Canonical bases arising from quantized enveloping algebras. (English) Zbl 0703.17008

Lusztig, George

J. Am. Math. Soc. 3, No. 2, 447-498 (1990).

Let \(U^+\) be the \(+\) part of the quantized enveloping algebra \(U\) associated to a root system. This is an algebra over the field \({\mathbb Q}(v)\) which for \(v=1\) specializes to the classical enveloping algebra \(U^+_ 1\) of the nilpotent radical of a Borel subalgebra in a semisimple Lie algebra. One of the main results of the paper is the construction of a canonical basis \(B\) of \(U^+\) as a \({\mathbb Q}(v)\)-vector space.
There is an analogy between the definition of \(B\) and the definition of a new basis for a Hecke algebra given in D. Kazhdan and G. Lusztig [Invent. Math. 53, 165–184 (1979; Zbl 0499.20035)]. The canonical basis \(B\) has a number of remarkable properties. One of them is that the product of two elements in \(B\) is a linear combination of elements in \(B\) with coefficients in \({\mathbb N}[v,v^{-1}]\). Another one is that \(B\) is well adapted to finite dimensional representations of \(U\). Namely, let \(L_ d\) be a finite dimensional simple \(U\)-module corresponding to the dominant weight \(d\) and let \(x_ 0\) be a lowest weight vector for it. Then it is shown that the set \(\{\delta x_ 0|\delta\) running through the set of all elements \(\delta\in B\) such that \(\delta x_ 0\neq 0\}\) forms a basis of \(L_ d\). It gives rise to a canonical basis in any finite dimensional simple module of the corresponding semisimple Lie algebra, that for type \(A\) should be closely related to the basis in C. De Concini and D. Kazhdan [Isr. J. Math. 40, 275–290 (1981; Zbl 0537.20006)]. It is also given a purely combinatorial formula for \(\dim L_ d\).

Reviewer: Vladimir L. Popov (Moskva)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
20C08	Hecke algebras and their representations
17B20	Simple, semisimple, reductive (super)algebras

Keywords:

quantized enveloping algebra; canonical basis; Hecke algebra

Citations:

Zbl 0499.20035; Zbl 0537.20006

References:

[1]	A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5 – 171 (French). · Zbl 0536.14011
[2]	A. D. Berenstein and A. V. Zelevinsky, Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys. 5 (1988), no. 3, 453 – 472. · Zbl 0712.17006 · doi:10.1016/0393-0440(88)90033-2
[3]	I. N. Bernšteĭn, I. M. Gel\(^{\prime}\)fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19 – 33 (Russian). · Zbl 0269.08001
[4]	N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). · Zbl 0186.33001
[5]	Corrado De Concini and David Kazhdan, Special bases for \?_{\?} and \?\?(\?), Israel J. Math. 40 (1981), no. 3-4, 275 – 290 (1982). · Zbl 0537.20006 · doi:10.1007/BF02761368
[6]	M. Dyer and G. Lusztig, Appendix to \( [{\text{L2}}]\), Geom. Dedicata (to appear).
[7]	Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71 – 103; correction, ibid. 6 (1972), 309 (German, with English summary). · Zbl 0232.08001 · doi:10.1007/BF01298413
[8]	David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165 – 184. · Zbl 0499.20035 · doi:10.1007/BF01390031
[9]	David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185 – 203. · Zbl 0461.14015
[10]	George Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3 (1990), no. 1, 257 – 296. · Zbl 0695.16006
[11]	George Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990), no. 1-3, 89 – 113. · Zbl 0714.17013 · doi:10.1007/BF00147341
[12]	G. Lusztig, Left cells in Weyl groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 99 – 111. · doi:10.1007/BFb0071433
[13]	G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), no. 2, 169 – 178. · Zbl 0473.20029 · doi:10.1016/0001-8708(81)90038-4
[14]	Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583 – 591. · Zbl 0735.16009 · doi:10.1007/BF01231516
[15]	A. V. Zelevinskiĭ, Two remarks on graded nilpotent classes, Uspekhi Mat. Nauk 40 (1985), no. 1(241), 199 – 200 (Russian). · Zbl 0577.20032