Irreducible representations of the osp(2,1) and spl(2,1) graded Lie algebras

Abstract

We illustrate through the examples of the osp(2,1) and spl(2,1) algebras the differences between the properties of the irreducible representations of simple graded Lie algebras and simple Lie algebras.

Disordered Dirac fermions: the marriage of three different approaches

“…where A = (4ǫγ) 1/4 is a free parameter corresponding to the arbitrary relative normalizations of the su(2) doublet (Q 1 , Q 4 ) and the two singlets (Q 2 , Q 3 ) in the four-dimensional representation of osp(2|2) [59,72]. With the normalization chosen in equation (4.3) the compact bosonic exponents e ±iφ have conformal dimension h φ = 1/4, and the non-compact bosonic exponents e ±ϕ have h ϕ = −1/4.…”

“…and the eigenvalues of Q 3 and B (called isospin and baryon number respectively) are used to classify the basis states of finite-dimensional representations [72,86]. A representation [b, q] contains at most four multiplets of states:…”

Doped Heisenberg chains: Spin-S generalizations of the supersymmetric t-J model

“…The 'typical' representations [b, s] of this algebra can be characterized by the eigenvalues of operators B and S 2 on the multiplet with largest total SU(2)-spin [11,12]. Their dimension is 8S and they can be decomposed into two spin-(S − 1/2) multiplets with charge b ± 1/2 and a spin-S and a spin-(S − 1) multiplet with charge b with respect to the SU(2)-subalgebra of gl(2|1).…”

“…Below we will construct a class of integrable vertex models from solutions to the Yang Baxter equation (YBE) which are invariant under the action of the graded Lie algebra gl(2|1) [11,12].…”

Interplay between a quantum impurity and a boundary field in the SUSYt–Jmodel

“…[23,24] and Appendix A). This particluar representation has already been used in constructing an integrable model of a doped spin s Heisenberg chain [25][26][27].…”