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Hall algebras and quantum groups. (English) Zbl 0735.16009

Let \(R\) be a finite dimensional representation-finite hereditary algebra of type \(\Delta\) over a field. Then there is a bijection between the isoclasses of \(R\)-modules of finite length and the functions \(a: \Phi^ +\to N_ 0\), where \(\Phi^ +\) is the set of all positive roots of \(\Delta\). For a commutative ring \(\Lambda\) and \(q\in\Lambda\), the author defines the Hall algebra \({\mathcal H}(R,\Lambda,q)\) which is a free \(\Lambda\)-module with basis formed by the isoclasses of \(R\)-modules of finite length and multiplication given by Hall polynomials counting some filtrations of such modules. The author also defines some canonical derivations \(\delta_ 1,\dots,\delta_ n\) (where \(n\) is the number of vertices of \(\Delta\)) from \({\mathcal H}(R,\Lambda,q)\) to itself, and then the skew polynomial ring \({\mathcal H}'(R,\Lambda,q)={\mathcal H}(R,\Lambda,q)[T_ i,\delta_ i]\). Then \({\mathcal H}'(R,\mathbb C,1)\) is isomorphic to the enveloping algebra \(U(\mathfrak b_ +)\), where \(\mathfrak b_ +\) is the Borel algebra of the semisimple Lie algebra \(\mathfrak g\) of type \(\Delta\). Finally, the author defines the completion \(\widehat{{\mathcal H}'(R)}=\displaystyle\lim_{{\longleftarrow\atop m}}{\mathcal H}'(R,\mathbb C[q]/(q-1)^ m,q)\) of the generic algebra \({\mathcal H}'(R,\mathbb C[q],q)\). The main results of the paper describe the algebra \(\widehat{{\mathcal H}'(R)}\) by generators and relations. As a consequence one obtains that \(\widehat{{\mathcal H}'(R)}\) is precisely Drinfeld’s quantization \(U_ h({\mathfrak b}_ +)\) of \(U(\mathfrak b_ +)\). In particular, \(\widehat{{\mathcal H}'(R)}\) is a Hopf algebra.

MSC:

16G30 Representations of orders, lattices, algebras over commutative rings
17B20 Simple, semisimple, reductive (super)algebras
17B35 Universal enveloping (super)algebras
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T05 Hopf algebras and their applications
16T20 Ring-theoretic aspects of quantum groups
16S36 Ordinary and skew polynomial rings and semigroup rings
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References:

[1] [D] Drinfeld, V.G.: Quantum groups. In: Proc. Int. Congr. Math. Berkeley 1986. Am. Math. Soc., 1987, pp. 798-820
[2] [DR] Dlab, V., Ringel, C.M.: On algebras of finite representation type. J. Algebra33, 306-394 (1975) · Zbl 0332.16014
[3] [M] Macdonald, I.G.: Symmetric functions and Hall polynomials. Clarendon Press: Oxford, 1979 · Zbl 0487.20007
[4] [R1] Ringel, C.M.: Hall algebras. In: Topics in Algebra. Banach Centre Publ. 26. Warszawa (To appear)
[5] [R2] Ringel, C.M.: Hall polynomials for the representation-finite hereditary algebras. Adv. Math. (To appear)
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