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Representations of quantum algebras. (English) Zbl 0724.17012
Let U be the quantum group associated to a Cartan matrix $$(a_{ij})$$ of finite type. Thus, U is a Hopf algebra over the field $$A={\mathbb{Q}}(v)$$ of rational functions in an indeterminate v. The authors study the representation theory of U by defining a kind of dual of U, the ‘coordinate algebra’ $${\mathcal A}[U]$$, and setting up a general theory of induction from suitable subalgebras. In the case of ‘generalized parabolic subalgebras’, it is shown that this induction procedure has the standard properties, e.g. Frobenius reciprocity, transitivity and the tensor identity. In the case of a ‘Borel subalgebra’, analogues of Serre’s theorem, Grothendieck’s theorem, Kempf’s vanishing theorem for dominant characters and Demazure’s character formula are obtained, and it is shown that the concepts and results about good and excellent filtrations extend to the quantum case.
Let $$U_{\Gamma}$$ be obtained from U by specializing $${\mathcal A}$$ into a field $$\Gamma$$. The authors develop a Borel-Weil-Bott theory for $$U_{\Gamma}$$. If the image $$\zeta$$ of v is not a root of unity, the theory is analogous to the classical case, while if $$\zeta$$ is a root of unity (and $$char(\Gamma)=0)$$ it is analogous to the classical modular theory. The authors establish a linkage principle and a translation principle for $$U_{\Gamma}$$ which enables them to define a kind of “Jantzen filtration” and prove a sum formula. It is also shown that (whether or not $$\zeta$$ is a root of unity) finite-dimensional $$U_{\Gamma}$$-modules are integrable.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 20G40 Linear algebraic groups over finite fields
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