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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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##### References:
 [1] Alev, J., Farkas, D.R.: Finite group actions on Poisson algebras. In: Duval, C., Guieu, L., Ovsienko, V. (eds.) The Orbit Method in Geometry and Physics (Marseille, 2000), Progress in Mathematics, vol. 213, pp. 9–28. Birkhäuser Boston, Boston, MA (2003) · Zbl 1079.16018 [2] Alev, J., Foissy, L.: Le groupe de traces de Poisson de la variété quotient $$\mathfrak{h}\oplus\mathfrak{h}^*/W$$ en rang 2. preprint posted on arXiv:math/0603142v2 (July) (2007) [3] Alev, J., Lambre, T.: Comparaison de l’homologie de Hochschild et de l’homologie de Poisson pour une déformation des surfaces de Klein. In: Algebra and Operator Theory (Tashkent 1997), pp. 25–38. Kluwer, Dordrecht (1998) · Zbl 0931.16007 [4] Bavula, V.V., Jordan, D.A.: Isomorphism problems and groups of automorphisms for generalized Weyl algebras. Trans. Amer. Math. Soc. 353, 769–794 (2001) · Zbl 0961.16016 · doi:10.1090/S0002-9947-00-02678-7 [5] Borho, W., Gabriel, P., Rentschler, R.: Primideale in Einhüllenden Auflösbarer Lie-Algebren. Lecture Notes in Mathematics, vol. 357. Springer, Berlin (1973) · Zbl 0293.17005 [6] Brown, K.A., Goodearl, K.R.: Lectures on Algebraic Quantum Groups (Advanced Courses in Mathematics CRM Barcelona). Birkhäuser, Basel (2002) [7] Brown, K.A., Gordon, I.: Poisson orders, symplectic reflection algebras and representation theory. J. Reine Angew. Math. 559, 193–216 (2003) · Zbl 1025.17007 · doi:10.1515/crll.2003.048 [8] Cho, E.H., Oh, S.-Q.: Primitive spectrum of quantum (2 $$\times$$ 2)-matrices and associated Poisson structure. Far East J. Math. Sci. 6, 251–259 (1998) · Zbl 0913.16004 [9] Dixmier, J.: Enveloping Algebras. Grad. Stud. Math. 11. American Mathematical Society, Providence, RI (1996) · Zbl 0867.17001 [10] Erdmann, K., Wildon, M.J.: Introduction to Lie Algebras. Springer, London (2006) · Zbl 1139.17001 [11] Farkas, D.R.: Modules for Poisson algebras. Comm. Algebra 28, 3293–3306 (2000) · Zbl 1018.17014 · doi:10.1080/00927870008827025 [12] Goodearl, K.R.: A Dixmier-Moeglin equivalence for Poisson algebras with torus actions. In: Algebra and its Applications (Athens, Ohio, 2005). Contemp. Math., vol. 419, pp. 131–154. American Mathematical Society, Providence, RI (2006) · Zbl 1147.17017 [13] Goodearl, K.R., Launois, S.: The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras. Preprint posted on arXiv:math/0705.3486v1 (2007) (May) [14] Goodearl, K.R., Letzter, E.S.: The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras. Trans. Amer. Math. Soc. 352, 1381–1403 (2000) · Zbl 0978.16040 · doi:10.1090/S0002-9947-99-02345-4 [15] Havlicek, M., Posta, S.: On the classification of irreducible finite-dimensional representations of $$U^\prime_q(so_3)$$ algebra. J. Math. Phys. 42, 472–500 (2001) · Zbl 1032.17022 · doi:10.1063/1.1328078 [16] Havlicek, M., Klimyk, A.V., Posta, S.: Representations of the cyclically q-deformed algebra so q (3). J. Math. Phys. 40, 2135–2161 (1999) · Zbl 0959.17015 · doi:10.1063/1.532856 [17] Hodges, T.J.: Noncommutative deformations of type-A Kleinian singularities. J. Algebra 161, 271–290 (1993) · Zbl 0807.16029 · doi:10.1006/jabr.1993.1219 [18] Hodges, T.J., Levasseur, T.: Primitive ideals of $$C_{q}[{\text SL}(n)]$$ . J. Algebra 168, 455–468 (1994) · Zbl 0814.17012 · doi:10.1006/jabr.1994.1239 [19] Ito, T., Terwilliger, P., Weng, C.: The quantum algebra U q (sl 2) and its equitable presentation. J. Algebra 298, 284–301 (2006) · Zbl 1090.17004 · doi:10.1016/j.jalgebra.2005.07.038 [20] Jordan, D.A.: Iterated skew polynomial rings and quantum groups. J. Algebra 156, 194–218 (1993) · Zbl 0809.16032 · doi:10.1006/jabr.1993.1070 [21] Jordan, D.A.: Primitivity in skew Laurent polynomial rings and related rings. Math. Z. 213, 353–371 (1993) · Zbl 0797.16037 · doi:10.1007/BF03025725 [22] Jordan, D.A.: Finite-dimensional simple modules over certain iterated skew polynomial rings. J. Pure Appl. Algebra 98, 45–55 (1995) · Zbl 0829.16017 [23] Jordan, D.A.: Down-up algebras and ambiskew polynomial rings. J. Algebra 228, 311–346 (2000) · Zbl 0958.16030 · doi:10.1006/jabr.1999.8264 [24] Jordan, D.A., Sasom, N.: Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms. Preprint posted on arXiv:math/0708.3923v1 (August 2007) [25] Jordan, D.A., Wells, I.E.: Invariants for automorphisms of certain iterated skew polynomial rings. Proc. Edinburgh Math. Soc. 39, 461–472 (1996) · Zbl 0864.16027 · doi:10.1017/S0013091500023221 [26] Joseph, A.: Quantum groups and their primitive ideals. Ergeb. Math., vol. 29(3). Springer Verlag, Berlin (1995) · Zbl 0808.17004 [27] Kassel, C.: L’homologie cyclique des algèbres enveloppantes. Invent. Math. 91, 221–251 (1988) · Zbl 0653.17007 · doi:10.1007/BF01389366 [28] Kassel, C.: Quantum Groups, Graduate Texts in Mathematics, vol. 155. Springer, New York (1995) · Zbl 0808.17003 [29] Kraft, H., Small, L.W.: Invariant algebras and completely reducible representations. Math. Res. Lett. 1, 45–55 (1994) · Zbl 0849.16036 [30] Lorenz, M.: Multiplicative Invariant Theory. Encyclopaedia of Mathematical Sciences, vol. 135. Springer, Berlin (2005) [31] Mathieu, O.: Bicontinuity of the Dixmier map. J. Amer. Math. Soc. 4, 837–863 (1991) · Zbl 0743.17013 · doi:10.1090/S0894-0347-1991-1115787-5 [32] Oh, S.-Q.: Symplectic ideals of Poisson algebras and the Poisson structure associated to quantum matrices. Comm. Algebra 27, 2163–2180 (1999) · Zbl 0936.16041 · doi:10.1080/00927879908826555 [33] Oh, S.-Q.: Poisson enveloping algebras. Comm. Algebra 27, 2181–2186 (1999) · Zbl 0936.16020 · doi:10.1080/00927879908826556 [34] Passman, D.S.: Infinite Crossed Products. Academic, San Diego (1989) · Zbl 0662.16001 [35] Sasom, N.: Reversible skew Laurent polynomial rings, rings of invariants and related rings. Ph.D. thesis, University of Sheffield, 2006, http://david-jordan.staff.shef.ac.uk/NSthesis.pdf [36] Smith, S.P.: Quantum groups: an introduction and survey for ring theorists. In: Montgomery, S., Small, L. (eds.) Noncommutative Rings (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., vol. 24, pp. 131–178. Springer, New York (1992) · Zbl 0744.16023
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