The highest coefficient of \(\text{det}H_ \eta\) and the center of the specialization at odd roots of unity for untwisted affine quantum algebras
The author studies the untwisted affine quantum algebra \({\mathfrak U}_{\mathfrak q}\) and its specializations at a primitive root of unity \(\varepsilon\). He calculates the highest coefficient of the determinant \(\text{det} H_\eta\), where \(H_\eta\) denotes a contravariant form defined on the Verma modules of \({\mathfrak U}_{\mathfrak q}\). To this aim he uses a type of Poincaré-Birkhoff-Witt basis and a reduction procedure, which gives the matrix of \(\text{det} H_\eta\) a triangular form. The multiplicity of \(\varepsilon\) in \(\text{det} H_\eta\) is calculated and related to a discussion of the center \({\mathfrak Z} ({\mathfrak U}_\varepsilon)\), which at last is described as a tensor product of its negative, null, and positive part. The paper is organized as follows: Part 1. Preliminaries, with the subsections 1. General setting, 2. Definitions, 3. Basis of type Poincaré-Birkhoff-Witt for \({\mathfrak U}_{\mathfrak q}\), 4. Integer form and specialization, 5. Verma modules, 6. The contravariant form H, 7. The ``Casimir'' operator C, 8. The classification of the untwisted affine Dynkin diagrams; Part 2. The determinant of the contravariant form, with the subsections 1. Some general facts, 2. The vectors \(E_\alpha\) with \(\alpha\in R^{re}_+\), 3. The vectors \(E_{(m, \delta,i)}\), 4. Triangularization of \(H_\eta\), 5. The highest coefficient \(b_\eta\) of \(\text{det} H_\eta\); Part 3. The center \({\mathfrak Z} ({\mathfrak U}_\varepsilon)\), with the subsections 1. The multiplicity of \(\varepsilon\) in \(\text{det} H_\eta\), 2. Relations between \(H_\eta\) and \({\mathfrak Z} ({\mathfrak U}_\varepsilon) \cap {\mathfrak U}^+_\varepsilon\), 3. Explicit description of \({\mathfrak Z} ({\mathfrak U}_\varepsilon) \cap {\mathfrak U}^+_\varepsilon\), 4. Explicit description of \({\mathfrak Z} ({\mathfrak U}_\varepsilon)\).
scientific article; zbMATH DE number 1013951