The rational Cherednik algebra H_{t,c}(W,h)

Use the data of Reflection data and parameters for a rational Cherednik algebra. The tensor algebra $T(\mathfrak{V})={⨁}_{n\ge 0}{\mathfrak{V}}^{\otimes n}$ is the unital associative algebra generated by $\mathfrak{V}$ with no relations. The smash product $T(\mathfrak{V})⋊\mathbb{C}W$ is the vector space $T(\mathfrak{V})\otimes \mathbb{C}W$ with multiplication determined by $(aw)(bv)=aw(b)(wv)$, for $a,b\in T(\mathfrak{V})$ and $w,v\in W$.

For $t\in \mathbb{C}$, define $H_{t,c}(W,\mathfrak{h})$ to be the quotient of this smash product by the two-sided ideal generated by $[x,x^{\mathrm{\prime }}]=0,[y,y^{\mathrm{\prime }}]=0,[y,x]=tx(y)1-{\sum }_{s\in S}{c}_s{\alpha }_s(y){\alpha }_s^{\vee }(x)s,$ for all $x,x^{\mathrm{\prime }}\in {\mathfrak{h}}^{\ast }$ and $y,y^{\mathrm{\prime }}\in \mathfrak{h}$. Here $[a,b]=ab-ba$. The product ${\alpha }_s(y){\alpha }_s^{\vee }(x)$ is the normalization from the data item; the parameter $c_s$ is constant on conjugacy classes. This quotient is an associative unital complex algebra by construction. The case $t=1$ is denoted $H_{1,c}$ and is used for category O.

This definition makes no PBW or noncollapse assertion. Those are proved in The PBW theorem for rational Cherednik algebras.

For $W=\{1\}$ and $\dim \mathfrak{h}=1$, the presentation is $[y,x]=t$. For the order-two reflection on a one-dimensional $\mathfrak{h}$, the normalization in the preceding definition gives $[y,x]=t-2cs$.