Rational Cherednik Algebras

sage.algebras.rational_cherednik_algebra.RationalCherednikAlgebra

A rational Cherednik algebra.

Let \(k\) be a field. Let \(W\) be a complex reflection group acting on a vector space \(\mathfrak{h}\) (over \(k\)). Let \(\mathfrak{h}^*\) denote the corresponding dual vector space. Let \(\cdot\) denote the natural action of \(w\) on \(\mathfrak{h}\) and \(\mathfrak{h}^*\). Let \(\mathcal{S}\) denote the set of reflections of \(W\) and \(\alpha_s\) and \(\alpha_s^{\vee}\) are the associated root and coroot of \(s\). Let \(c = (c_s)_{s \in W}\) such that \(c_s = c_{tst^{-1}}\) for all \(t \in W\).

The rational Cherednik algebra is the \(k\)-algebra \(H_{c,t}(W) = T(\mathfrak{h} \oplus \mathfrak{h}^*) \otimes kW\) with parameters \(c, t \in k\) that is subject to the relations:

\[\begin{split}\begin{aligned} w \alpha & = (w \cdot \alpha) w, \\ \alpha^{\vee} w & = w (w^{-1} \cdot \alpha^{\vee}), \\ \alpha \alpha^{\vee} & = \alpha^{\vee} \alpha + t \langle \alpha^{\vee}, \alpha \rangle + \sum_{s \in \mathcal{S}} c_s \frac{\langle \alpha^{\vee}, \alpha_s \rangle \langle \alpha^{\vee}_s, \alpha \rangle}{ \langle \alpha^{\vee}, \alpha \rangle} s, \end{aligned}\end{split}\]

where \(w \in W\) and \(\alpha \in \mathfrak{h}\) and \(\alpha^{\vee} \in \mathfrak{h}^*\).

INPUT:

  • ct – a finite Cartan type
  • c – the parameters \(c_s\) given as an element or a tuple, where the first entry is the one for the long roots and (for non-simply-laced types) the second is for the short roots
  • t – the parameter \(t\)
  • base_ring – (optional) the base ring
  • prefix – (default: ('a', 's', 'ac')) the prefixes

Todo

Implement a version for complex reflection groups.

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