# 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](background.md#canbas)). 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](background.md#canbas)) 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: Q Grp Automs (ex-eb13a8)`

```magma
> 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

```