副玻色子代数中玻色子化的变体。霍普夫和超霍普夫结构
Variants of bosonisation in Parabosonic algebra. The Hopf and super-Hopf structures
摘要：
有限自由度或无限自由度的仿声子代数被认为是一个$\mathbb{Z}_{2}$ -分次结合代数，并证明了它是一个$\mathbb{Z}_{2}$ -分次（或：超）Hopf代数.本文直接建立了抛物子代数的超Hopf代数结构，而不涉及它与$osp(1/2n)$李超代数结构的关系.超Hopf代数的概念等价地描述为辫子monoidal范畴${}_{\mathbb{CZ}_{2}}\mathcal{M}$中的Hopf代数。综述了将辫子monoidal范畴${}_{H}\mathcal{M}$（其中$H$是拟三角Hopf代数）中的Hopf代数转换为普通Hopf代数的玻色化技术.在本文中，我们证明了，对于抛物子代数$P_{B}$，除了应用玻色化技术的原始超Hopf代数，玻色化建设也实现了使用两个运营商，相关的抛物子总数运营商。这两种技术都将相同的超Hopf代数$P_{B}$转换为普通的Hopf代数，从而产生具有普通Hopf结构的$P_{B}$的两种不同变体。
英文摘要：
Parabosonic algebra in finite or infinite degrees of freedom is considered as a $\mathbb{Z}_{2}$-graded associative algebra, and is shown to be a $\mathbb{Z}_{2}$-graded (or: super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the $osp(1/2n)$ Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category ${}_{\mathbb{CZ}_{2}}\mathcal{M}$. The bosonisation technique for switching a Hopf algebra in the braided monoidal category ${}_{H}\mathcal{M}$ (where $H$ is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper we prove that for the parabosonic algebra $P_{B}$, beyond the application of the bosonisation technique to the original super-Hopf algebra, a bosonisation-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra $P_{B}$ to an ordinary Hopf algebra, producing thus two different variants of $P_{B}$, with ordinary Hopf structure.
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