Quantum group structure in the Fock space resolutions of ŝl(n) representations. (English) Zbl 0704.17009
The authors are looking for a two-sided resolution of an irreducible highest weight module for the affine Kac-Moody algebra \(\hat sl(n)\) in terms of Fock space modules analogous to the Bernstein-Gel’fand-Gel’fand resolution by Verma modules. A Fock space realization of \(\hat sl(n)\) is given generalizing M. Wakimoto’s realization of \(\hat sl(2)\) [Commun. Math. Phys. 104, 605-609 (1986; Zbl 0587.17009)]. Intertwining operators on Fock space modules are built as appropriate contour integrals over products of “screening operators”. Within the contour integrals the algebra of screening operators satisfies the quantum group identities of \({\mathcal U}_ q({\mathfrak n}_+)\). There is a map from the singular vectors in a “quantum Verma module” to intertwining operators. In case \(n=3\) the intertwiners between Fock space modules are used to construct a complex of \(\hat sl(3)\) Fock space modules. A lack of necessary understanding of quantum Verma modules prevents the construction of the complex in the general case.
There are several conjectures: First, that the construction generalizes to get a complex for arbitrary n. Second, that the constructed complex is in fact a resolution, that the cohomology is concentrated in zero dimension equal to the irreducible highest weight module. It is stated in a footnote, that the last conjecture was answered by B. L. Feigin and F. V. Frenkel in the affirmative (Moscow Preprint 1989) using geometrical methods and restricting the considerations to the case of integrable highest weight modules.
The physical background is two-dimensional conformal field theory in a free field realization.
There are several conjectures: First, that the construction generalizes to get a complex for arbitrary n. Second, that the constructed complex is in fact a resolution, that the cohomology is concentrated in zero dimension equal to the irreducible highest weight module. It is stated in a footnote, that the last conjecture was answered by B. L. Feigin and F. V. Frenkel in the affirmative (Moscow Preprint 1989) using geometrical methods and restricting the considerations to the case of integrable highest weight modules.
The physical background is two-dimensional conformal field theory in a free field realization.
Reviewer: H.Boseck
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
Keywords:
highest weight module; affine Kac-Moody algebra; Fock space modules; Intertwining operators; quantum group; quantum Verma modules; two- dimensional conformal field theoryCitations:
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