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Derivations and automorphism of certain quantum algebras. (Dérivations et automorphismes de quelques algèbres quantiques.) (French) Zbl 0760.17003
As its title indicates, this paper is concerned with automorphisms and derivations of certain quantum algebras. More precisely:
(i). Let $$A$$ be the multiparametric quantum affine algebra, i.e. $$A$$ is generated by variables $$x_{i}$$, $$1\leq i \leq n$$, subject to the commutation rules $$x_{i}x_{j}= q_{ij}x_{j}x_{i}$$, where $$q_{ij}q_{ji} = 1$$, $$q_{ii} = 1$$. Then all the derivations of $$A$$ are determined. The analysis is carried further in some special cases, and this is used to obtain information about the automorphisms of $$A$$. For example, Aut($$A$$) is determined if $$q_{ij} = q$$ for $$i< j$$ and $$q$$ is not a root of 1.
(ii). In a similar way, the automorphisms and derivations of the algebra of $$2\times 2$$ quantum matrices and of the quantum enveloping algebra of $$s\ell(2)$$ (again $$q$$ is different from a root of 1) are determined.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W20 Automorphisms and endomorphisms 16W25 Derivations, actions of Lie algebras
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##### References:
 [1] Jacobson N., Lectures in Abstract Algebra III · Zbl 0124.27002 [2] DOI: 10.1017/S030500410005266X · Zbl 0362.17008 [3] Manin Y.I., Quantum groups and non commutative geometry (1988) · Zbl 0724.17006 [4] Rentschler R., C.R.A.S 267 pp 384– (1968) [5] Smith S.P., An introduction and survey for ring theorists · Zbl 0744.16023 [6] Van Der Kulk W., Nieuw Arch.Wisk 1 pp 33– (1953)
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