On the algebra of symmetries of Laplace and Dirac operators
On the algebra of symmetries of Laplace and Dirac operators
复制标题
关于拉普拉斯和狄拉克算子的对称代数
DOI：
10.1007/s11005-018-1065-0
复制
发表时间：
2017
影响因子：
1.2
通讯作者：
J. Van der Jeugt
中科院分区：
文献类型：
--
作者：
H. De Bie;R. Oste;J. Van der Jeugt
我们考虑经典拉普拉斯算子的推广，其中包括根据与有限反射群（称为 Dunkl 算子）相关的微分-差分算子定义的 Laplace-Dunkl 算子。对于这个类拉普拉斯算子，我们以广义角动量算子的形式确定了一组与之交换的对称性，并给出了对称代数的代数关系。在这种情况下，广义狄拉克算子被定义为类拉普拉斯算子的平方根。我们明确地确定了一系列分级算子，这些算子根据其程度与类狄拉克算子进行交换或反交换。这些对称算子生成的代数被证明是标准角动量代数和最近定义的高阶 Bannai-Ito 代数的推广。
We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.