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NATO ASI series. Series B : Physics

The Structure of Finite Dimensional Affine Hecke Algebra Quotients and their Realization in 2D Lattice Models

Dan Lévy

Published Jan 1, 1993• 1 citation
Publisher
DOI
10.1007/978-1-4899-1612-9_16

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LATTICE ALGEBRAS AND THE HIDDEN SYMMETRY OF THE 2D ISING MODEL IN A MAGNETIC FIELD

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Lattice algebras are defined and used to study the symmetries of 2D lattice models. New and interesting examples of such algebras are provided by the affine Hecke algebra, owing to the possibility of constructing braid generators out of its generators. I propose an Ansatz for the braid generators and derive some solutions. A particular finite-dimensional quotient is shown to be a natural generalization of the Temperley-Lieb-Jones algebra. It is used to give a unified picture of known and unknown symmetries of the spin-½ xxz model with boundary terms. The Ising model in an external magnetic field is also a representation of this quotient.

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On a two-parameter generalization of Hecke algebras and the XY chain in a magnetic field

Paul Martin

The structure and representation theory of a two-parameter generalization of Hecke algebras, denoted HL(q,η), which plays an important role in the construction of the L-site XY chain in a magnetic field is investigated. In particular, the structure of the XY chain representation of the algebra, on C2⊗L, and hence the structure of the centralizer algebra EndHL(q,η)(C2⊗L) for the XY chain representation, is analyzed. It is proved that the full algebra never reduces to the ordinary (one-parameter) Hecke algebra HL(q) in any specialization of the parameters, but contains the Hecke algebra as a quotient in a certain specialization of one of the parameters. This corrects an earlier claim of Hinrichsen and Rittenberg [Phys. Lett. B 275, 350 (1992)]. Bounds are established on the generic dimension of the centralizer algebra 4 ≤ dim(EndHL(q,η)(C⊗L2)) ≤ 8 (the upper bound is realized in every known case). Hence, it is argued that the structure of the centralizer algebra is independent of L, which is consistent with a result of Hinrichsen and Rittenberg, but that it is isomorphic to two commuting copies (rather than one copy) of the Pauli matrices. This contrasts sharply with the dimension of EndHL(q)(C⊗L2), which grows without bound as L increases.

Journal of Mathematical Physics•Nov 1, 1992•2 citations
Lattice algebras are defined and used to study the symmetries of 2D lattice models. New and interesting examples of such algebras are provided by the affine Hecke algebra, owing to the possibility of constructing braid generators out of its generators. I propose an Ansatz for the braid generators and derive some solutions. A particular finite-dimensional quotient is shown to be a natural generalization of the Temperley-Lieb-Jones algebra. It is used to give a unified picture of known and unknown symmetries of the spin-½ xxz model with boundary terms. The Ising model in an external magnetic field is also a representation of this quotient.
The structure and representation theory of a two-parameter generalization of Hecke algebras, denoted HL(q,η), which plays an important role in the construction of the L-site XY chain in a magnetic field is investigated. In particular, the structure of the XY chain representation of the algebra, on C2⊗L, and hence the structure of the centralizer algebra EndHL(q,η)(C2⊗L) for the XY chain representation, is analyzed. It is proved that the full algebra never reduces to the ordinary (one-parameter) Hecke algebra HL(q) in any specialization of the parameters, but contains the Hecke algebra as a quotient in a certain specialization of one of the parameters. This corrects an earlier claim of Hinrichsen and Rittenberg [Phys. Lett. B 275, 350 (1992)]. Bounds are established on the generic dimension of the centralizer algebra 4 ≤ dim(EndHL(q,η)(C⊗L2)) ≤ 8 (the upper bound is realized in every known case). Hence, it is argued that the structure of the centralizer algebra is independent of L, which is consistent with a result of Hinrichsen and Rittenberg, but that it is isomorphic to two commuting copies (rather than one copy) of the Pauli matrices. This contrasts sharply with the dimension of EndHL(q)(C⊗L2), which grows without bound as L increases.