Bannai–Ito algebras and the osp(1;2) superalgebra

Bie, Hendrik De; Genest, Vincent X.; Vijver, Wouter van de; Vinet, Luc

doi:10.1007/978-3-319-69164-0_52

Cited by 7 publications

(12 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…which are the structure relations of B(n) given in (1.1) [2]. Underneath this quick derivation of (6.3) is the fact that the entire algebra B(n) is generated by the Casimirs associated to 2-subsets since the general relations are inferred from those of B(3).…”

Section: The Higher Rank Bannai-ito Algebra As a Commutantmentioning

confidence: 95%

See 1 more Smart Citation

The dual pair Pin(2n)×osp(1|2), the Dirac

Gaboriaud

Vinet

et al. 2018

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

The Bannai-Ito algebra can be defined as the centralizer of the coproduct embedding of osp(1|2) in osp(1|2) ⊗n . It will be shown that it is also the commutant of a maximal Abelian subalgebra of o(2n) in a spinorial representation and an embedding of the Racah algebra in this commutant will emerge. The connection between the two pictures for the Bannai-Ito algebra will be traced to the Howe duality which is embodied in the P in(2n)×osp(1|2) symmetry of the massless Dirac equation in R 2n . Dimensional reduction to R n will provide an alternative to the Dirac-Dunkl equation as a model with Bannai-Ito symmetry.(1.1)The dual pair P in(2n) × osp(1|2), the Dirac equation and the Bannai-Ito algebra

show abstract

Section: The Higher Rank Bannai-ito Algebra As a Commutantmentioning

confidence: 95%

“…The Bannai-Ito algebra B(n) can be presented in terms of generators and relations [1,2]. Let [n] = {1, 2, .…”

Section: Introductionmentioning

confidence: 99%

The dual pair Pin(2n)×osp(1|2), the Dirac

Gaboriaud

Vinet

et al. 2018

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

show abstract

“…The Z 2 -grading of osp(1|2) can be encoded by the grading involution P satisfying (5) [P, E ± ] = 0 , [P, H] = 0 , {P, F ± } = 0 and P 2 = 1 .…”

Section: Properties Of the Lie Superalgebra Osp(1|2)mentioning

confidence: 99%

“…This was actually achieved using models of osp(1|2) given in terms of Dirac-Dunkl operators [6], [7]. For reviews of these algebras and some of their applications see [4], [5].…”

Section: Introductionmentioning

confidence: 99%

Bannai–Ito algebras and the universal R-matrix of $$\pmb {\mathfrak {osp}}(1|2)$$

2019

Self Cite

View full text Add to dashboard Cite

The Bannai-Ito algebra BI(n) is viewed as the centralizer of the action of osp(1|2) in the n-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the universal R-matrix of osp(1|2). The specific structure of the osp(1|2) embeddings to which the centralizing elements are attached as Casimir elements is explained. With the generators defined, the structure relations of BI(n) are derived from those of BI(3) by repeated action of the coproduct and using properties of the R-matrix and of the generators of the symmetric group S n .

show abstract

“…is central in BI. The applications to the Racah problems for su (2), su (1,1), sl −1 (2) and the connections to the Laplace-Dunkl and Dirac-Dunkl equations on the 2-sphere have been explored in [3,6,8,11,12,13,14,15,17,19,20,23]. For more information and recent progress, see [2,4,5,7,9,10,22].…”

Section: Introductionmentioning

confidence: 99%

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

Huang¹

2020

SIGMA

View full text Add to dashboard Cite

Assume that F is a field with char F = 2. The Racah algebra 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. 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 : → BI 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 can be considered as an F-subalgebra of BI. Moreover we show that any Casimir element of 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.

show abstract

Bannai–Ito algebras and the osp(1;2) superalgebra

Cited by 7 publications

References 11 publications

The dual pair Pin(2n)×osp(1|2), the Dirac

The dual pair Pin(2n)×osp(1|2), the Dirac

Bannai–Ito algebras and the universal R-matrix of $$\pmb {\mathfrak {osp}}(1|2)$$

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

Contact Info

Product

Resources

About