an:05642304
Zbl 1188.16022
Jordan, David A.; Sasom, Nongkhran
Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms
EN
J. Algebra Appl. 8, No. 5, 733-757 (2009).
00256008
2009
j
16S36 16W20 17B63 16S80 16W22 16W10
skew Laurent polynomial rings; Poisson automorphisms; rings of invariants; deformations of Poisson algebras
The authors consider the skew Laurent polynomial ring \(S=R[x^{\pm 1};\alpha]\), where \(\alpha\) is an automorphism of \(R\), and study involutions \(\theta\) on \(S\) such that \(\theta(x)=x^{-1}\) and the restriction \(\theta|_R\) is an involution \(\gamma\) of \(R\). They show that such \(\theta\) exists if and only if \(\gamma\alpha\gamma^{-1}=\alpha^{-1}\), in which case they say that \(\theta\) is a reversing automorphism and \(S\) is a reversible skew Laurent polynomial ring. The concept of reversibility arises in dynamical systems and the theory of flows.
The authors study invariants for reversing automorphisms and then apply their results to two principal examples: the localization at the powers of a normal element of the enveloping algebra of the two-dimensional non-Abelian Lie algebra and the coordinate ring of the quantum torus. Both these rings are deformations of Poisson algebras over the base field \(\mathbb{F}\) and in each case the ring of \(\theta\)-invariants is a deformation of the coordinate ring of a surface in \(\mathbb{F}^3\) and is a factor of a deformation of \(\mathbb{F}[x_1,x_2,x_3]\) for a Poisson bracket determined by the appropriate surface. Both deformations are examples of algebras determined by noncommutative potentials.
Volodymyr Mazorchuk (Uppsala)