Transposed Poisson structures on Block and Witt type Lie algebras

A transposed Poisson algebra  is a triple $(\mathcal{L},\cdot,[\cdot,\cdot])$ consisting of a vector space $\mathcal{L}$ with two bilinear operations $\cdot$ and $[\cdot,\cdot]$, such that

1. $(\mathcal{L},\cdot)$ is a commutative associative algebra;
2. $(\mathcal{L},[\cdot,\cdot])$ is a Lie algebra;
3. the "transposed" Leibniz law holds: $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y]$ for all
$x,y,z\in \mathcal{L}$.

A transposed Poisson algebra structure on a Lie algebra $(\mathcal{L},[\cdot,\cdot])$ is a (commutative associative) multiplication $\cdot$ on $\mathcal{L}$ such that $(\mathcal{L},\cdot,[\cdot,\cdot])$ is a transposed Poisson algebra.

In this talk we will present results based on two joint works  with Ivan Kaygorodov (Universidade da Beira Interior). In the first work, we describe transposed Poisson algebra structures on Block Lie algebras $\mathcal{B}(q)$ and Block Lie superalgebras $\mathcal{S}(q)$, where $q$ is an arbitrary complex number. In the second work, we solve the same problem for Witt type algebras $V(f)$.

Start Date
Venue
FC1 029
End Date

Speaker

Mykola Khrypchenko

Speaker's Institution

Centro de Matemática da Universidade do Porto

Area

Algebra, Combinatorics and Number Theory