Mathematics

The aim of this paper is to introduce and study BiHom-Poisson algebras, in particular Non-BiHom-Commutative BiHom-Poisson algebras. We discuss their representation theory and Semi-direct product. Furthermore, we characterize admissible BiHom-Poisson algebras. Finally, we establish the classification of 2-dimensional BiHom-Poisson algebras.

Let K be a field such that char(K)∤k and char(K)∤k+1 . We describe all (k+1) -potent matrices over the group of upper triangular matrix. In the case that K is a finite field we show how to compute the number of these elements in triangular matrix groups and use this formula to compute the number of (k+1) -potent elements in the Incidence Algebra I(X,K) where X is a finite poset.

The present paper is devoted to study 2-local superderivations on the super Virasoro algebra and the super W(2,2) algebra. We prove that all 2-local superderivations on the super Virasoro algebra as well as the super W(2,2) algebra are (global) superderivations.

In this paper we provide the basic setup for a project, initiated by Felix Rehren, aiming at classifying all 2 -generated primitive axial agebras of Monster type (α,β) . We first revise Rehren's construction of an initial object in the cathegory of primitive n -generated axial algebras giving a formal one, filling some gaps and, though correcting some inaccuracies, confirm Rehren's results. Then we focus on 2 -generated algebras which naturally part into three cases: the two critical cases α=2β and α=4β , and the generic case (i.e. all the rest). About these cases, which will be dealt in detail in subsequent papers, we give bounds on the dimensions (the generic case already treated by Rehen) and classify all 2-generated primitive axial algebras of Monster type (α,β) over Q(α,β) for α and β algebraically independent indeterminates over Q . Finally we restrict to the 2 -generated Majorana algebras (i.e. when α= 1 4 and β= 1 32 ), showing that these fall precisely into the nine isomorphism types of the Norton-Sakuma algebras.

In this paper we prove that 2 -generated primitive axial algebras of Monster type (2β,β) over a ring R in which 2 and β are invertible can be generated as R -module by 8 vectors. We then completely classify 2 -generated primitive axial algebras of Monster type (2β,β) over any field of characteristic other than 2 .

The Jordan algebra of the symmetric matrices of order two over a field K has two natural gradings by Z 2 , the cyclic group of order 2. We describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2. We exhibit bases for these identities in each of the two cases. In one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from Invariant theory. Moreover we extend the latter grading to a Z 2 -grading on B n , the Jordan algebra of a symmetric bilinear form in a vector space of dimension n ( n=1 , 2, \dots, ∞ ). We call this grading the \textsl{scalar} one since its even part consists only of the scalars. As a by-product we obtain finite bases of the Z 2 -graded identities for B n . In fact the last result describes the weak Jordan polynomial identities for the pair ( B n , V n ) .

This paper initiates the study of 2-local derivations on Lie algebras over fields of prime characteristic. Let g be a simple Jacobson-Witt algebra W n over a field of prime characteristic p with cardinality no less than p n . In this paper, we study properties of 2-local derivations on g , and show that every 2-local derivation on g is a derivation.

In this paper, new block representations of Moore-Penrose inverses for arbitrary complex 2?2 block matrices are given. The approach is based on block representations of orthogonal projection matrices.

The main goal of this paper is to introduce the notion of 3 -L-dendriform algebras which are the dendriform version of 3 -pre-Lie algebras. In fact they are the algebraic structures behind the O -operator of 3 -pre-Lie algebras. They can be also regarded as the ternary analogous of L-dendriform algebras. Moreover, we study the generalized derivations of 3 -L-dendriform algebras. Finally, we explore the spaces of quasi-derivations, the centroids and the quasi-centroids and give some properties.

Axial algebras are a recently introduced class of non-associative algebra, with a naturally associated group, which generalise the Griess algebra and some key features of the moonshine VOA. Sakuma's Theorem classifies the eight 2 -generated axial algebras of Monster type. In this paper, we compute almost all the 3 -generated axial algebras whose associated Miyamoto group is minimal 3 -generated (this includes the minimal 3 -generated algebras). We note that this work was carried out independently to that of Mamontov, Staroletov and Whybrow and extends their result by computing more algebras and not assuming primitivity, or an associating bilinear form.