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

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Abstract: We construct an action of the extended double affine braid group

m a t h c a l d d o t B

on the quantum toroidal algebra

U_{q} (m a t h f r a k g_{m a t h r m t o r})

in all untwisted types. In the simply laced cases, using this action and certain involutions of

m a t h c a l d d o t B

we obtain automorphisms and anti-automorphisms of

U_{q} (m a t h f r a k g_{m a t h r m t o r})

which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements

C

and

k_{0}^{a_{0}} d o t s k_{n}^{a_{n}}

up to inverse. This generalises existing results in type

A

due to Miki.

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