450 research outputs found
An action of the free product on the -Onsager algebra and its current algebra
Recently Pascal Baseilhac and Stefan Kolb introduced some automorphisms
, of the -Onsager algebra , that are roughly
analogous to the Lusztig automorphisms of . We
use and a certain antiautomorphism of to obtain an
action of the free product on
as a group of (auto/antiauto)-morphisms. The action forms a
pattern much more symmetric than expected. We show that a similar phenomenon
occurs for the associated current algebra . We give some
conjectures and problems concerning and .Comment: 15 page
The Universal Askey-Wilson Algebra and DAHA of Type
Let denote a field, and fix a nonzero such that
. The universal Askey-Wilson algebra is the associative
-algebra defined by generators and relations in the following way.
The generators are , , . The relations assert that each of
, ,
is central in . The universal
DAHA of type is the associative -algebra
defined by generators and relations (i)
; (ii) is central; (iii)
. We display an injection of -algebras
that sends ,
, . For the map
we compute the image of the three central elements mentioned above. The
algebra has another central element of interest, called the Casimir
element . We compute the image of under . We describe
how the Artin braid group acts on and as a group of
automorphisms. We show that commutes with these actions. Some
related results are obtained
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