The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2020
• 1. Verfasser: Huang, Hau-Wen
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2020
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/210773
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra. Hau-Wen Huang. SIGMA 16 (2020), 075, 15 pages

Assume that is a field with char ≠ 2. The Racah algebra ℜ is a unital associative -algebra defined by generators and relations. The generators are A, B, C, D, and the relations assert that [A, B]=[B, C]=[C, A]=2D, and each of [A, D]+AC−BA, [B, D]+BA−CB, [C, D]+CB−AC is central in ℜ. The Bannai-Ito algebra is a unital associative -algebra generated by X, Y, Z, and the relations assert that each of {X, Y}−Z, {Y, Z}−X, {Z, X}−Y is central in I. It was discovered that there exists an -algebra homomorphism ζ: ℜ → that sends A↦(2X−3)(2X+1)/16, B↦(2Y−3)(2Y+1)16, C↦(2Z−3)(2Z+1)/16. We show that ζ is injective and therefore ℜ can be considered as an -subalgebra of . Moreover, we show that any Casimir element of ℜ can be uniquely expressed as a polynomial in {X, Y} − Z, {Y, Z} − X, {Z, X} − Y, and X + Y + Z with coefficients in .