PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Let 𝔮 be a finite-dimensional Lie algebra. The symmetric algebra (𝔮) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (𝔮) to any finite order automorphism ϑ of 𝔮. We study related Poisson-commutative subalgebras (𝔮; ϑ) of 𝒮(𝔮) and associated Lie algebra contractions of 𝔮. To obtain substantial results, we have to assume that 𝔮 = 𝔤 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If 𝔤 = 𝔥⊕⋯⊕𝔥 (sum of k copies), where 𝔥 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (𝔮; ϑ) is polynomial and maximal. Furthermore, we quantise this (𝔤; ϑ) using a Gaudin subalgebra in the enveloping algebra 𝒰(𝔤).

VERTEX OPERATOR ALGEBRAS AND WEAK JACOBI FORMS

Let V be a strongly regular vertex operator algebra. For a state h ∈ V1 satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr M qL(0)-c/24ζh(0) (M a V-module) is a vector-valued weak Jacobi form of weight 0 and a certain index 〈h, h〉/2. We discuss refinements and applications of this result when V is holomorphic, in particular we prove that if g = eh(0) is a finite-order automorphism then Tr V qL(0)-c/24g is a modular function of weight 0 on a congruence subgroup of SL 2(ℤ).

Lie Tori and Their Fixed Point Subalgebras

We study the fixed point subalgebra of a centerless irreducible Lie torus under a certain finite order automorphism. We investigate which axioms of a Lie torus hold for the fixed points and which do not. We relate our study to some recent results about the fixed points of extended affine Lie algebras under a class of finite order automorphisms.

Topological types of Klein surfaces with a maximum order automorphism

If X is a Klein surface (KS) with boundary, of algebraic genus p, and Φ is an automorphism of order N, May [8] proved that N ≤ 2p + 2 when X is orientable and p is even, and N ≤ 2p otherwise.He proved also that the unique topological type of an orientable KS having an orientation-preserving automorphism of maximum order is a surface with one boundary component when p is even, with two boundary components when p is odd.

Real order-automorphism groups

By using the concept of tame embeddings of chains, a characterization is given of the subobjects of the lattice-ordered groups of order-automorphisms of the chains of rational and real numbers.