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- W4315705930 abstract "We introduce the emph{universal algebra} of two Poisson algebras $P$ and $Q$ as a commutative algebra $A:={mathcal P} (P, , Q )$ satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional Poisson algebra $P$ and several of its applications are highlighted. For any Poisson $P$-module $U$, we construct a functor $U otimes - colon {}_{A} {mathcal M} to {}_Q{mathcal P}{mathcal M}$ from the category of $A$-modules to the category of Poisson $Q$-modules which has a left adjoint whenever $U$ is finite dimensional. Similarly, if $V$ is an $A$-module, then there exists another functor $ - otimes V colon {}_P{mathcal P}{mathcal M} to {}_Q{mathcal P}{mathcal M}$ connecting the categories of Poisson representations of $P$ and $Q$ and the latter functor also admits a left adjoint if $V$ is finite dimensional. If $P$ is $n$-dimensional, then ${mathcal P} (P) := {mathcal P} (P, , P)$ is the initial object in the category of all commutative bialgebras coacting on $P$. As an algebra, ${mathcal P} (P)$ can be deescribed as the quotient of the polynomial algebra $k[X_{ij} , | , i, j = 1, cdots, n]$ through an ideal generated by $2 n^3$ non-homogeneous polynomials of degree $leq 2$. Two applications are provided. The first one describes the automorphisms group ${rm Aut}_{rm Poiss} (P)$ as the group of all invertible group-like elements of the finite dual ${mathcal P} (P)^{rm o}$. Secondly, we show that for an abelian group $G$, all $G$-gradings on $P$ can be explicitly described and classified in terms of the universal coacting bialgebra ${mathcal P} (P)$." @default.
- W4315705930 created "2023-01-12" @default.
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- W4315705930 date "2023-01-10" @default.
- W4315705930 modified "2023-09-26" @default.
- W4315705930 title "Universal constructions for Poisson algebras. Applications" @default.
- W4315705930 doi "https://doi.org/10.48550/arxiv.2301.03807" @default.
- W4315705930 hasPublicationYear "2023" @default.
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