Anyonic realization of the quantum affine Lie superalgebras

Frappat, L.; Sciarrino, A.; Sciuto, S.; Sorba, P.

doi:10.1088/0305-4470/30/3/015

Cited by 20 publications

(30 citation statements)

References 9 publications

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“…Finally, let us mention that it is also possible to obtain a supersymmetric version of the construction (see ref. [3] for the case U q ( A(m, n))).…”

Section: Discussionmentioning

confidence: 99%

Anyonic realization of the quantum affine Lie algebra

et al. 1996

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We give a realization of quantum affine Lie algebras U q ( A N −1 ) and U q ( C N ) in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter q being related to the statistical parameter ν of the anyons by q = e iπν . In the limit of the deformation parameter going to one we recover the Feingold-Frenkel [1] fermionic construction of undeformed affine Lie algebras.

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“…Finally, let us mention that it is also possible to obtain a supersymmetric version of the construction (see ref. [3] for the case U q ( A(m, n))).…”

Section: Discussionmentioning

confidence: 99%

Anyonic realization of the quantum affine Lie algebra

et al. 1996

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“…where π j := (j, j + 1) ∈ S n for 1 ≤ j ≤ n−1. A permutation π ∈ S n can be represented (not in a unique way, in general) as a reduced product of a minimal number of adjacent transpositions, i.e., in the form (12) with a minimal m. This number m is then called the length of π, denoted by |π|. It is well known that |π| is equal to the number of inversions of π, i.e., the number of 1 ≤ i < j ≤ n such that π(i) > π(j).…”

Section: Operator P Nmentioning

confidence: 99%

“…Let us mention that anyon systems have also been considered in the discrete setting, i.e., when T ⊂ N, see e.g. [12,23,13]. It should be, however, mentioned that, when discussing the anyons in the discete setting, Goldin and Majid [13] dropped the assumption that the annihilation operator is adjoint of the creation operator, and proved an anyonic exclusion principle for their model.…”

Section: Introductionmentioning

confidence: 99%

Fock representations of Q-deformed commutation relations

Bożejko

Lytvynov

Wysoczański

2017

Journal of Mathematical Physics

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We consider Fock representations of the Q-deformed commutation relationsHere T := R d (or more generally T is a locally compact Polish space), the function Q : T 2 → C satisfies |Q(s, t)| ≤ 1 and Q(s, t) = Q(t, s), andσ being a fixed reference measure on T . In the case where |Q(s, t)| ≡ 1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev (1995). These generalized statistics contain anyon statistics as a special case (with T = R 2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H := L 2 (T → C, σ) is constructed. An explicit form of the orthogonal projection of H ⊗n onto the n-particle space F n (H) is derived. A scalar product in F n (H) is given by an operator P n ≥ 0 in H ⊗n which is strictly positive on F n (H). We realize the smeared operators ∂ † t and ∂ t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂ † s ∂ † t = Q(t, s)∂ † t ∂ † s , ∂ s ∂ t = Q(t, s)∂ t ∂ s , valid for those s, t ∈ T for which |Q(s, t)| = 1.

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“…For further discussions of anyons in mathematical physics literature (including the discrete setting), see e.g. [13,15,17,19,33,[36][37][38][39][40][41] and the references therein. We also refer to the paper [8] which deals with a Fock representation of the commutations relations identified by a sequence of self-adjoint operators in a Hilbert space which have norm ≤ 1 and which satisfy the braid relations.…”

Section: Fock Space Representation Of the Anyon Commutation Relationsmentioning

confidence: 99%

“…By (17), τ is gauge-invariant if and only if S (k,n) ≡ 0 for k = n. Thus, a gauge-invariant state is completely determined by S (n,n) (n ∈ N).…”

Section: Gauge-invariant Quasi-free States On the Ccr And Car Algebrasmentioning

confidence: 99%

Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations

Lytvynov

2016

Commun. Math. Phys.

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Anyonic realization of the quantum affine Lie superalgebras

Cited by 20 publications

References 9 publications

Anyonic realization of the quantum affine Lie algebra

Anyonic realization of the quantum affine Lie algebra

Fock representations of Q-deformed commutation relations

Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations

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