TY - JOUR

T1 - A connection behind the Terwilliger algebras of H(D,2) and [Formula presented]

AU - Huang, Hau Wen

AU - Wen, Chia Yi

N1 - Publisher Copyright:
© 2023 Elsevier Inc.

PY - 2023/11/15

Y1 - 2023/11/15

N2 - The universal enveloping algebra U(sl2) of sl2 is a unital associative algebra over C generated by E,F,H subject to the relations [H,E]=2E,[H,F]=−2F,[E,F]=H. The distinguished central element [Formula presented] is called the Casimir element of U(sl2). The universal Hahn algebra H is a unital associative algebra over C with generators A,B,C and the relations assert that [A,B]=C and each of α=[C,A]+2A2+B,β=[B,C]+4BA+2C is central in H. The distinguished central element Ω=4ABA+B2−C2−2βA+2(1−α)B is called the Casimir element of H. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism ♮:H→U(sl2) that sends [Formula presented] We determine the image of ♮ and show that the kernel of ♮ is the two-sided ideal of H generated by β and 16Ω−24α+3. By pulling back via ♮ each U(sl2)-module can be regarded as an H-module. For each integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl2)-module Ln up to isomorphism. We show that the H-module Ln (n≥1) is a direct sum of two non-isomorphic irreducible H-modules.

AB - The universal enveloping algebra U(sl2) of sl2 is a unital associative algebra over C generated by E,F,H subject to the relations [H,E]=2E,[H,F]=−2F,[E,F]=H. The distinguished central element [Formula presented] is called the Casimir element of U(sl2). The universal Hahn algebra H is a unital associative algebra over C with generators A,B,C and the relations assert that [A,B]=C and each of α=[C,A]+2A2+B,β=[B,C]+4BA+2C is central in H. The distinguished central element Ω=4ABA+B2−C2−2βA+2(1−α)B is called the Casimir element of H. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism ♮:H→U(sl2) that sends [Formula presented] We determine the image of ♮ and show that the kernel of ♮ is the two-sided ideal of H generated by β and 16Ω−24α+3. By pulling back via ♮ each U(sl2)-module can be regarded as an H-module. For each integer n≥0 there exists a unique (n+1)-dimensional irreducible U(sl2)-module Ln up to isomorphism. We show that the H-module Ln (n≥1) is a direct sum of two non-isomorphic irreducible H-modules.

KW - Askey–Wilson relations

KW - Halved cubes

KW - Hypercubes

KW - Lie algebras

KW - Terwilliger algebras

UR - http://www.scopus.com/inward/record.url?scp=85167424841&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2023.07.019

DO - 10.1016/j.jalgebra.2023.07.019

M3 - 期刊論文

AN - SCOPUS:85167424841

SN - 0021-8693

VL - 634

SP - 456

EP - 479

JO - Journal of Algebra

JF - Journal of Algebra

ER -