Automorphisms of quantum toroidal algebras from an action of the extended double affine braid group

Abstract

We first construct an action of the extended double affine braid group $\mathcal{\ddot{B}}$ on the quantum toroidal algebra $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of $\mathcal{\ddot{B}}$ we produce automorphisms and anti-involutions of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements $C$ and $k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$ up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type $A$ due to Miki which have been instrumental in the study of the structure and representation theory of $U_{q}(\mathfrak{sl}_{n+1,\mathrm{tor}})$.

Figures (6)

• Figure 1: Illustration of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ and its quantum affine subalgebras $\mathcal{U}_{h}$ and $\mathcal{U}_{v}$
• Figure 2: Illustration of $\mathcal{\ddot{B}}$ and its extended affine braid subgroups $\mathcal{B}_{h}$ and $\mathcal{B}_{v}$
• Figure 3: Illustrations of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ displaying the two generating sets
• Figure 4: The untwisted affine Dynkin diagrams. Black labels are vertex numbers, blue labels are $a_{i}$ values, and in the non-symmetric cases $a_{i}^{\vee}$ values are in red
• Figure 5: The twisted affine Dynkin diagrams. Black labels are vertex numbers, blue labels are $a_{i}$ values, and red labels are $a_{i}^{\vee}$ values

Theorems & Definitions (38)

• Definition 3.1
• Definition 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6
• Definition 3.7
• Proposition 3.8
• Remark 3.9
• Example 3.10