# Super‐Hopf realizations of Lie superalgebras: Braided Paraparticle extensions of the Jordan‐Schwinger map

@article{Kanakoglou2010SuperHopfRO,
title={Super‐Hopf realizations of Lie superalgebras: Braided Paraparticle extensions of the Jordan‐Schwinger map},
author={Konstantinos Kanakoglou and Costas Daskaloyannis and Alfredo Herrera-Aguilar},
journal={arXiv: Mathematical Physics},
year={2010},
volume={1256},
pages={193-200}
}
• Published 20 July 2010
• Mathematics
• arXiv: Mathematical Physics
The mathematical structure of a mixed paraparticle system (combining both parabosonic and parafermionic degrees of freedom) commonly known as the Relative Parabose Set, will be investigated and a braided group structure will be described for it. A new family of realizations of an arbitrary Lie superalgebra will be presented and it will be shown that these realizations possess the valuable representation‐theoretic property of transferring invariably the super‐Hopf structure. Finally two classes…
7 Citations
On a class of Fock-like representations for Lie Superalgebras
• Mathematics
• 2011
Utilizing Lie superalgebra (LS) realizations via the Relative Parabose Set algebra $P_{BF}$, combined with earlier results on the Fock-like representations of $P_{BF}^{(1,1)}$, we proceed to the
Lie algebras and Yang-Baxter equations
At the previous congress (CRM 6), we reviewed the construction of Yang-Baxter operators from associative algebras, and presented some (colored) bialgebras and Yang-Baxter systems related to them. The
Z2 and Klein graded Lie algebras
In this master's thesis, we recall the definitions and basic results for Lie superalgebras. We specify the definition for Klein graded Lie algebras and, motivated by well known results for Lie
Yang-Baxter operators from (G, \theta)-Lie algebras
• Mathematics
• 2010
The (G, \theta)-Lie algebras are structures which unify the Lie algebras and Lie superalgebras. We use them to produce solutions for the quantum Yang-Baxter equation. The constant and the
YANG-BAXTER OPERATORS FROM (G,)-LIE ALGEBRAS
• Mathematics
• 2010
The (G,�)-Lie algebras are structures which unify the Lie algebras and Lie superalgebras. We use them to produce solutions for the quantum Yang–Baxter equation. The constant and the
ON JORDAN (CO)ALGEBRAS
We present new results about Jordan algebras and Jordan coalgebras, and we discuss about their connections with the Yang-Baxter equations.
Towards applications of graded Paraparticle algebras
• Mathematics
HNPS Proceedings
• 2019
An outline is sketched, of applications of the ideas and the mathematical methods presented at the 19th symposium of the Hnps in Thessaloniki, May 2010

## References

SHOWING 1-5 OF 5 REFERENCES
Mixed Paraparticles, Colors, Braidings and a new class of Realizations for Lie superalgebras
• Mathematics
• 2009
A rigorous algebraic description of the notion of realization, specialized in the case of Lie superalgebras is given. The idea of the Relative Parabose set $P_{BF}$ is recalled together with some
Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan Schwinger map
• Mathematics
• 2000
The aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the well-known Hopf algebraic structure of the Lie
Universal R-matrices for finite Abelian groups -- a new look at graded multilinear algebra
The universal R-matrices and, dually, the coquasitriangular structures of the group Hopf algebra of a finite Abelian group (resp. of an arbitrary Abelian group) are determined. This is used to
Hopf algebras and their actions on rings
Definitions and examples Integrals and semisimplicity Freeness over subalgebras Action of finite-dimensional Hopf algebras and smash products Coradicals and filtrations Inner actions Crossed products
The Theory of Lie Superalgebras: An Introduction
Preparatory remarks.- Formal constructions.- Simple Lie superalgebras.- A survey of some further developments.