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

Abstract

Assume that 𝔽 is a field with char𝔽 ≠ 2. The Racah algebra ℜ is a unital associative 𝔽-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 𝔅𝔍 is a unital associative 𝔽-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 𝔅I. It was discovered that there exists an 𝔽-algebra homomorphism ζ: ℜ → 𝔅𝔍 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 𝔽-subalgebra of 𝔅𝔍. 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 𝔽.