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This is a Quantitative Study study.

October 20, 2025Open Access

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

Key Points

The explicit construction of quadratic Poisson brackets is detailed for the space (sl(N)+C)*, enhancing existing structures.
Compatibility of newly defined brackets with the standard Lie-Poisson bracket on gl(N)* is shown, pointing to broader implications.
Using specific tensors and advanced techniques like Poisson bivectors, it provides innovative approaches to the algebraic structure and relations.
Case studies, particularly for N=3, offer insights into practical applications and theoretical explorations of the bracket structures.

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.

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Cite This Study

Panasyuk et al. (Sun,) studied this question.

synapsesocial.com/papers/68f5fcce8d54a28a75cf1b86 https://doi.org/https://doi.org/10.48550/arxiv.2509.24054

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