# Catalog Of Crystals¶

Let $$I$$ be an index set and let $$(A,\Pi,\Pi^\vee,P,P^\vee)$$ be a Cartan datum associated with generalized Cartan matrix $$A = (a_{ij})_{i,j\in I}$$. An abstract crystal associated to this Cartan datum is a set $$B$$ together with maps

$e_i,f_i \colon B \to B \cup \{0\}, \qquad \varepsilon_i,\varphi_i\colon B \to \ZZ \cup \{-\infty\}, \qquad \mathrm{wt}\colon B \to P,$

subject to the following conditions:

1. $$\varphi_i(b) = \varepsilon_i(b) + \langle h_i, \mathrm{wt}(b) \rangle$$ for all $$b \in B$$ and $$i \in I$$;
2. $$\mathrm{wt}(e_ib) = \mathrm{wt}(b) + \alpha_i$$ if $$e_ib \in B$$;
3. $$\mathrm{wt}(f_ib) = \mathrm{wt}(b) - \alpha_i$$ if $$f_ib \in B$$;
4. $$\varepsilon_i(e_ib) = \varepsilon_i(b) - 1$$, $$\varphi_i(e_ib) = \varphi_i(b) + 1$$ if $$e_ib \in B$$;
5. $$\varepsilon_i(f_ib) = \varepsilon_i(b) + 1$$, $$\varphi_i(f_ib) = \varphi_i(b) - 1$$ if $$f_ib \in B$$;
6. $$f_ib = b'$$ if and only if $$b = e_ib'$$ for $$b,b' \in B$$ and $$i\in I$$;
7. if $$\varphi_i(b) = -\infty$$ for $$b\in B$$, then $$e_ib = f_ib = 0$$.