Presentation by Borel subalgebras and Chevalley generators for quantum enveloping algebras

We provide an alternative approach to the Faddeev-Reshetikhin-Takhtajan presentation of the quantum group U_q(g), with L-operators as generators and relations ruled by an R-matrix. We look at U_q(g) as being generated by the quantum Borel subalgebras U_q(b_+) and U_q(b_-), and we use the standard presentation of the latters as quantum function algebras. When g = gl(n) these Borel quantum function algebras are generated by the entries of a triangular q-matrix, thus eventually U_q(gl(n)) is generated by the entries of an upper triangular and a lower triangular q-matrix, which share the same diagonal. The same elements generate over the ring of Laurent polynomials the unrestricted integer form of U_q(gl(n)) of De Concini and Procesi, which we present explicitly, together with a neat description of the associated quantum Frobenius morphisms at roots of 1. All this holds, mutatis mutandis, for g = sl(n) too.Comment: AMS-TeX, 17 pages. Final version appeared in "Proceedings of the Edinburgh Mathematical Society

Global splittings and super Harish-Chandra pairs for affine supergroups

This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, I discuss the "splitting" properties of affine supergroups, i.e. special kinds of factorizations they may admit - either globally, or pointwise. Second, I present a new contribution to the study of affine supergroups by means of super Harish-Chandra pairs (a method already introduced by Koszul, and later extended by other authors). Namely, I provide an explicit, functorial construction \Psi which, with each super Harish-Chandra pair, associates an affine supergroup that is always globally strongly split (in short, gs-split) - thus setting a link with the first part of the paper. On the other hand, there exists a natural functor \Phi from affine supergroups to super Harish-Chandra pairs: then I show that the new functor \Psi - which goes the other way round - is indeed a quasi-inverse to \Phi, provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) \Phi and \Psi are equivalences between the categories of gs-split affine supergroups and of super Harish-Chandra pairs. Such a result was known in other contexts, such as the smooth differential or the complex analytic one, or in some special cases, via different approaches: the novelty in the present paper lies in that I construct a different functor \Psi and thus extend the result to a much larger setup, with a totally different, more geometrical method (very concrete indeed, and characteristic free). The case of linear algebraic groups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper.Comment: La-TeX file, 48 pages. Final revised version, *after correcting the galley proofs* - to appear in "Transactions of the AMS

Algebraic supergroups of Cartan type

I present a construction of connected affine algebraic supergroups G_V associated with simple Lie superalgebras g of Cartan type and with g-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the supergroups G_V that I introduced. In particular, the supergroup constructed in this way associated with g := W(n) and its standard representation is described somewhat more in detail. In addition, *** an "Erratum" is added here *** after the main text to fix a mistake which was kindly pointed out to the author by prof. Masuoka after the paper was published: this "Erratum" is accepted for publication in "Forum Mathematicum", it appears here in its final form (but prior to proofreading). In it, I also explain more in detail the *Existence Theorem* for algebraic supergroups of Cartan type which comes out of the main result in the original paper.Comment: Main file: La-TeX file, 47 pages, already published (see below). Erratum: La-TeX file, 6 pages, to appear (see below). For the main file, the original publication is available at www.degruyter.com (cf. the journal reference here below

Dual Affine Quantum Groups

Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction - via formal Hopf algebras - we construct a new quantum group $U_q(\hat{\mathfrak{h}})$, dual of $U_q(\hat{\mathfrak{g}})$. Studying its restricted and unrestricted integer forms and their specializations at roots of 1 (in particular, their classical limits), we prove that $U_q(\hat{\mathfrak{h}})$ yields quantizations of $\hat{\mathfrak{h}}$ and $\hat{G}^\infty$ (the formal group attached to $\hat{\mathfrak{g}}$), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.Comment: 36 pages, AMS-TeX file. This the author's final version, corresponding to the pronted journal version. arXiv admin note: text overlap with arXiv:q-alg/951102

On the radical of Brauer algebras

The radical of the Brauer algebra B_f^x is known to be non-trivial when the parameter x is an integer subject to certain conditions (with respect to f). In these cases, we display a wide family of elements in the radical, which are explicitly described by means of the diagrams of the usual basis of B_f^x . The proof is by direct approach for x=0, and via classical Invariant Theory in the other cases, exploiting then the well-known representation of Brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation. This also gives a great part of the radical of the generic indecomposable B_f^x-modules. We conjecture that this part is indeed the whole radical in the case of modules, and it is the whole part in a suitable step of the standard filtration in the case of the algebra. As an application, we find some more precise results for the module of pointed chord diagrams, and for the Temperley-Lieb algebra - realised inside B_f^1 - acting on it.Comment: AMS-TeX file, 2 figures (in EPS format), 25 pages. This is the final version, to appear in "Mathematische Zeitschrift". Comparing to the previous one, it has been streamlined and shortened - yet the mathematical content stands the same. The list of references has been update

Geometrical Meaning of R-matrix Action for Quantum Groups at Roots of 1

The present work splits in two parts: first, we perform a straightforward generalization of results from [Re], proving autoquasitriangularity of quantum groups $U_q(\frak{g})$ and their unrestricted specializations at roots of 1, in particular the function algebra $F[H]$ of the Poisson group $H$ dual of $G$; second, as a main contribution, we prove the convergence of the (specialized) $R$--matrix action to a birational automorphism of a 2$\ell$--fold ramified covering of the specialization of $U_q(\frak{g})$ at a primitive $\ell$--th root of 1, and of a 2-fold ramified covering of $H$, thus giving a geometric content to the notion of triangularity (or autoquasitriangularity) for quantum groups.Comment: 23 pages, AMS-TeX C, Version 3.0; final author's version, as appeared in the printed pape

Duality functors for quantum groupoids

We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum groupoids", Comm. Math. Phys. 216 (2001), 539-581], we provide suitable notions of "quantum groupoids". For these objects, we detail somewhat in depth the formalism of linear duality; this yields several fundamental antiequivalences among (the categories of) the two basic kinds of "quantum groupoids". On the other hand, we develop a suitable version of a "quantum duality principle" for quantum groupoids, which extends the one for quantum groups - dealing with Hopf algebras - originally introduced by Drinfeld (cf. [V. G. Drinfeld, "Quantum groups", Proc. ICM (Berkeley, 1986), 1987, pp. 798-820], sec. 7) and later detailed in [F. Gavarini, "The quantum duality principle", Annales de l'Institut Fourier 53 (2002), 809-834].Comment: La-TeX file, 47 pages. Final version, after galley proofs correction, published in "Journal of Noncommutative Geometry". Compared with the previously posted version, we streamlined the whole presentation, we fixed a few details and we changed a bit the list of reference