Calogero Operator and Lie Superalgebras
Calogero Operator and Lie Superalgebras
复制标题
Calogero 算子和李超代数
DOI：
10.1023/a:1015968505753
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发表时间：
2002
影响因子：
1
通讯作者：
A. Sergeev
中科院分区：
文献类型：
--
作者：
A. Sergeev
AbstractWe construct a supersymmetric analogue of the Calogero operator
$$\mathcal{S}\mathcal{L}$$
, which depends on the parameter k. This analogue is related to the root system of the Lie superalgebra
$$gl{\text{(}}n{\text{|}}m{\text{)}}$$
. It becomes the standard Calogero operator for m = 0 and becomes the operator constructed by Veselov, Chalykh, and Feigin up to changing the variables and the parameter k for m = 1. For k = 1 and 1/2, the operator
$$\mathcal{S}\mathcal{L}$$
is the radial part of the second-order Laplace operator for the symmetric superspaces corresponding to the respective pairs
$$(gl \oplus gl,gl) and(gl,osp){\text{ }}$$
. We show that for any m and n, the supersymmetric analogues of the Jack polynomials constructed by Kerov, Okounkov, and Olshanskii are eigenfunctions of the operator
$$\mathcal{S}\mathcal{L}$$
. For k = 1 and 1/2, the supersymmetric analogues of the Jack polynomials coincide with the spherical functions on the above superspaces. We also study the algebraic analogue of the Berezin integral.
AbstractWe construct a supersymmetric analogue of the Calogero operator
$$\mathcal{S}\mathcal{L}$$
, which depends on the parameter k. This analogue is related to the root system of the Lie superalgebra
$$gl{\text{(}}n{\text{|}}m{\text{)}}$$
. It becomes the standard Calogero operator for m = 0 and becomes the operator constructed by Veselov, Chalykh, and Feigin up to changing the variables and the parameter k for m = 1. For k = 1 and 1/2, the operator
$$\mathcal{S}\mathcal{L}$$
is the radial part of the second-order Laplace operator for the symmetric superspaces corresponding to the respective pairs
$$(gl \oplus gl,gl) and(gl,osp){\text{ }}$$
. We show that for any m and n, the supersymmetric analogues of the Jack polynomials constructed by Kerov, Okounkov, and Olshanskii are eigenfunctions of the operator
$$\mathcal{S}\mathcal{L}$$
. For k = 1 and 1/2, the supersymmetric analogues of the Jack polynomials coincide with the spherical functions on the above superspaces. We also study the algebraic analogue of the Berezin integral.