Eigenvalues of Casimir invariants for Uq[osp(m∣n)]

Dancer, K.A.; Gould, M. D.; Links, Jon

doi:10.1063/1.2137712

Cited by 4 publications

(8 citation statements)

References 23 publications

(35 reference statements)

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“…The various relations that can be deduced from equation (27) are no longer inductive in form, so can not be used to directly construct the σba for direct comparison. They can, however, be shown to be consistent with relations (22) and (23), as demonstrated in [30]. Thus the conditions obtained by considering the intertwining property for the lowering generators,…”

Section: The Intertwining Propertysupporting

confidence: 76%

“…Then the inductive relations (23) are applied to these simple operators to find the remaining values of σba . One of many equivalent ways of doing this is given below.…”

Section: The R-matrix For the Vector Representationmentioning

confidence: 99%

“…One immediate application of the Lax operator is that it affords a means to construct the Casimir invariants of the associated quantum superalgebra. This construction for U q [osp(m|n)], as well as a derivation of the eigenvalues of the Casimir invariants when acting on irreducible highest weight modules, can be found in [23].…”

Section: Introductionmentioning

confidence: 99%

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Lax Operator for the Quantised Orthosymplectic Superalgebra U q [osp(m|n)]

Dancer

Gould

Links

2007

Algebr Represent Theor

Self Cite

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Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasitriangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang-Baxter equation. Applying the vector representation π, which acts on the vector module V , to the left-hand side of a universal R-matrix gives a Lax operator. In this Communication a Lax operator is constructed for the quantised orthosymplectic superalgebras Uq[osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang-Baxter equation acting on V ⊗ V ⊗ W , where W is an arbitrary Uq[osp(m|n)] module. The case W = V is studied as an example.

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Section: The Intertwining Propertysupporting

confidence: 76%

“…Then the inductive relations (23) are applied to these simple operators to find the remaining values of σba . One of many equivalent ways of doing this is given below.…”

Section: The R-matrix For the Vector Representationmentioning

confidence: 99%

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Lax Operator for the Quantised Orthosymplectic Superalgebra U q [osp(m|n)]

Dancer

Gould

Links

2007

Algebr Represent Theor

Self Cite

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“…The quantum superalgebra U q [osp(m|n)] is a q-deformation of the classical orthosymplectic superalgebra. A brief explanation of U q [osp(m|n)] is given below, with more details to be found in [10]. Throughout we use n = 2k and l = ⌊ m 2 ⌋, so m = 2l or m = 2l + 1.…”

Section: Introductionmentioning

confidence: 99%

Generalized Perk–Schultz models: solutions of the Yang–Baxter equation associated with quantized orthosymplectic superalgebras

Mehta¹,

Dancer²,

Gould³

et al. 2005

J. Phys. A: Math. Gen.

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The Perk-Schultz model may be expressed in terms of the solution of the Yang-Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra Uq[sl(m|n)], with a multiparametric co-product action as given by Reshetikhin. Here we present analogous explicit expressions for solutions of the Yang-Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras Uq[osp(m|n)]. In this manner we obtain generalisations of the Perk-Schultz model. 1 product decomposition. Here, we will explicitly formulate R-matrices for the case V (δ 1 ) ⊗ V (δ 1 ) for U q [osp(m|n)], in both the twisted and untwisted cases by explicitly computing the elementary intertwiners. We mention that formal expressions for the solutions of the Yang-Baxter equation associated with fundamental representations of superalgebras are given in [8], which may also be used to determine explicit expressions for the R-matrices (e.g. [9]). An alternative approach is to use the Lax operator method as described in [10,11].Once the explicit R-matrices have been obtained, we will introduce the Reshetikhin twist [5] in order to generate more general R-matrices with additional free parameters. These results can be used to obtain classes of integrable Hamiltonians describing systems of interacting fermions, with potential applications in condensed matter systems (cf. [12]).

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“…We note that the eigenvalues of higher order central elements C pkq V are computed explicitly in [LZ93,DGL05] for quantum supergroups U q pgl m|n q and U q posp m|2n q, where V is the natural representation. With the eigenvalue formula, it is shown in [Li10] that the centre of U q pgl n q is generated by C pkq V for 1 ď k ď n. In a sequel to this paper, we will prove an analogue of this result for quantum groups of types B, C and D.…”

Section: Introductionmentioning

confidence: 99%

Explicit generators and relations for the centre of the quantum group

Dai,

Zhang

2021

Preprint

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For the standard Drinfeld-Jimbo quantum group Uqpgq associated with a simple Lie algebra g, we construct explicit generators of the centre ZpUqpgqq, and determine the relations satisfied by the generators. For g of type Anpn ě 2q, D 2k`1 pk ě 2q or E 6 , the centre ZpUqpgqq is isomorphic to a quotient of a polynomial algebra in multiple variables, which is described in a uniform manner for all cases. For g of any other type, ZpUqpgqq is generated by n "rankpgq algebraically independent elements.

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Eigenvalues of Casimir invariants for Uq[osp(m∣n)]

Cited by 4 publications

References 23 publications

Lax Operator for the Quantised Orthosymplectic Superalgebra U q [osp(m|n)]

Lax Operator for the Quantised Orthosymplectic Superalgebra U q [osp(m|n)]

Generalized Perk–Schultz models: solutions of the Yang–Baxter equation associated with quantized orthosymplectic superalgebras

Explicit generators and relations for the centre of the quantum group

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