An algebra A not encountered in either the usual algebraic varieties or supervarieties is introduced. A is a graded and deformed version of the quaternions, with structure similar to that of a Jordan-Lie superalgebra as defined by Okubo and Kamiya, but it is shown to be neither that of a purely associative Lie superalgebra, nor that of a purely antiassociative Jordan-Lie superalgebra. Rather, it exhibits a novel kind of associativity, here called `ordered graded associativity', that is somewhat `in between' pure associativity and pure antiassociativity. In addition to graded associativity, the generators of A obey graded commutation relations encountered in both the usual Lie superalgebras and in graded Jordan-Lie algebras. They also satisfy new graded Jacobi identities that combine characteristics of the Jacobis obeyed by the generators of ungraded Lie, graded Lie and graded Jordan-Lie algebras. Mainly due to these three features, A is called a `mixed' Jordan-Lie superalgebra. The present paper defines A and compares it with the Jordan-Lie superalgebra defined by Okubo and Kamiya.
On general Lagrangian formulations for arbitrary mixed-symmetric higher-spin fermionic fields on Minkowski backgrounds
无约束拉格朗日公式的细节（是NPB 862（2012）270，[arXiv：1110.5044[hep-th]]，Phys. of Part.和Nucl.43（2012）689，[arXiv：1202.4710 [hep-th]]）被综述用于在任意维Minkowski时空上传播的费米粒子，并且由服从具有$k$行的Young表$Y(s_1,...,s_k)$的庞加莱群的酉不可约半整数高自旋表示描述。该过程是基于构造的Verma模块，并找到辅助振子实现的正交辛$osp(1|2k)$超代数编码的第二类运营商的约束子系统的HS对称超代数。应用一个通用的BRST-BFV方法允许再现规范不变的拉格朗日与可约规范对称性描述的自由动力学的无质量和大量的费米子场的任何自旋与适当数量的规范和Stukelberg场。一般构造具有导出仅含完整约束的拉格朗日量的明显可能性。
Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model
