Transposed Poisson Structures on Generalized Witt Algebras and Block Lie Algebras

Abstract

Abstract We describe transposed Poisson structures on generalized Witt algebras $$W(A,V,\langle \cdot ,\cdot \rangle )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and Block Lie algebras L ( A , g , f ) over a field F of characteristic zero, where $$\langle \cdot ,\cdot \rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> and f are non-degenerate. More specifically, if $$\dim (V)&gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , then all the transposed Poisson algebra structures on $$W(A,V,\langle \cdot ,\cdot \rangle )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are trivial; and if $$\dim (V)=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , 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 $$\mathcal {B}(q)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .

Topics & Concepts