TY - JOUR

T1 - Finite-dimensional irreducible modules of the racah algebra at characteristic zero

AU - Huang, Hau Wen

AU - Bockting-Conrad, Sarah

N1 - Publisher Copyright:
© 2020, Institute of Mathematics. All rights reserved.

PY - 2020

Y1 - 2020

N2 - Assume that F is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative 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 ℜ. In this paper we discuss the finite-dimensional irreducible R-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional R-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible R-modules.

AB - Assume that F is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative 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 ℜ. In this paper we discuss the finite-dimensional irreducible R-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional R-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible R-modules.

KW - Irreducible modules

KW - Quadratic algebra

KW - Racah algebra

KW - Tridiagonal pairs

KW - Universal property

UR - http://www.scopus.com/inward/record.url?scp=85084837563&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2020.018

DO - 10.3842/SIGMA.2020.018

M3 - 期刊論文

AN - SCOPUS:85084837563

SN - 1815-0659

VL - 16

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

M1 - 018

ER -