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

2025/09/28 by Andriy Panasyuk, Panasyuk, Andriy, T. Skrypnyk +1
Mathematics · #Advanced Algebra and Geometry #Finite Group Theory Research #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.2509.24054

Abstract

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.

Citations

• On linear-quadratic Poisson pencils on trivial central extensions of\n semisimple Lie algebras
• Torsion free sheaves on Weierstrass cubic curves and the classical\n Yang-Baxter equation

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