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

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.