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This paper constructs quadratic Poisson brackets on (sl(N)+C), compatible with the standard Lie-Poisson bracket on gl(N), suggesting links to Sklyanin algebras.
In the present paper, using two constant tensors c and b on sl(N)⊗ 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)+C)^* which is compatible with the standard Lie--Poisson bracket on gl(N)^* (sl(N)⊕ 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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Panasyuk et al. (2025) studied this question.
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