Mathematics

Let F be a field of characteristic zero and let E be the Grassmann algebra of an infinite dimensional F -vector space L . In this paper we study the superalgebra structures (that is the Z 2 -gradings) that the algebra E admits. By using the duality between superalgebras and automorphisms of order 2 we prove that in many cases the Z 2 -graded polynomial identities for such structures coincide with the Z 2 -graded polynomial identities of the "typical" cases E ∞ , E k ∗ and E k where the vector space L is homogeneous. Recall that these cases were completely described by Di Vincenzo and Da Silva in \cite{disil}. Moreover we exhibit a wide range of non-homogeneous Z 2 -gradings on E that are Z 2 -isomorphic to E ∞ , E k ∗ and E k . In particular we construct a Z 2 -grading on E with only one homogeneous generator in L which is Z 2 -isomorphic to the natural Z 2 -grading on E , here denoted by E can .

An important step in the determination of the automorphism group of the quantum torus of rank n (or twisted group algebra of Z n ) is the determination of its so-called non-scalar automorphisms. We present a new algorithimic approach towards this problem based on the bivector representation ??2 :GL(n,Z)?�GL(( n 2 ),Z) of GL(n,Z) and thus compute the non-scalar automorphism group Aut( Z n ,λ) in several new cases. As an application of our ideas we show that the quantum polynomial algebra (multiparameter quantum affine space of rank n ) has only scalar (or toric) automorphisms provided that the torsion-free rank of the subgroup generated by the defining multiparameters is no less than ( n?? 2 )+1 thus improving an earlier result. We also investigate the question: when is a multiparameter quantum affine space free of so-called linear automorphisms other than those arising from the action of the n -torus ( F ??) n .

It is well known that the set of isomorphism classes of extensions of groups with abelian kernel is characterized by the second cohomology group. In this paper we generalise this characterization of extensions to a natural class of extensions of monoids, the cosetal extensions. An extension k: N -> G, e: G -> H, where k is the inclusion and e is the quotient , is cosetal if for all g,g' in G in which e(g) = e(g'), there exists a (not necessarily unique) n in N such that g = k(n)g'. These extensions generalise the notion of special Schreier extensions, which are themselves examples of Schreier extensions. Just as in the group case where a semidirect product could be associated to each extension with abelian kernel, we show that to each cosetal extension (with abelian group) kernel, we can uniquely associate a weakly Schreier split extension. The characterization of weakly Schreier split extensions is combined with a suitable notion of a factor set to provide a cohomology group granting a full characterization of cosetal extensions, as well as supplying a Baer sum.

This paper is devoted to the calculation of Batalin-Vilkovisky algebra structures on the Hochschild cohomology of skew Calabi-Yau generalized Weyl algebras. We firstly establish a Van den Bergh duality at the level of complex. Then based on the results of Solotar et al., we apply Kowalzig and Krähmer's method to the Hochschild homology of generalized Weyl algebras, and translate the homological information into cohomological one by virtue of the Van den Bergh duality, obtaining the desired Batalin-Vilkovisky algebra structures. Finally, we apply our results to quantum weighted projective lines and Podleś quantum spheres, and the Batalin-Vilkovisky algebra structures for them are described completely.

BiHom-Akivis algebras are introduced. The BiHom-commutator-BiHom-associator algebra of a regular BiHom-algebra is a BiHom-Akivis algebra. It is shown that BiHom-Akivis algebras can be obtained from Akivis algebras by twisting along two algebra endomorphisms. It is pointed out that a BiHom-Akivis algebra associated to a regular BiHom-alternative algebra is a BiHom-Malcev algebra.

The goal of this paper is to introduce BiHom-Poisson color algebras and give various constructions by using some specific maps such as morphisms. We introduce averaging operator and element of centroid for BiHom-Poisson color algebras and point out that BiHom-Poisson color algebras are closed under averaging operators and elements of centroid. We also show that any regular BiHom-associative color algebra leads to BiHom-Poisson color algebra via the commutator bracket. Then we prove that any BiHom-Poisson color algebra together with Rota-Baxter operator or multiplier give rises to another BiHom-Poisson color algebra. Next, we show that tensor product of any BiHom-associative color algebra and any BiHom-Poisson color algebra is also a BiHom- Poisson color algebra

The purpose of this paper is to determine skew-symmetric biderivations Bider s (L,V) and commuting linear maps Com(L,V) on a Hom-Lie algebra (L,α) having their ranges in an (L,α) -module (V,ρ,β) , which are both closely related to Cent(L,V) , the centroid of (V,ρ,β) . Specifically, under appropriate assumptions, every δ∈ Bider s (L,V) is of the form δ(x,y)= β −1 γ([x,y]) for some γ∈Cent(L,V) , and Com(L,V) coincides with Cent(L,V) . Besides, we give the algorithm for describing Bider s (L,V) and Com(L,V) respectively, and provide several examples.

The purpose of this paper is to introduce the notion of noncommutative BiHom-pre-Poisson algebra. Also we establish the bimodules and matched pairs of noncommutative BiHom-(pre)-Poisson algebras and related relevant properties are also given. Finally, we exploit the notion of O -operator to illustrate the relations existing between noncommutative BiHom-Poisson and noncommutative BiHom pre-Poisson algebras.

In this paper we discuss the properties of the biordered set obtained from a complemented modular lattice and defines an operation using the sandwich elements of the biordered set. Further we describe a biordered subset satisfying certain conditions, so that the complemented modular lattice admits a homogeneous basis. Finally analogous to von Neumann coordinatization theorem we describe the coordinatization theorem for complemented modular lattice using the biordered set of idempotents.

In this paper, we compute all possible differential structures of a 3 -dimensional DG Sklyanin algebra A , which is a connected cochain DG algebra whose underlying graded algebra A # is a 3 -dimensional Sklyanin algebra S a,b,c . We show that there are three major cases depending on the parameters a,b,c of the underlying Sklyanin algebra S a,b,c : (1) either a 2 ≠ b 2 or c≠0 , then ∂ A =0 ; (2) a=−b and c=0 , then the 3 -dimensional DG Sklyanin algebra is actually a DG polynomial algebra; and (3) a=b and c=0 , then the DG Sklyanin algebra is uniquely determined by a 3×3 matrix M . It is worthy to point out that case (2) has been systematically studied in \cite{MGYC} and case (3) is just the DG algebra A O −1 ( k 3 ) (M) in \cite{MWZ}. We solve the problem on how to judge whether a given 3 -dimensional DG Sklyanin algebra is Calabi-Yau.