基于根系的有理Calogero模型观测量超代数上的超迹数

基于根系的有理Calogero模型观测量超代数上的超迹

Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

基于根系R的有理Calogero模型的观测量的超代数是由N个待定项中的多项式、微分-差分Dunkl算子和根系R生成的Coxeter群G的群代数生成的结合超代数。证明了该超代数具有QR超迹，其中QR是Coxeter群G的共轭类中特征值不等于-1的类的个数.

Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

If $G$ is a finite Coxeter group, then symplectic reflection algebra $H:=H_{1,η}(G)$ has Lie algebra $\mathfrak {sl}_2$ of inner derivations and can be decomposed under spin: $H=H_0 \oplus H_{1/2} \oplus H_{1} \oplus H_{3/2} \oplus ...$. We show that if the ideals $\mathcal I_i$ ($i=1,2$) of all the vectors from the kernel of degenerate bilinear forms $B_i(x,y):=sp_i(x\cdot y)$, where $sp_i$ are (super)traces on $H$, do exist, then $\mathcal I_1=\mathcal I_2$ if and only if $\mathcal I_1 \bigcap H_0=\mathcal I_2 \bigcap H_0$.

Klein operator and the Numbers of independent Traces and Supertraces on the Superalgebra of Observables of Rational Calogero Model based on the Root System

In the Coxeter group W(R) generated by the root system R, let T(R) be the number of conjugacy classes having no eigenvalue 1 and let S(R) be the number of conjugacy classes having no eigenvalue -1. The algebra H{R) of observables of the rational Calogero model based on the root system R possesses T(R) independent traces, the same algebra considered as an associative superalgebra with respect to a certain natural parity possesses S(R) even independent supertraces and no odd trace or supertrace. The numbers T(R) and S(R) are determined for all irreducible root systems (hence for all root systems…

变形量子Calogero-Moser问题与李超代数

Deformed quantum Calogero-Moser problems and Lie superalgebras

引入了与逆成分李超代数的根系有关的变形量子Calogero-Moser-Sutherland问题。该构造基于V. Serganova提出的广义根系的概念。对于经典级数，我们找到了量子积分的递推公式，这意味着这些问题的可积性。研究了相应的量子积分代数，给出了变形参数通值的庞加莱级数的显式表达式。

对称李超代数与变形量子Calogero-Moser问题

Symmetric Lie superalgebras and deformed quantum Calogero-Moser problems

研究了对称李超代数和相应的球函数的表示论与变形量子Calogero-Moser系统理论的关系。在对称对g=gl（n，2 m），k=osp（n，2 m）的特殊情况下，我们建立了球典型不可约g-模的投射覆盖与Calogero-Moser积分代数D_{n,m}作用于相应的Laurent拟不变量A_{n,m}的有限维广义特征空间之间的自然双射.

具有多个不变双线性型的单李超代数的一个例子

An example of simple Lie superalgebra with several invariant bilinear forms

给出了具有二维超迹空间的结合超代数A。证明了（i）它是单的，（ii）它的交换子[A，A}是单李超代数，（iii）这个交换子至少有2维非退化双线性不变形式空间.