Symmetry Integrability and Geometry-Methods and Applications | |
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero | |
article | |
Hau-Wen Huang1  Sarah Bockting-Conrad2  | |
[1] Department of Mathematics, National Central University;Department of Mathematical Sciences, DePaul University | |
关键词: Racah algebra; quadratic algebra; irreducible modules; tridiagonal pairs; universal property; | |
DOI : 10.3842/SIGMA.2020.018 | |
来源: National Academy of Science of Ukraine | |
【 摘 要 】
Assume that ${\mathbb F}$ is an algebraically closed field with characteristic zero. The Racah algebra $\Re$ is the unital associative ${\mathbb F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$, $D$ and the relations assert that $[A,B]=[B,C]=[C,A]=2D$ and that each of $[A,D]+AC-BA$, $[B,D]+BA-CB$, $[C,D]+CB-AC$ is central in $\Re$. In this paper we discuss the finite-dimensional irreducible $\Re$-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional $\Re$-module and its universal property. We additionally give the necessary and sufficient conditions for $A$, $B$, $C$ to be diagonalizable on finite-dimensional irreducible $\Re$-modules.
【 授权许可】
Unknown
【 预 览 】
Files | Size | Format | View |
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RO202106300000708ZK.pdf | 398KB | download |