Fabio Gavarini

QUANTIZATION OF POISSON GROUPS

Let Gbe a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfeld structure of Poisson group; let Hbe its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups U M q, (h) — dual of U M q, (g) — which yield infinitesimal quantization of Hand G � ; we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for Hwhat was known for G � , completing the quantization of the pair (G � ,H � ).

THE GLOBAL QUANTUM DUALITY PRINCIPLE

Abstract Let R be an integral domain, let ħ ∈ R{0} be such that 𝕂 := R/ħR is a field, and let ℋ𝒜 be the category of torsionless (or flat) Hopf algebras over R. We call H ∈ ℋ𝒜 a ‘quantized function algebra’ (= QFA), resp. ‘quantized restricted universal enveloping algebra’ (= QrUEA), at ħ if—roughly speaking—H/ħH is the function algebra of a connected Poisson group, resp. the (restricted, if R/ħR has positive characteristic) universal enveloping algebra of a (restricted) Lie bialgebra. Extending a result of Drinfeld, we establish an ‘inner’ Galois correspondence on ℋ𝒜, via two endofunctors, ( )∨ and ( )′, of ℋ𝒜 such that H ∨ is a QrUEA and H′ is a QFA (for all H ∈ ℋ𝒜). In addition: (a) the image of ( )∨, resp. of ( )′, is the full subcategory of all QrUEAs, resp. of all QFAs; (b) if p := Char(𝕂) = 0, the restrictions ( )∨|QFAs and ( )′|QrUEAs yield equivalences inverse to each other; (c) if p = 0, starting from a QFA over a Poisson group G, resp. from a QrUEA over a Lie bialgebra 𝔤, the functor ( )∨, resp. ( )′, gives a QrUEA, resp. a QFA, over the dual Lie bialgebra, resp. the dual Poisson group. Several, far-reaching applications are developed in detail in [F. Gavarini, The global quantum duality principle: theory, examples, and applications, preprint 2003, http://arxiv.org/abs/math.QA/0303019] and [F. Gavarini, The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries, J. Algebra 285 (2005), 399–437] and [F. Gavarini, Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms, Commun. Math. Phys. 253 (2005), 121–155].

Quantun function algebras as quantum enveloping algebras

Inspired by a result in (Ga), we locate three integer forms of Fq(SL(n + 1)) over k ( q;q 1 ) , with a presentation by generators and relations, which for q = 1 specialize to U(h), where h is the Lie bialgebra of the Poisson Lie group dual to SL(n + 1). In sight of this we prove two PBW-like theorems for Fq(SL(n + 1)), both related to the classical PBW theorem for U(h).

On the Construction of Chevalley Supergroups

We give a description of the construction of Chevalley supergroups, providing some explanatory examples.We avoid the discussion of the A(1, 1), P(3) and Q(n) cases, for which our construction holds, but the exposigetion becomes more complicated. We shall not in general provide complete proofs for our statements, instead we will make an effort to convey the key ideas underlying our construction. A fully detailed account of our work is scheduled to appear in [Fioresi and Gavarini, Chevalley Supergroups, preprint arXiv:0808.0785 Memoirs of the AMS (2008) (to be published).

A PBW BASIS FOR LUSZTIG'S FORM OF UNTWISTED AFFINE QUANTUM GROUPS

Letbe an untwisted ane Kac-Moody algebra over the field C, and let Uq(ˆ) be the associated quantum enveloping algebra; let Uq(ˆ) be the Lusztigs integer form of Uq(ˆ), generated by q-divided powers of Chevalley generators over a suitable subring R of C(q). We prove a Poincare-Birkho-Witt like theorem for Uq(ˆ), yielding a basis over R made of ordered products of q-divided powers of suitable quantum root vectors.

CHEVALLEY SUPERGROUPS OF TYPE D(2;1;a)

I present a construction a` la Chevalley of affine supergroups associated with simple Lie superalgebras of (classical) type D(2,1;a), for any possible value of the parameter a - in particular, including non-integral values of a. This extends the similar work performed in [R. Fioresi, F. Gavarini, Chevalley Supergroups, Memoirs of the AMS 215 (2012), no. 1014 - arXiv:0808.0785v8 [math.RA]], where all other simple Lie superalgebras of classical type were considered. The case of simple Lie superalgebras of Cartan type is dealt with in [F. Gavarini, Algebraic supergroups of Cartan type, Forum Mathematicum (to appear), 92 pages - arXiv:1109.0626v5 [math.RA], so this work completes the program of constructing connected affine supergroups associated with any simple Lie superalgebra.

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. nIn 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.

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

Abstract: The present work splits in two parts: first, we perform a straightforward generalization of results from [Re], proving that quantum groups n

PRESENTATION BY BOREL SUBALGEBRAS AND CHEVALLEY GENERATORS FOR QUANTUM ENVELOPING ALGEBRAS

uqMg

The crystal duality principle: From Hopf algebras to geometrical symmetries

and their unrestricted specializations at roots of 1, in particular the function algebra F[H] of the Poisson group H dual of G, are braided; second, as a main contribution, we prove the convergence of the (specialized) R-matrix action to a birational automorphism of a n