# zbMATH — the first resource for mathematics

Derivations of the algebra $$U_{q}^{+}(B_2)$$. (Dérivations de l’algèbre $$U_{q}^{+}(B_2)$$.) (French) Zbl 1222.17013
Let $$\mathfrak g$$ be a simple Lie algebra of type $$B_2$$ over an algebraically closed field $$\mathbb K$$ of characteristic 0 and let $$q$$ be an element of $${\mathbb K}^{\ast}$$ which is not a root of unity. In the paper under review the authors study the derivations of the quantized enveloping algebra $$U=U_q^+({\mathfrak g})$$ of the positive nilpotent part of $$\mathfrak g$$. The algebra $$U$$ is generated by two elements $$e_1$$ and $$e_2$$ satisfying the Serre relations $$e_1^2e_2+(q^2+q^{-2})e_1e_2e_1+e_2e_1^2=0$$ and $$e_2^3e_1-(q^2+1+q^{-2})e_2^2e_1e_2+(q^2+1+q^{-2})e_2e_1e_2^2-e_1e_2^3=0$$.
The main result states that as a module over the centre $$Z(U)$$ of $$U$$ the algebra of derivations of $$U$$ has the form $$\text{Der}(U)=Z(U)\gamma_1\oplus Z(U)\gamma_2\oplus \text{DerInn}(U)$$, where $$\text{DerInn}(U)$$ is the algebra of inner derivations and the derivations $$\gamma_1$$ and $$\gamma_2$$ are defined by $$\gamma_1(e_1)=0$$, $$\gamma_1(e_2)=e_2$$ and $$\gamma_2(e_1)=e_1$$, $$\gamma_2(e_2)=-e_2$$.
##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16S30 Universal enveloping algebras of Lie algebras 16S36 Ordinary and skew polynomial rings and semigroup rings 16T20 Ring-theoretic aspects of quantum groups 16W25 Derivations, actions of Lie algebras 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
Full Text: