The Anticommutator Spin Algebra, Its Representations and Quantum Group Invariance

Abstract

We define a 3-generator algebra obtained by replacing the commutators by anticommutators in the defining relations of the angular momentum algebra. We show that integer spin representations are in one to one correspondence with those of the angular momentum algebra. The half-integer spin representations, on the other hand, split into two representations of dimension j + 1 2 . The anticommutator spin algebra is invariant under the action of the quantum group SOq(3) with q = −1.

The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra

“…Due to the importance of the oscillator algebra, it is worth recording this reduction in some detail. Most algebraic results connected to this skewed addition of three quantum harmonic oscillators have interestingly been obtained previously in [8,14,1,3]. In the case µ 1 = µ 2 = µ 3 = 0, the Bannai-Ito (3.7) algebra becomes…”

“…The exact expression for the Racah coefficients is finally obtained up to a phase factor using the orthogonality relation of the BI polynomials. In section 5, we discuss the degenerate case of the Bannai-Ito algebra corresponding to the anticommutator spin algebra [8,14,1,3]. We conclude by explaining that the operators K 1 and K 2 , together with their anticommutator K 3 , form a Leonard triple.…”

Supersymmetry of the Quantum Rotor

“…which also satisfies Q 2 = H. Other operators, associated to different coproducts of the Lie superalgebra os p (1,2) can also be constructed; see [10] for more details.…”

The Classification of Finite-Dimensional Irreducible Modules of Bannai/Ito Algebra

“…In the case = = = 0, the Bannai/Ito algebra becomes xy + yx = z yz + zy = x zx + xz = y This algebra has been studied in [1,7] as an anti-commutator version of the classical Lie algebra su 2 . In [1] Arik and Kayserilioglu called this the anti-commutator spin algebra.…”

“…In [1] Arik and Kayserilioglu called this the anti-commutator spin algebra. In [2] Brown classified the finite-dimensional irreducible modules for the anti-commutator spin algebra.…”