Reflection Groups and Rigidity of Quadratic Poisson Algebras

In this paper, we study the invariant theory of quadratic Poisson algebras. Let G be a finite group of the graded Poisson automorphisms of a quadratic Poisson algebra A. When the Poisson bracket of A is skew-symmetric, a Poisson version of the Shephard-Todd-Chevalley theorem is proved stating that the fixed Poisson subring A^G is skew-symmetric if and only if G is generated by reflections. For many other well-known families of quadratic Poisson algebras, we show that G contains limited or even no reflections. This kind of Poisson rigidity result ensures that the corresponding fixed Poisson subring A^G is not isomorphic to A as Poisson algebras unless G is trivial.

[1]  P. Polo,et al.  A Rigidity Theorem for Finite Group Actions on Enveloping Algebras of Semisimple Lie Algebras , 1995 .

[2]  Luigi Ferraro,et al.  Differential graded algebra over quotients of skew polynomial rings by normal elements , 2019, Transactions of the American Mathematical Society.

[3]  M. Audard,et al.  Detection of the Neupert Effect in the Corona of an RS Canum Venaticorum Binary System by XMM-Newton and the Very Large Array , 2002, astro-ph/0206080.

[4]  A. Tikaradze.  The Weyl algebra as a fixed ring , 2017, Advances in Mathematics.

[5]  Jason Gaddis,et al.  The Zariski cancellation problem for Poisson algebras , 2019, Journal of the London Mathematical Society.

[6]  J. J. Zhang,et al.  Shephard–Todd–Chevalley Theorem for Skew Polynomial Rings , 2008, 0806.3210.

[7]  A. Polishchuk.  Algebraic geometry of Poisson brackets , 1997 .

[8]  Chelsea M. Walton,et al.  Poisson geometry of PI three‐dimensional Sklyanin algebras , 2017, Proceedings of the London Mathematical Society.

[9]  Xingting Wang,et al.  Homological unimodularity and Calabi–Yau condition for Poisson algebras , 2016, 1608.00172.

[10]  R. Przybysz.  On one class of exact Poisson structures , 2001 .

[11]  U. Washington,et al.  Rigidity of graded regular algebras , 2007, 0706.0662.

[12]  C. Chevalley.  Invariants of Finite Groups Generated by Reflections , 1955 .

[13]  Kei-ichi Watanabe.  Certain invariant subrings are Gorenstein. II , 1974 .

[14]  U. Umirbaev.  Universal enveloping algebras and universal derivations of Poisson algebras , 2011, 1102.0366.

[15]  Larry Smith,et al.  Polynomial Invariants of Finite Groups , 1995 .

[16]  渡辺 敬一.  Certain invariant subrings are Gorenstein , 1976 .

[17]  Certain invariant subrings are Gorenstein. II , 1974 .

[18]  Jason Gaddis.  The isomorphism problem for quantum affine spaces, homogenized quantized Weyl algebras, and quantum matrix algebras , 2016, 1605.08711.

[19]  Jia-Feng Lu,et al.  Universal enveloping algebras of Poisson Ore extensions , 2014, 1403.5852.

[20]  Sei-Qwon Oh.  Poisson enveloping algebras , 1999 .

[21]  A. Borel.  Essays in the history of Lie groups and algebraic groups , 2001 .

[22]  G. C. Shephard,et al.  Finite Unitary Reflection Groups , 1954, Canadian Journal of Mathematics.

[23]  Milen Yakimov,et al.  Poisson geometry and representations of PI 4-dimensional Sklyanin algebras , 2018, Selecta Mathematica.

[24]  M. Vancliff.  The Defining Relations of Quantum n × n Matrices , 1995 .

[25]  L. Topley,et al.  The orbit method for Poisson orders , 2017, Proceedings of the London Mathematical Society.

[26]  S. P. Smith.  Can the Weyl algebra be a fixed ring , 1989 .

[27]  Kenneth A. Brown,et al.  Poisson orders, symplectic reflection algebras and representation theory , 2002 .

[28]  James J. Zhang,et al.  Unimodular graded Poisson Hopf algebras , 2017, Bulletin of the London Mathematical Society.

[29]  Sei-Qwon Oh.  Poisson Polynomial Rings , 2006 .

[30]  B. G. R. Bossoto,et al.  A-Poisson structures , 2010, 1010.3398.

[31]  M. Kontsevich,et al.  Automorphisms of the Weyl Algebra , 2005, math/0512169.

[32]  J. J. Zhang,et al.  INVARIANT THEORY OF FINITE GROUP ACTIONS ON DOWN-UP ALGEBRAS , 2013, 1308.0579.

[33]  S.-Q. Wang,et al.  Twisted Poincar\'{e} duality between Poisson homology and Poisson cohomology , 2014, 1404.5177.

[34]  David J. Benson,et al.  Polynomial invariants of finite groups , 1993 .

[35]  K. Goodearl.  Semiclassical Limits of Quantized Coordinate Rings , 2008, 0812.1612.

[36]  J. Dixmier.  Sur les algèbres de Weyl , 1968 .

[37]  D. Farkas,et al.  Finite Group Actions on Poisson Algebras , 2003 .

[38]  A. Tikaradze.  Derived invariants of the fixed ring of enveloping algebras of semisimple Lie algebras , 2019, Mathematische Zeitschrift.