Shapovalov determinant for loop superalgebras

Yetter-Drinfeld modules over Nichols systems and their reflections

“…In particular, for Nichols systems of diagonal type we will obtain the result, that such an induced objects irreducibility can be characterized by a polynomial that is given by the positive roots of its Nichols system (Theorem 5.2.18, Theorem 5.2.19). This polynomial (Definition 5.1.25) also appears in [KM99], [BK02], [LL08], but most notably in a similar context in [HY10], and is known as the Shapovalov determinant. While in [HY10] a similar result was obtained with a lot of technicality and explicit calculations, here, in a more general context, we will obtain this polynomial from the Shapovalov morphism, which is the reason we chose this name.…”