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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
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