# Double Affine Hecke Algebras of Rank 1 and the $Z 3$-Symmetric Askey-Wilson Relations - Mathematics > Rings and Algebras

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Abstract: We consider the double affine Hecke algebra$H=Hk 0,k 1,k^\vee 0,k^\vee 1;q$ associated with the root system$C^\vee 1,C 1$. We display three elements $x$, $y$, $z$ in $H$ that satisfyessentially the $Z 3$-symmetric Askey-Wilson relations. We obtain the relationsas follows. We work with an algebra $\hat H$ that is more general than $H$,called the universal double affine Hecke algebra of type $C 1^\vee,C 1$. Anadvantage of $\hat H$ over $H$ is that it is parameter free and has a largerautomorphism group. We give a surjective algebra homomorphism ${\hat H} \to H$.We define some elements $x$, $y$, $z$ in $\hat H$ that get mapped to theircounterparts in $H$ by this homomorphism. We give an action of Artin-s braidgroup $B 3$ on $\hat H$ that acts nicely on the elements $x$, $y$, $z$; onegenerator sends $x\mapsto y\mapsto z \mapsto x$ and another generatorinterchanges $x$, $y$. Using the $B 3$ action we show that the elements $x$,$y$, $z$ in $\hat H$ satisfy three equations that resemble the $Z 3$-symmetricAskey-Wilson relations. Applying the homomorphism ${\hat H}\to H$ we find thatthe elements $x$, $y$, $z$ in $H$ satisfy similar relations.

Autor: Tatsuro Ito, Paul Terwilliger

Fuente: https://arxiv.org/