Kashiwara Operators

Falpha(m, i): AlgQUEElt, RngIntElt -> AlgQUEElt

Given a monomial \(m\) in \(U^-\) for some quantized enveloping algebra \(U\), i.e., \(m\) must be a monomial in the first \(n\) generators of \(U\), where \(n\) is the number of positive roots of the corresponding root datum, returns another monomial in the negative part of \(U\) that is obtained by applying the \(i\)-th Kashiwara operator \(\tilde{F}_i\) to \(m\) (see Section The Canonical Basis). Here \(i\) must lie between \(1\) and the rank of the root datum.

Ealpha(m, i): AlgQUEElt, RngIntElt -> AlgQUEElt

Given a monomial \(m\) in \(U^-\) for some quantized enveloping algebra \(U\), i.e., \(m\) must be a monomial in the first \(n\) generators of \(U\), where \(n\) is the number of positive roots of the corresponding root datum, return \(\tilde{E}_i(m)\) (see Section The Canonical Basis) if the \(i\)-th Kashiwara operator \(\tilde{E}_i\) is applicable to \(m\). Otherwise the zero element of \(U\) is returned. Here \(i\) must lie between \(1\) and the rank of the root datum.

Example: Kashiwara Operators on Monomials (ex-eb13a8)

> R:= RootDatum("F4");
> U:= QuantizedUEA(R);
> m:= U.1*U.5*U.10*U.18*U.24;
> m;
F_1*F_5*F_10*F_18*F_24
> Falpha(m, 3);
F_1*F_6*F_7*F_10*F_18*F_24
> Ealpha(m, 4);
F_1*F_4*F_5*F_7*F_9*F_18*F_24
> Ealpha(m, 2);
0

Run in calculator (opens in a new tab)Copy input