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 -