84 research outputs found

### 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

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