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14 June 2024
 
  » arxiv » 2302.00403

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Transposed Poisson structures on generalized Witt algebras and Block Lie algebras
Ivan Kaygorodov ; Mykola Khrypchenko ;
Date 1 Feb 2023
AbstractWe describe transposed Poisson structures on generalized Witt algebras $W(A,V, langle cdot,cdot angle )$ and Block Lie algebras $L(A,g,f)$ over a field $F$ of characteristic zero, where $langle cdot,cdot angle$ and $f$ are non-degenerate. More specifically, if $dim(V)>1$, then all the transposed Poisson algebra structures on $W(A,V,langle cdot,cdot angle)$ are trivial; and if $dim(V)=1$, 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 $mathcal{B}(q)$.
Source arXiv, 2302.00403
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