On split regular BiHom-Poisson superalgebras

Abstract

The paper introduces the class of split regular BiHom-Poisson superalgebras, which is a natural generalization of split regular Hom-Poisson algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Poisson superalgebras A is of the form A = U + α I α with U a subspace of a maximal abelian subalgebra H and any I α , a well described ideal of A, satisfying [I α ,. Under certain conditions, in the case of A being of maximal length, the simplicity of the algebra is characterized.the case of Hom-Lie algebras, the relevant structure for a tensor theory is a Hom-Poisson algebra structure. A Hom-Poisson algebra has simultaneously a Hom-Lie algebra structure and a Hom-associative algebra structure, satisfying the Hom-Leibniz identity in [18]. In [20], Wang, Zhang and Wei characterized Hom-Leibniz superalgebras and Hom-Leibniz Poisson superalgebras, and presented the methods to construct these superalgebras.A BiHom-algebra is an algebra in such a way that the identities defining the structure are twisted by two homomorphisms φ and ψ. This class of algebras was introduced from a categorical approach in [13] which as an extension of the class of Hom-algebras. If the two linear maps are the same automorphisms, BiHom-algebras will be return to Hom-algebras. These algebraic structures include BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras. The representation theory of BiHom-Lie algebras was introduced by Cheng and Qi in [12], in which, BiHom-cochain complexes, derivation, central extension, derivation extension, trivial representation and adjoint representation of BiHom-Lie algebras were studied. More applications of BiHom-algebras, BiHom-Lie superalgebras, BiHom-Lie colour algebras and BiHom-Novikov algebras can be found in ([16], [21], [1], [14]).The class of the split algebras is specially related to addition quantum numbers, graded contractions and deformations. For instance, for a physical system which displays a symmetry, it is interesting to know the detailed structure of the split decomposition, since its roots can be seen as certain eigenvalues which are the additive quantum numbers characterizing the state of such system. Determining the structure of split algebras will become more and more meaningful in the area of research in mathematical physics. Recently,[2][3][4][5][6][7][8], [9]-[11], [22]-[23]), the structure of different classes of split algebras have been determined by the techniques of connections of roots. The purpose of this paper is to consider the class of split regular BiHom-Poisson superalgebras, which is a natural extension of split regular BiHom-Lie superalgebras and split Hom-Lie superalgebras.In Section 2, we prove that such an arbitrary split regular BiHom-Poisson superalgebras A is of the form A = U + α I α with U a subspace of a maximal abelian subalgebra H and any I α , a well described ideal of A, satisfying [I α , I β ] + I α I β = 0 if [α] = [β].In Section 3, we ...

Transposed BiHom-Poisson algebras

“…Roughly speaking, a BiHom-associative algebra (or Lie algebra) is an algebra (or Lie algebra) such that the associativity (or Jacobi condition) is twisted by two (commuting) endomorphisms, for details see [10], which can be seen as an extension of Hom-type algebra [13] arising in quasi-deformations of Lie algebras of vector fields. Now there are so many research related to BiHom-type algebras, see refs [5,11,12,[15][16][17][18][19][20][21][23][24][25][26][27][28]. In [21], the authors introduced the notion of BiHom-Poisson algebras and gave a necessary and sufficient condition under which BiHom-Novikov-Poisson algebras (which are twisted generalizations of Novikov-Poisson algebras [30] and Hom-Novikov-Poisson algebras [31]) give rise to BiHom-Poisson algebras.…”

On split regular BiHom-Leibniz superalgebras

“…Determining the structure of split algebras will become more and more meaningful in the area of research in mathematical physics. Recently,[4][5][6][7][8][9][10][11][12][13][14][15], [19]), the structure of different classes of split algebras have been determined by the techniques of connections of roots. The purpose of this paper is to consider the structure of split regular BiHom-Leiniz superalgebras by the techniques of connections of roots based on some work in [15] and [27].…”