TY - JOUR

T1 - The universal enveloping algebra of and the Racah algebra

AU - Bockting-Conrad, Sarah

AU - Huang, Hau Wen

N1 - Publisher Copyright:
© 2019, © 2019 Taylor & Francis Group, LLC.

PY - 2020/3/3

Y1 - 2020/3/3

N2 - Let (Formula presented.) denote a field with (Formula presented.) The Racah algebra (Formula presented.) is the unital associative (Formula presented.) -algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert that (Formula presented.) and each of the elements (Formula presented.) is central in (Formula presented.) Additionally the element (Formula presented.) is central in (Formula presented.) In this paper, we explore the relationship between the Racah algebra (Formula presented.) and the universal enveloping algebra (Formula presented.) Let a, b, c denote mutually commuting indeterminates. We show that there exists a unique (Formula presented.) -algebra homomorphism (Formula presented.) that sends (Formula presented.) where x, y, z are the equitable generators for (Formula presented.) We additionally give the images of (Formula presented.) and certain Casimir elements of (Formula presented.) under (Formula presented.) We also show that the map (Formula presented.) is an injection and thus provides an embedding of (Formula presented.) into (Formula presented.) We use the injection to show that (Formula presented.) contains no zero divisors.

AB - Let (Formula presented.) denote a field with (Formula presented.) The Racah algebra (Formula presented.) is the unital associative (Formula presented.) -algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert that (Formula presented.) and each of the elements (Formula presented.) is central in (Formula presented.) Additionally the element (Formula presented.) is central in (Formula presented.) In this paper, we explore the relationship between the Racah algebra (Formula presented.) and the universal enveloping algebra (Formula presented.) Let a, b, c denote mutually commuting indeterminates. We show that there exists a unique (Formula presented.) -algebra homomorphism (Formula presented.) that sends (Formula presented.) where x, y, z are the equitable generators for (Formula presented.) We additionally give the images of (Formula presented.) and certain Casimir elements of (Formula presented.) under (Formula presented.) We also show that the map (Formula presented.) is an injection and thus provides an embedding of (Formula presented.) into (Formula presented.) We use the injection to show that (Formula presented.) contains no zero divisors.

KW - Casimir elements

KW - Lie algebras

KW - Racah algebra

KW - quadratic algebras

KW - universal enveloping algebras

UR - http://www.scopus.com/inward/record.url?scp=85075129023&partnerID=8YFLogxK

U2 - 10.1080/00927872.2019.1670199

DO - 10.1080/00927872.2019.1670199

M3 - 期刊論文

AN - SCOPUS:85075129023

SN - 0092-7872

VL - 48

SP - 1022

EP - 1040

JO - Communications in Algebra

JF - Communications in Algebra

IS - 3

ER -