Rational Calogero models based on rank-2 root systems: supertraces on the superalgebras of observables

An invariant supertrace for the category of representations of Lie superalgebras

In this paper we give a re-normalization of the supertrace on the category of representations of Lie superalgebras of type I, by a kind of modified superdimension. The genuine superdimensions and supertraces are generically zero. However, these modified superdimensions are non-zero and lead to a kind of supertrace which is non-trivial and invariant. As an application we show that this new supertrace gives rise to a non-zero bilinear form on a space of invariant tensors of a Lie superalgebra of type I. The results of this paper are completely classical results in the theory of Lie superalgebras…

Type-I Quantum Superalgebras, $q$-Supertrace and Two-variable Link Polynomials

A new general eigenvalue formula for the eigenvalues of Casimir invariants, for the type-I quantum superalgebras, is applied to the construction of link polynomials associated with {\em any} finite dimensional unitary irrep for these algebras. This affords a systematic construction of new two-variable link polynomials asociated with any finite dimensional irrep (with a real highest weight) for the type-I quantum superalgebras. In particular infinite families of non-equivalent two-variable link polynomials are determined in fully explicit form.

A型中的Weyl广群、Young图与Borel子代数

The Weyl groupoid in Type A, Young diagrams and Borel subalgebras

本文在 $m>n$、互素且 $\kappa=-n/m$ 时，用周长 $n$、长度 $m$ 的旋转圆柱上的 Young 图统一刻画 Weyl 广群轨道与仿射 Lie 超代数 Borel 子代数轨道的图同构。