The equal-parameter double-affine Hecke algebra of type A1

Use the rank-one root datum, affine actions, field $K$, and Laurent polynomials $B_{i,m}$ from The rank-one root datum and parameter conventions for DAHA and The rank-one DAHA Bernstein quotients are Laurent polynomials. Let $F$ be the free associative unital $K$-algebra on $Z,Z^{-1},T_0,T_1,\pi$. Define the equal-parameter double-affine Hecke algebra

where $I$ is the two-sided ideal generated by the following relations:

1. $Z{Z}^{-1}=Z^{-1}Z=1$.
2. ${\pi }^2=1$ and $\pi T_0\pi =T_1$ (hence also $\pi T_1\pi =T_0$).
3. $(T_i-t)(T_i+t^{-1})=0$ for $i=0,1$.
4. For every $m\in \mathbb{Z}$ and $i\in \{0,1\}$,

where $s_1(Z^m)=Z^{-m}$ and $s_0(Z^m)=q^{-2m}{Z}^{-m}$. 5. $\pi Z\pi =q^{-1}{Z}^{-1}$.

This is the rank-one root-datum DAHA presentation: $T_0,T_1$ are the affine Hecke generators, $Z$ is the weight-lattice generator, and $\pi$ is the length-zero symmetry. There is no finite braid relation between $T_0$ and $T_1$ because the affine $A_1$ Coxeter label is infinite. The scalar normalization is $(T_i-t)(T_i+t^{-1})=0$; $q$ controls translations and is independent of $t$. The quotient is a definition, not a claim about a PBW basis or a faithful polynomial representation.

The fundamental translation is $y=\pi s_1$, acting by the $q$-shift $z\mapsto qz$; its DAHA lift is $Y=\pi T_1$. The quadratic relation makes $T_1$ invertible, with $T_1^{-1}=T_1-(t-t^{-1})$.

The existence of the free algebra quotient uses the standard universal property of generators and two-sided relations in The free associative R-algebra on a set and descent of relations. The local proof of polynomial representation faithfulness is separate.