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arXiv · 2509.24054

On compatible linear and quadratic Poisson brackets on $gl(N)$

Andriy Panasyuk·Taras Skrypnyk
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Abstract

In the present paper, using two constant tensors $c$ and $b$ on $sl(N)\otimes sl(N)$ satisfying certain linear-quadratic equation and a technique of Poisson bivectors and Schouten brackets, we explicitly construct quadratic Poisson bracket on the space $ (sl(N)+\mathbb{C})^*$ which is compatible with the standard Lie--Poisson bracket on $gl(N)^* \simeq (sl(N)\oplus \mathbb{C})^*$. The case of $N=3$ is considered in details. The relation of the proposed brackets with the generalized classical Sklyanin algebras is explained.

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Andriy Panasyuk, Taras Skrypnyk. 2025-09-28. On compatible linear and quadratic Poisson brackets on $gl(N)$. https://arxiv.org/abs/2509.24054

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