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Abstract
Assume that F is a field with char F ≠ 2. The Racah algebra ℜ is a unital associative Falgebra 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 BI is a unital associative Falgebra 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 Falgebra homomorphism ζ: ℜ → BI that sends A ↦→^{(2X}^{−3)(2X+1)}_{16}, B ↦→^{(2Y}^{−3)(2Y}_{16}^{+1)}, C ↦→^{(2Z−3)(2Z+1)}_{16} . We show that ζ is injective and therefore ℜ can be considered as an Fsubalgebra of BI. 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.
Original language  English 

Article number  075 
Journal  Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 
Volume  16 
DOIs  
State  Published  2020 
Keywords
 Bannai–Ito algebra
 Casimir elements
 Racah algebra
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 1 Finished

The Unversial Racah Algebra and Its Applications(3/4)
Huang, H.W. (PI)
1/08/19 → 31/07/20
Project: Research