Ref HAL: hal-02961331_v1
DOI: 10.1016/j.nuclphysb.2020.115176
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Free Fermions on vertices of distance-regular graphs are considered. Bipartitions are defined by taking as one part all vertices at a given distance from a reference vertex. The ground state is constructed by filling all states below a certain energy. Borrowing concepts from time and band limiting problems, algebraic Heun operators and Terwilliger algebras, it is shown how to obtain, quite generally, a block tridiagonal matrix that commutes with the entanglement Hamiltonian. The case of the Hadamard graphs is studied in detail within that framework and the existence of the commuting matrix is shown to allow for an analytic diagonalization of the restricted two-point correlation matrix and hence for an explicit determination of the entanglement entropy.

Commentaires: 24 pages, 37 ref.

Ref HAL: hal-02961322_v1
DOI: 10.1063/5.0008372
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Commentaires: 5 pages, 27 ref.

Ref HAL: hal-02961315_v1
DOI: 10.1142/9789811210679_0013
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Ref HAL: hal-02885777_v1
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We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation on a set of topological objects that we call fused braids. We use these algebras to prove a Schur-Weyl duality theorem for any tensor products of any symmetrised powers of the natural representation of U q (gl N). Then we proceed to the study of the fused Hecke algebras and in particular, we describe explicitely the irreducible representations and the branching rules. Finally, we aim to an algebraic description of the centralisers of the tensor products of U q (gl N)-representations under consideration. We exhibit a simple explicit element that we conjecture to generate the kernel from the fused Hecke algebra to the centraliser. We prove this conjecture in some cases and in particular, we obtain a description of the centraliser of any tensor products of any finite-dimensional representations of U q (sl 2).

Commentaires: 52 pages, 22 ref.

Ref HAL: hal-02890142_v1
Ref Arxiv: 1908.04806
DOI: 10.1088/1751-8121/ab604e
Ref. & Cit.: NASA ADS
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A description of the embedding of a centrally extended Askey–Wilson algebra, AW(3), in Uq(sl2) 3 is given in terms of the universal R-matrix of Uq(sl2). The generators of the centralizer of Uq(sl2) in its three-fold tensor product are naturally expressed through conjugations of Casimir elements with R. They are seen as the images of the generators of AW(3) under the embedding map by showing that they obey the AW(3) relations. This is achieved by introducing a natural coaction also constructed with the help of the R-matrix.

Ref HAL: hal-02890130_v1
Ref Arxiv: 1909.06426
DOI: 10.1007/s11005-019-01249-w
Ref. & Cit.: NASA ADS
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Résumé:

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 Sn.

Commentaires: 10 pages, 15 ref.

Ref HAL: hal-02065989_v1
Ref INSPIRE: 1722614
DOI: 10.1007/s00220-019-03299-6
WoS: 000459776400008
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Résumé:

In the continuity of our previous paper (Crampe et al. in Commun Math Phys 349:271, 2017, arXiv:1509.05516 ), we define three new algebras, ${\mathcal{A}_{\mathfrak{n}}(a,b,c)}$ , ${\mathcal{B}_{\mathfrak{n}}}$ and ${\mathcal{C}_{\mathfrak{n}}}$ , that are close to the braid algebra. They allow to build solutions to the Yang-Baxter equation with spectral parameters. The construction is based on a baxterisation procedure, similar to the one used in the context of Hecke or BMW algebras. The ${\mathcal{A}_{\mathfrak{n}}(a,b,c)}$ algebra depends on three arbitrary parameters, and when the parameter a is set to zero, we recover the algebra ${\mathcal{M}_{\mathfrak{n}}(b,c)}$ already introduced elsewhere for purpose of baxterisation. The Hecke algebra (and its baxterisation) can be recovered from a coset of the ${\mathcal{A}_{\mathfrak{n}}(0,0,c)}$ algebra. The algebra ${\mathcal{A}_{\mathfrak{n}}(0,b,-b^2)}$ is a coset of the braid algebra. The two other algebras ${\mathcal{B}_{\mathfrak{n}}}$ and ${\mathcal{C}_{\mathfrak{n}}}$ do not possess any parameter, and can be also viewed as a coset of the braid algebra.