It is given a way of computing Casimir eigenvalues for Weyl orbits as well as for irreducible representations of Lie algebras. A kappa(s) number of polinomials which depend on rank N are obtained explicitly for A_N Casimir operators of order s where kappa(s) is the number of partitions of s into positive integers except 1. It is also emphasized that these eigenvalue polinomials prove useful in obtaining formulas to calculate weight multiplicities and in explicit calculations of the whole cohomology ring of Classical and also Exceptional Lie algebras.
We study cut-and-join operators for spin Hurwitz partition functions. We provide explicit expressions for these operators in terms of derivatives in $p$-variables without straightforward matrix realization, which is yet to be found. With the help of these expressions spin cut-and-join operators can be calculated directly and algorithmically. The reason why it is possible is the connection between mentioned operators and specially chosen Casimir operators that are easy to compute. An essential part of the connection involves shifted Q-Schur functions.
We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in adjoint representation. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T^{\otimes 2} in the case when T is the adjoint representation. These projectors and characteristic identities are considered from the viewpoint of the universal description of the simple Lie algebras in terms of the Vogel parameters.
We find the characteristic identities for the split Casimir operator in the defining and adjoint representations of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras. These identities are used to build the projectors onto invariant subspaces of the representation $T^{\otimes 2}$ of the $osp(M|N)$ and $s\ell(M|N)$ Lie superalgebras in the cases when $T$ is the defining and adjoint representations. For defining representations, the $osp(M|N)$- and $s\ell(M|N)$-invariant solutions of the Yang-Baxter equation are expressed as rational functions of the split Casimir operator. For the adjoint repres…
We construct characteristic identities for the split (polarized) Casimir operators of the simple Lie algebras in defining (minimal fundamental) and adjoint representations. By means of these characteristic identities, for all simple Lie algebras we derive explicit formulae for invariant projectors onto irreducible subrepresentations in T^{\otimes 2} in two cases, when T is the defining and the adjoint representation. In the case when T is the defining representation, these projectors and the split Casimir operator are used to explicitly write down invariant solutions of the Yang-Baxter equatio…
Canonically relativistic quantum mechanics: Casimir field equations of the quaplectic group
量子动力学群李群的酉不可约表示的希尔伯特空间与量子态空间相同。代数的厄米特表示是可观的。表示Casimir不变算子集的本征值方程定义了系统的场方程。一类一般的动力群是半直积K * s N，其表示由麦基理论给出。齐次群K必须是正规群N的自同构的子群。原型动力群是庞加莱群。由卡西米尔算符的表象定义的场方程定义了物理学的基本方程;克莱因-戈登方程、狄拉克方程、麦克斯韦方程等等。本文探讨了一个更一般的动力群候选者，也是一个半直积，但其中的“翻译”正常子群N现在是海森堡群。海森堡群的相关自同构是辛群。这与对正交度量的要求一起导致齐次群K的伪酉群。给出了这个群的物理意义和动机，称之为四元群，并确定了该代数的厄米特不可约表示。与庞加莱群一样，群的选择定义了与量子粒子态相一致的表示的希尔伯特空间。给出了Casimir算子表示的本征值方程--场方程，并对其进行了研究。该理论体现了玻恩互易原理和一个新的相对性原理。
For every simple Lie algebra $\mathfrak{g}$ we consider the associated Takiff algebra $\mathfrak{g}^{}_{\ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $\mathfrak{g}$. We use a matrix presentation of $\mathfrak{g}^{}_{\ell}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${\rm U}(\mathfrak{g}^{}_{\ell})$. A similar matrix presentation for the affine Kac--Moody algebra $\widehat{\mathfrak{g}}^{}_{\ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the cent…
Some Subtleties in the Relationships among Heat Kernel Invariants, Eigenvalue Distributions, and Quantum Vacuum Energy
卡西米尔物理学（以及许多其他领域）中的一个常用工具是本征值密度的渐近（高频）展开，用作各种格林函数渐近行为计算的输入或输出。在这里，我们澄清了这个主题的一些细微之处和潜在的混乱方面。特别地，我们展示了Kolomeisky等人[Phys. Rev. A 87（2013）042519]的最新观测结果如何符合谱函数的分布渐近性的既定框架。
On a finite graph, we prove that trace of holonomies determine an intertwining relation between merge-and-split generators on collections of geodesic loops ensembles and Casimir operators on unitary connections. By adding a deformation part to the generator on loops, this result is extended to the Casimir operator modified in order to be self adjoint with respect to Yang- Mills measure.
In this paper, we introduce a new family of functions to construct Schrödinger operators with embedded eigenvalues. This particularly allows us to construct discrete Schrödinger operators with arbitrary prescribed sets of eigenvalues.
We fnd the asymptotics of eigenvalues of polynomially compact zero order pseudodiferential operators, the motivating example being the Neumann- Poincare operator in linear elasticity.
