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Finite-dimensional simple Poisson modules. (English) Zbl 1239.17019
The paper under review concerns the Poisson representation theory of Poisson algebras. In section 2, the author provides the necessary definitions of Poisson algebras, subalgebras, ideals, (iso)morphisms, modules over a Poisson algebra and module homomorphisms.
Let \(A=\mathbb{C}[x,y,z]\), \(f\in A\), and define \(\{,\}_f\) on \(A\) by the relations \(\{x,y\}_f=\partial f/\partial z, \{y,z\}_f=\partial f/\partial z, \{z,x\}_f=\partial f/\partial y\) and such that \(f\) is in the Poisson center of \(A\). Such a Poisson bracket will be called exact and determined by \(f\). Let also \(T\) be \(\mathbb{C}\)-algebra, \(t\in T\) such that \(B:=T/tT\) is non-trivial and commutative. Then \(T\) is called a quantization of the Poisson algebra \(B\) and every algebra of the form \(B_\lambda:=T/(t-\lambda)T,\;\lambda\in\mathbb{C}\)m, will be called a deformation.
The main result of the paper comes in section 3 and states the following: Let \((A,\{,\})\) be a finitely generated Poisson algebra, \((M,\{,\}_M)\) a finite-dimensional (henceforth f.d.) simple (i.e., having no proper Poisson submodules) Poisson \(A\)-module and \(J=\text{ann}_A(M)\) be the annihilator of \(M\) in \(A\) (as an ordinary module). Then there is a simple module \(M^\ast\) of the Lie algebra \(J/J^2\) s.t \(M=M^\ast\) as \(\mathbb{C}\)-vector spaces, and \(\forall j\in J,\;m\in M,\;[j+J^2,m]_{M^\ast}=\{j,m\}_M\). Secondly, let \(J\) be a Poisson maximal ideal of \(A\), that is a maximal ideal of \(A\) for the underlying multiplicative structure, which is also Poisson ideal for the Poisson structure of \(A\), and let \(N\) be a f.d. simple \(J/J^2\)-module. Then there exist a simple Poisson \(A\)-module \(N^\dagger\) and a Lie homomorphism \(f:A\to J/J^2\) such that the Poisson bracket \(\{,\}_{N^\dagger}\) is exact and determined by \(f\), and \(\text{ann}_A(N^\dagger)=J\). In fact, if \(M\) is a simple Poisson module with \(\dim(M)<\infty\), then \(M^{\ast\dagger}=M\). Conversely, for all Poisson maximal ideals \(J\) and all finite-dimensional simple \(J/J^2\)-modules \(N\), one has \(N^{\dagger\ast}=N\). This way, there is a bijection from the set of isomorphism classes of f.d. simple Poisson modules over \(A\) to the set of pairs \((J,\widehat{N})\), where \(J\) is a Poisson maximal ideal of \(A\) and \(\widehat{N}\) is the isomorphism class of the f.d. simple \(J/J^2\)-module \(\Gamma(\widehat{M})=(\text{ann}_A(M),\widehat{M^\ast})\).
The second part of the paper covers some concrete examples on the correspondence of f.d. simple modules over deformations. The author further studies how the f.d. simple Poisson modules behave when, using a finite group \(G\) of Poisson automorphisms of \(A\), one passes from \(A\) to \(A^G\). The examples treated are those of the Kleinian singularity of type \(A_1\), the invariants of the 2-torus, the invariants of \(\mathbb{C}[x^{\pm 1},y]\), the algebra \(U_q(\mathfrak{sl}_2)\), 1-dimensional simple Poisson modules and some other invariants of the torus with different behaviour than the ones considered previously.
In the last section the author considers Poisson algebras arising in Lie theory. Namely, let \(\mathfrak g\) be a simple Lie algebra, \(\mathfrak h\) a Cartan subalgebra and \(W\) the Weyl group. Let \(V:=\mathfrak h\oplus\mathfrak h^\ast\), and \(S(V)\) be the symmetric algebra of \(V\). Motivated by the work of J. Alev and L. Foissy [Commun. Algebra 37, No. 1, 368–388 (2009; Zbl 1287.17043)] computing the \(0\)-th Poisson homology of \(S(V)^W\), for \(\mathfrak g=A_2, B_2, G_2\) and \(\dim\mathfrak h=2\), the author discusses the results of the paper in these examples.

MSC:
17B63 Poisson algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16S80 Deformations of associative rings
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
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