Transposed Poisson Structures on Generalized Witt Algebras and Block Lie Algebras

Abstract

AbstractWe describe transposed Poisson structures on generalized Witt algebras $$W(A,V,\langle \cdot ,\cdot \rangle )$$ W ( A , V , ⟨ · , · ⟩ ) and Block Lie algebras L(A, g, f) over a field F of characteristic zero, where $$\langle \cdot ,\cdot \rangle $$ ⟨ · , · ⟩ and f are non-degenerate. More specifically, if $$\dim (V)>1$$ dim ( V ) > 1 , then all the transposed Poisson algebra structures on $$W(A,V,\langle \cdot ,\cdot \rangle )$$ W ( A , V , ⟨ · , · ⟩ ) are trivial; and if $$\dim (V)=1$$ dim ( V ) = 1 , then such structures are, up to isomorphism, mutations of the group algebra structure on FA. The transposed Poisson algebra structures on L(A, g, f) are in a one-to-one correspondence with commutative and associative multiplications defined on a complement of the square of L(A, g, f) with values in the center of L(A, g, f). In particular, all of them are usual Poisson structures on L(A, g, f). This generalizes earlier results about transposed Poisson structures on Block Lie algebras $$\mathcal {B}(q)$$ B ( q ) .