TY - JOUR

T1 - The racah algebra as a subalgebra of the bannai–ito Algebra

AU - Huang, Hau Wen

N1 - Publisher Copyright:
© 2020, Institute of Mathematics. All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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)(2Y16+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.

AB - 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)(2Y16+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.

KW - Bannai–Ito algebra

KW - Casimir elements

KW - Racah algebra

UR - http://www.scopus.com/inward/record.url?scp=85090520715&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2020.075

DO - 10.3842/SIGMA.2020.075

M3 - 期刊論文

AN - SCOPUS:85090520715

SN - 1815-0659

VL - 16

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

M1 - 075

ER -