BiHom-H-伪代数及其构造（英文）
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摘要
首先,给出了BiHom-结合H-伪代数的定义与例子,一个BiHom-结合H-伪代数由一个H-伪代数(A,μ)和满足BiHom-结合律的2个映射α,β∈Hom_H(A,A)构成,其为BiHom-结合代数和结合H-伪代数的推广.然且,介绍了名为Yau扭曲的方法,该方法是由一个结合H-伪代数(A,μ)和2个H-伪代数同态α,β构造BiHom-结合H-伪代数(A,(■α)μ,α,β).同时,介绍了Yau扭曲的推广形式,即由一个BiHom-结合H-伪代数(A,μ,α~～,β~～)和2个映射α,β∈Hom_H(A,A)构造BiHom-结合H-伪代数(A,μ(α■β),α~～α,β~～β).最后,给出了在2个BiHom-结合H-伪代数的张量积空间A■B上构造BiHom-结合H-伪代数的方法.
The definition and an example of BiHom-associative H-pseudoalgebra are given. A BiHom-H-pseudoalgebra is an H-pseudoalgebra(A,μ) with two maps α,β∈Hom_H(A,A) satisfying the BiHom-associative law which generalizes BiHom-associative algebras and associative H-pseudoalgebras. Secondly, a method which is called the Yau twist of constructing BiHom-associative H-pseudoaglebra(A,(■α)μ,α,β) from an associative H-pseudoalgebra(A,μ) and two maps of H-pseudoalgebras α,β, is introduced. Thirdly, a generalized form of the Yau twist is discussed. It concerns constructing a BiHom-associative H-pseudoalgebra(A,μ(α■β),α~～α,β~～β) from a BiHom-associative H-pseudoalgebra(A,μ,α~～,β~～) and two maps α,β∈Hom_H(A,A). Finally, a method of constructing BiHom-associative H-pseudoalgebra on tensor product space A■B of two BiHom-associative H-pseudoalgebras is given.
The definition and an example of BiHom-associative H-pseudoalgebra are given. A BiHom-H-pseudoalgebra is an H-pseudoalgebra(A,μ) with two maps α,β∈Hom_H(A,A) satisfying the BiHom-associative law which generalizes BiHom-associative algebras and associative H-pseudoalgebras. Secondly, a method which is called the Yau twist of constructing BiHom-associative H-pseudoaglebra(A,(■α)μ,α,β) from an associative H-pseudoalgebra(A,μ) and two maps of H-pseudoalgebras α,β, is introduced. Thirdly, a generalized form of the Yau twist is discussed. It concerns constructing a BiHom-associative H-pseudoalgebra(A,μ(α■β),α~～α,β~～β) from a BiHom-associative H-pseudoalgebra(A,μ,α~～,β~～) and two maps α,β∈Hom_H(A,A). Finally, a method of constructing BiHom-associative H-pseudoalgebra on tensor product space A■B of two BiHom-associative H-pseudoalgebras is given.
引文
[1] Graziani G,Makhlouf A,Menini C,et al.BiHom-associative algebras,BiHom-Lie algebras and BiHom-bialgebras[J].Symmetry,Integrability and Geometry:Methods and Applications,2015,11:086-1-086-34.DOI:10.3842/sigma.2015.086.
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[3] Makhlouf A,Silvestrov S.Hom-algebras and Hom-coalgebras[J].Journal of Algebra and Its Applications,2010,9(4):553-589.DOI:10.1142/s0219498810004117.
[4] Yau D.Hom-algebras and homology[J].Journal of Lie Theory,2009,19(2):409-421.
[5] Bakalov B,D'Andrea A,Kac V G.Theory of finite pseudoalgebras[J].Advances in Mathematics,2001,162(1):1-140.DOI:10.1006/aima.2001.1993.
[6] Kac V.Vertex algebras for beginners[M].Providence,Rhode Island:American Mathematical Society,1998.DOI:10.1090/ulect/010.
[7] Dorfman I.Dirac structures and integrability of nonlinear evolution equations [M].New York:John Wiley & Sons,1993.
[8] Gel'Fand I M,Dorfman I Y.Hamiltonian operators and infinite-dimensional Lie algebras[J].Functional Analysis and Its Applications,1982,15(3):173-187.DOI:10.1007/bf01089922.
[9] Xu X P.Equivalence of conformal superalgebras to Hamiltonian superoperators[J].Algebra Colloquium,2001,8(1):63-92.
[10] Sun Q X.Generalization of H-pseudoalgebraic structures[J].Journal of Mathematical Physics,2012,53(1):012105.DOI:10.1063/1.3665708.
[11] Wu Z X.Leibniz H-pseudoalgebras[J].Journal of Algebra,2015,437:1-33.DOI:10.1016/j.jalgebra.2015.04.019.
[2] Makhlouf A,Silvestrov S D.Hom-algebra structures[J].Journal of Generalized Lie Theory and Applications,2008,2(2):51-64.DOI:10.4303/jglta/s070206.
[3] Makhlouf A,Silvestrov S.Hom-algebras and Hom-coalgebras[J].Journal of Algebra and Its Applications,2010,9(4):553-589.DOI:10.1142/s0219498810004117.
[4] Yau D.Hom-algebras and homology[J].Journal of Lie Theory,2009,19(2):409-421.
[5] Bakalov B,D'Andrea A,Kac V G.Theory of finite pseudoalgebras[J].Advances in Mathematics,2001,162(1):1-140.DOI:10.1006/aima.2001.1993.
[6] Kac V.Vertex algebras for beginners[M].Providence,Rhode Island:American Mathematical Society,1998.DOI:10.1090/ulect/010.
[7] Dorfman I.Dirac structures and integrability of nonlinear evolution equations [M].New York:John Wiley & Sons,1993.
[8] Gel'Fand I M,Dorfman I Y.Hamiltonian operators and infinite-dimensional Lie algebras[J].Functional Analysis and Its Applications,1982,15(3):173-187.DOI:10.1007/bf01089922.
[9] Xu X P.Equivalence of conformal superalgebras to Hamiltonian superoperators[J].Algebra Colloquium,2001,8(1):63-92.
[10] Sun Q X.Generalization of H-pseudoalgebraic structures[J].Journal of Mathematical Physics,2012,53(1):012105.DOI:10.1063/1.3665708.
[11] Wu Z X.Leibniz H-pseudoalgebras[J].Journal of Algebra,2015,437:1-33.DOI:10.1016/j.jalgebra.2015.04.019.