Generalized quantum groups, including super and elliptic ones, and Weyl groupoids

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In 2006, Yamane, with Heckenberger, introduced axioms of Weyl groupoids and showed that Weyl groupoids satisfy a Matsumoto-type theorem. In 2008, using this result, they gave a factorization formula for the Shapovalov determinants of generalized quantum groups, including quantum superalgebras, by using the longest elements of the Weyl groupoids. This study is epoch-making since it was achieved by using intensively a special situation happened at roots of unity. Under some special conditions, simlar theorems were shown by several people, but their proofs were made by using the Shapovalov's original result for the simple Lie algebras.