Drinfeld Double for Infinitesimal BiHom-bialgebras

Nonhomogeneous associative Yang-Baxter equations

“…The relations among nonhomogeneous associative (BiHom-)Yang-Baxter pair, double curved Rota-Baxter (BiHom-)system, double curved weak BiHom-pseudotwistor and BiHom-associative algebra are discussed (see Theorems 2.5, 2.10 and 2.16). Equivalent characterizations of (quasitriangular) covariant BiHom-bialgebra are also given (see Theorem 2.23), which can recover [23,Theorem 3.5] and [5,Proposition 3.10]. And also we prove that the solution to nonhomogeneous associative BiHom-Yang-Baxter equation (abhYBe) can be provided by unitary quasitriangular covariant BiHom-bialgebra (Proposition 2.33).…”

Coquasitriangular infinitesimal BiHom-bialgebras and related structures

“…(2) If further (A, µ, 1, α, β) is a unitary BiHom-associative algebra, then the 8-tuple (A, µ, [20,21,25]. If further α = β = ψ = ω = id, then one obtains Joni and Rota's infinitesimal bialgebras [12].…”

“…We recall from [23] the definition of (anti)quasitriangular unitary λ-infBH-bialgebra. For convenience, we follow the notations in [17] or [25]. Let (A, µ, α, β) be a unitary BiHomassociative algebra, ψ, ω : A −→ A be linear maps and r ∈ A ⊗ A.…”

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.…”