem of the existence of an actor in the category of alternative algebras.

Multiplicative δ-derivation in alternative algebras

Every multiplicative [Formula: see text]-derivation of an alternative algebra [Formula: see text] is additive if there exists an idempotent [Formula: see text] in [Formula: see text] satisfying the following conditions: (i) [Formula: see text] implies [Formula: see text]; (ii) [Formula: see text] implies [Formula: see text]; (iii) [Formula: see text] implies [Formula: see text] for [Formula: see text]. In particular, every [Formula: see text]-derivation of a prime alternative algebra with a nontrivial idempotent is additive. This generalizes the known result obtained by Rodrigues, Guzzo and Ferreira for [Formula: see text]-derivations. As an application, we apply multiplicative [Formula: see text]-derivation to an alternative complex algebra [Formula: see text] of all [Formula: see text] complex matrices to see how it decomposes into a sum of [Formula: see text]-inner derivation and a [Formula: see text]-derivation on [Formula: see text] given by an additive derivation [Formula: see text] on [Formula: see text].

The Integral Quantum Loop Algebra of $\mathfrak {gl}_{n}$

We will construct the Lusztig form for the quantum loop algebra of $\mathfrak {gl}_{n}$ by proving the conjecture [4, 3.8.6] and establish partially the Schur–Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine $\mathfrak {gl}_{n}$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine $\mathfrak {sl}_{n}$ and construct its canonical basis to provide an alternative algebra structure related to a conjecture of Lusztig in [29, §9.3], which has been already proved in [34].

Hom-Lie Triple System and Hom-Bol Algebra Structures on Hom-Maltsev and Right Hom-Alternative Algebras

Every multiplicative Hom-Maltsev algebra has a natural multiplicative Hom-Lie triple system structure. Moreover, there is a natural Hom-Bol algebra structure on every multiplicative Hom-Maltsev algebra and on every multiplicative right (or left) Hom-alternative algebra.

RIGHT NUCLEUS IN GENERALIZED RIGHT ALTERNATIVE RINGS

Some properties of the right nucleus in generalized right alternative rings have been presented in this paper. In a generalized right alternative ring R which is finitely generated or free of locally nilpotent ideals, the right nucleus Nr equals the center C. Also, if R is prime and Nr ¹ C, then the associator ideal of R is locally nilpotent. Seong Nam [5] studied the properties of the right nucleus in right alternative algebra. He showed that if R is a prime right alternative algebra of char. ≠ 2 and Right nucleus Nr is not equal to the center C, then the associator ideal of R is locally nilpotent. But the problem arises when it come with the study of generalized right alternative ring as the ring dose not absorb the right alternative identity. In this paper we consider our ring to be generalized right alternative ring and try to prove the results of Seong Nam [5]. At the end of this paper we give an example to show that the generalized right alternative ring is not right alternative.