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## Abstract

Assume that F is a field with char F ≠ 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 BI 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 → 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 F-subalgebra 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 |
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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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