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October 20, 20250 citationsOpen Access

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

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APAndriy PanasyukTST. Skrypnyk

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. (2025) studied this question.

synapsesocial.com/papers/68f5fcce8d54a28a75cf1b86https://doi.org/10.48550/arxiv.2509.24054
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