Some remarks on periodic gradings
Abstract
Let $\mathfrak q$ be a finite-dimensional Lie algebra, $\vartheta\in Aut(\mathfrak q)$ a finite order automorphism, and $\mathfrak q_0$ the subalgebra of fixed points of $\vartheta$. Using $\vartheta$ one can construct a pencil $\mathcal P$ of compatible Poisson brackets on $S(\mathfrak q)$, and thereby a `large' Poisson-commutative subalgebra $Z(\mathfrak q,\vartheta)$ consisting of $\mathfrak q_0$-invariants in $S(\mathfrak q)$. We study one particular bracket $\{\,\,,\,\}_{\infty}\in\mathcal P$ and the related Poisson centre ${\mathcal Z}_\infty$. It is shown that ${\mathcal Z}_\infty$ is a polynomial ring, if $\mathfrak q$ is reductive.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.00599
- arXiv:
- arXiv:2405.00599
- Bibcode:
- 2024arXiv240500599Y
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Algebraic Geometry
- E-Print:
- To appear in Proceedings of the 14th Ukraine Algebra Conference, Contemporary Mathematics. arXiv admin note: text overlap with arXiv:2102.10065