On the computation of fusion over the affine Temperley-Lieb algebra. (English) Zbl 1402.81214
Summary: Fusion product originates in the algebraization of the operator product expansion in conformal field theory. N. Read and H. Saleur [ibid. 777, No. 3, 316–351 (2007; Zbl 1200.81136)] introduced an analogue of fusion for modules over associative algebras, for example those appearing in the description of 2d lattice models. The article extends their definition for modules over the affine Temperley-Lieb algebra \(\mathsf{TL}_n^{\mathsf{a}}\).
Since the regular Temperley-Lieb algebra \(\mathsf{TL}_n\) is a subalgebra of the affine \(\mathsf{TL}_n^{\mathsf{a}}\), there is a natural pair of adjoint induction-restriction functors \((\uparrow^a_r, \downarrow_r^a)\). The existence of an algebra morphism \(\phi : \mathsf{TL}_n^{\mathsf{a}} \rightarrow \mathsf{TL}_n\) provides a second pair of adjoint functors \((\Uparrow_a^r, \Downarrow_a^r)\). Two fusion products between \(\mathsf{TL}^{\mathsf{a}}\)-modules are proposed and studied. They are expressed in terms of these four functors. The action of these functors is computed on the standard, cell and irreducible \(\mathsf{TL}_n^{\mathsf{a}}\)-modules. As a byproduct, the Peirce decomposition of \(\mathsf{TL}_n^{\mathsf{a}}(q + q^{- 1})\), when \(q\) is not a root of unity, is given as direct sum of the induction \(\uparrow_r^a \mathsf{S}_{n, k}\) of standard \(\mathsf{TL}_n\)-modules to \(\mathsf{TL}_n^{\mathsf{a}}\)-modules. Examples of fusion products of various pairs of affine modules are given.
Since the regular Temperley-Lieb algebra \(\mathsf{TL}_n\) is a subalgebra of the affine \(\mathsf{TL}_n^{\mathsf{a}}\), there is a natural pair of adjoint induction-restriction functors \((\uparrow^a_r, \downarrow_r^a)\). The existence of an algebra morphism \(\phi : \mathsf{TL}_n^{\mathsf{a}} \rightarrow \mathsf{TL}_n\) provides a second pair of adjoint functors \((\Uparrow_a^r, \Downarrow_a^r)\). Two fusion products between \(\mathsf{TL}^{\mathsf{a}}\)-modules are proposed and studied. They are expressed in terms of these four functors. The action of these functors is computed on the standard, cell and irreducible \(\mathsf{TL}_n^{\mathsf{a}}\)-modules. As a byproduct, the Peirce decomposition of \(\mathsf{TL}_n^{\mathsf{a}}(q + q^{- 1})\), when \(q\) is not a root of unity, is given as direct sum of the induction \(\uparrow_r^a \mathsf{S}_{n, k}\) of standard \(\mathsf{TL}_n\)-modules to \(\mathsf{TL}_n^{\mathsf{a}}\)-modules. Examples of fusion products of various pairs of affine modules are given.
MSC:
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 17B65 | Infinite-dimensional Lie (super)algebras |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81T27 | Continuum limits in quantum field theory |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 16D10 | General module theory in associative algebras |
Citations:
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