Scientific Interests
of Fabio GAVARINI
home page of Fabio GAVARINI
9 July 2018
Research main guidelines
My scientific activity till now has been centered around representation theory, Lie theories and their quantization: Lie algebras, Lie groups, algebraic groups and their representations, invariant theory, homogeneous spaces, quantum theories, etc. Within this very broad framework, my research work has focused upon three specific topics: algebraic groups and their representations, quantum groups, Hopf algebras and their generalizations. Hereafter I sketch a rough outline of my results about all this (
Remark: the numbers in square brackets below refer to the subsequent list of publications).
Algebraic Groups, Representations and related topics:
In the broad field of algebraic group theory and its representationtheoretical side, I mainly focused upon two main trends: classical invariant theory, and (algebraic) supergroup theory.
Classical Invariant Theory: In invariant theory for classical groups a central role is played by the socalled SchurBrauerWeyl duality. This strictly relates the irreducible representations of such a group to the irreducible representations of the algebra
A(m) which centralizes the action of the group itself on the
mth tensor power of
V, where
V is the natural representation (over a field) of the group. For
GL(V) or
SL(V), the algebra
A(m) is a quotient of the group algebra of the symmetric group on
m elements; when realizing the latter as set of graphs, and its product as a "graphical composition", one obtains a combinatorial description of
A(m). For the orthogonal or symplectic groups instead,
A(m) is a quotient of the socalled
Brauer algebra, which has as basis a semigroup of graphs which extends the symmetric group, and still admits a suitable combinatorial description. For the last one hundred years this theory has been, and still is nowadays, of central importance for the representation theory of classical groups; it cyclically comes back under specialists' investigations, both because of its intrinsic interest and because several longstanding classical problems still remain unsolved.
In this context, the paper
[
2]
makes use of the above mentioned duality to study the submodule
T_{k,m}(V) of all tensors of valence
k
in the
mth tensor power of
V; understanding
the structure of this space was exactly one of the unsolved problems
mentioned right now. The result in
[
2]
is a description of
T_{k,m}(V) as a representation,
for the Brauer algebra, induced from a simpler one. As a byproduct this
description  essentially combinatorial  provides a new proof of
Littlewood's restriction formula (for
m big enough) which
describes the restriction to
SP(V), or to
O(V),
of an irreducible
GL(V)representation. The analysis and
results of this work are improved in
[
5],
where a more subtle combinatorial description of the Brauer algebra and
of its indecomposable modules yields a new proof of a stronger version
of Littlewood's restriction formula: in particular, the original result
for the orthogonal group is improved in this work, for the first time
since Littlewood's article of 1944. With similar techniques, in
[
26]
a great part of the radical of the Brauer algebra is described, and
similarly for its indecomposable modules.
Supergroups:
In classical geometry, the symmetry groups of interest are the Lie groups (in the differential setup) or the algebraic groups (in the algebraic framework). In supergeometry, these are replaced by Lie supergroups or algebraic supergroups: objects of both types can be defined with the language of topology (which better stress the underlying aspects of classical geometry), or in terms of functors of points (which better serves a wider generalizations). These supergroups are in close relation with Lie superalgebras via the superanalogue of Lie's theorems relating Lie groups and Lie algebras in the classical setting. Nevertheless, the study of Lie superalgebras is somehow easier  and it has been pushed forward much more  than that of supergroups. In particular, the classification (and structure theory) of supergroups is much less advanced than the Lie superalgebras one: indeed, even the construction of examples is somewhat more problematic.
In this context, with the papers
[
29],
[
30],
[
31],
[
32]
and
[
33]
one moves a step forward in the direction of a (sort of) classification program for "simple" algebraic supergroups (roughly speaking, say). Indeed, these works provide an existence theorem for all connected algebraic supergroups whose tangent Lie superalgebra is simple. We recall that such Lie superalgebras (simple, finitedimensional), whose classification is wellknown, split into two classes:
those of
classical type  which are the superanalogue (in a sense), of simple finitedimensional complex Lie algebras, or of affine complex KacMoody algebras  and those of
Cartan type.
In detail, in the works
[
29]
and
[
33]
one obtains such an existence result for the
classical case, by means of a direct, concrete construction, which mimics the classical one by Chevalley that provides all connected simple algebraic groups whose tangent Lie algebra is semisimple. Indeed, one starts with a classical Lie superalgebra and a simple representation of it: the desired algebraic supergroups are then given as subgroups (in the general linear supergroup over the representation space) generated by "oneparameter supersubgroups" attached to root vectors in the Lie superalgebra itself. In particular, this construction provides a unifying approach to most of the algebraic supergroups already known in literature; in addition, it yields an explicit recipe to produce new examples. Moreover, these supergroups are built in as "super groupschemes" over
Z. On the other hand in
[
30]
 proceedings of a conference  one gives a short, thoughtout presentation of this very construction, and in addition one shows also how one can extend the just described method to other contexts.
Some particular "Chevalley supergroups" (namely those of type
D(2,1;a), now considered as complex Lie supergroups) are investigated again in
[
41].
Roughly speaking, the Lie superalgebras of type
D(2,1;a) form a oneparameter family whose elements are (Lie superalgebras that are) simple for all but a finite number of values of the parameter: in this work we show that the construction of the associated Lie supergroups also makes sense for those "singular" values of the parameter, leading to nonsimple supergroups that we describe in some detail. One can realize this in different ways (that yield different results in the singular case), five of these are presented in detail; while doing this, we also compare the approach by Scheunert with that by Kac (more largely followed in literature).
As a next step, in
[
32]
one proves sort of a converse result of those in
[
29]:
namely, one shows that every connected algebraic supergroup whose tangent Lie superalgebra is classical is necessarily isomorphic to a Chevalley supergroup of the type considered in
[
29].
With largely similar techniques and strategy, in
[
31]
one proves a similar existence and uniqueness  up to isomorphisms  theorem for (connected) algebraic supergroups whose tangent Lie superalgebra is of Cartan type; in particular, the case of type
W(n) is treated a bit more in detail. The same topic is treated again in
[
37],
where a mistake in
[
31]
is amended and a key step of the main construction is clarified more in depth.
Finally, in
[
36]
I dwell on the more general problem of studying an (affine) algebraic supergroup via its associated super HarishChandra pair  namely, the datum of its underlying classical algebraic group and its tangent Lie superalgebra. This is a key theme in supergroup theory, which is addressed by several authors in different ways; I myself present yet another approach, tightly related with the existence of "global splittings" for (affine) supergroups. Roughly, for a supervariety this is the property to split into direct product of a classical (algebraic) variety and a totally odd supervariety  in short, a global factorization of type "
even x
odd": although this does not necessarily hold for supervarieties in general (it always works locally, but globally may fail), it is true instead  under mild conditions  for affine supergroups. The same topic is treated again in
[
39]
but considering Lie supergroups (of either real smooth, real analytic or complex holomorphic type) instead of algebraic supergroups: namely, the equivalence of Lie supergroups and super HarishChandra pairs is proved giving a new functorial recipe to construct, out of a given super HarishChandra pair, a suitably devised Lie supergroup. The underlying idea of this work is (essentially) the same as in
[
36],
yet technicalities are different because of the different framework.
Quantum Groups: Quantum Groups
are algebraic deformations, in the category of Hopf algebras, either of
universal enveloping algebras of Lie algebras  hence they are called
quantized universal enveloping algebras (QUEA in the sequel) 
or of function algebras of algebraic groups or Lie groups  then they
are called
quantized function algebras (QFA in the sequel).
Introduced in 1985 as "quantum symmetries", they later prove themselves
very interesting also in the study of algebraic groups in positive
characteristic and their representations, or for the theory of knot
(and link) invariants and many other topics. Moreover, they are
intrinsically related to the geometric theory that one gets from
them as semiclassical limits or, conversely, which they quantize:
namely, the theory of Poisson (Lie or algebraic) groups and of Lie
bialgebras, and more in general the geometry of Poisson varieties.
My contributions in this field divide into three main trends.
QFA=QUEA: The
"equivalence" between QUEAs and QFAs is a first trend.
In
[
0]
and
[
3],
starting from the most famous quantum groups, built upon semisimple Lie algebras, with standard Poisson structure on the associated groups, I introduce quantum groups for the dual Poisson groups: the data I start from now are the QUEAs by Jimbo (and Drinfeld) and their integral forms, both restricted and unrestricted. Taking the first or the second ones the corresponding semiclassic limits respectively are universal enveloping algebras of Lie bialgebras or function algebras of Poisson groups: this is the basic idea, which applies as well to the case of Hopf algebras dual to the QUEAs I was starting from, thus yielding quantum groups "duals" to the above ones. The analogue of this work for affine KacMoody algebras is done in
[
7]:
here the key tool is a theorem of PoincaréBirkhoffWitt
(PBW) type for restricted integral forms of the affine QUEAs; such a
result is stated and proved in
[
6].
In
[
4]
I perform, dually, a similar construction to that of
[
0],
and
[
3],
yet more concrete; but now I start from the QFAs associated to
SL(n) or
GL(n), for which it is available a
wellknown presentation by generators and relations by means of
"
qmatrices"; another result in this paper is a PBW
theorem for the QFA over
SL(n).
All these results then get improved in
[
23]
and
[
24].
For the
SL(n) case, looking at the
corresponding QUEA as a QFA leads to find an alternative presentation:
this is the content of
[
19],
where such a presentation is given in terms of "
qmatrices".
In particular, this yields an alternative approach to the wellknown
presentation by
Loperators due to Faddeev, Reshetikhin
and Takhtajan.
A further expansion of some aspects of
[
4]
is the article
[
25]:
in it some theorems of PoincaréBirkhoffWitt type are proved
for the QFA associated to
Mat(n), to
GL(n) or
to
SL(n), and for their specializations at roots of 1.
As a corollary, one obtains also some interesting results about
the structure of Frobenius algebra for these QFA at
roots of 1.
Finally, another development of all these constructions is performed in
[
40],
where
multiparametric quantum groups are taken into account, along with their integral forms and their specializations at roots of 1. Somehow related with this is
[
42];
here
deformations of (standard) quantum groups are studied, which can be realized either by twisting the comultiplication or by a 2cocycle deformation of the multiplication: both procedures lead to
multiparameter quantum groups, and  although the approaches are somehow "dual" to each other  the final outcome is (essentially) the same in either case.
QDP: The
quantum
duality principle, or QDP in the sequel, is the guiding idea of the
second trend, and explains the results of the first one. In the original
formulation by Drinfeld, the QDP provides an category equivalence between
QUEAs and QFAs, for quantum groups defined (as Hopf algebras) over
k[[
h]], where
k is a field, and topologically
complete: in
[
12]
I give a complete and rigorous proof  the first one in literature 
of this result. In
[
E1]
and
[
22]
instead I push forward this idea, formulating a much stronger version
of the QDP for Hopf algebras defined over very general rings and without
additional topological assumptions. Namely, I prove that Drinfeld's
recipes do define two endofunctors of the category of such Hopf algebras,
which realize a Galois correspondence in which these functors have as
images the subcategory of the QUEAs and of the QFAs respectively; moreover,
QUEAs and QFAs are exactly the subcategories of those objects which are
fixed by the composition of the two functors. Of course, as the contexts
are different the techniques which are involved in
[
12]
and in
[
E1],
[
22]
are rather different.
While
[
E1]
is a widely expanded essay, enriched with several examples and applications,
[
22]
is the journal article which treats just the main, central result of
[
E1],
namely the theorem expressing the stronger version of the QDP explained
above. Both
[
9]
and
[
11]
 proceedings of conferences  are short versions of
[
E1],
each one enriched with an original example. On the other hand,
[
E2]
instead  notes for a summer school  is a survey of the results of
[
E1]
by means of several explicit examples and applications.
More in general, a direct application of the QDP to Hopf algebras defined over a field leads to the
crystal duality principle, or CDP in short. This can also be obtained by classical means  i.e., without involving quantum groups  so it reads like a chapter of classical Hopf algebra theory. The works dealing with these are
[
15],
[
16]
and
[
17]:
for more details, see section
CDP in "Hopf Algebras and related structures" below.
A further development is
[
18],
where we state and prove a QDP for homogeneous spaces, or for the
corresponding subgroups. As an application we compute an explicit
quantization of an important Poisson structure on the space of Stokes
matrices; a shorter, seminal version of this work is
[
21],
where some other applications and examples are presented. Moreover,
a version of this work in terms of
global quantum groups
is developed in
[
35],
where in addition we consider different versions of "quantization" for subgroups;
in this way also noncoisotropic subgroups can be taken into account, but our results
show that in the end the coisotropic ones necessarily play a key role.
Finally, these ideas are extended to the context of
projective homogeneous spaces,
studying the example of Grassmann varieties in
[
28]
and the general case in
[
27].
In yet another direction, in
[
34]
we explore the possibility to extend all this lot of ideas to the framework of "quantum groupoids", i.e. quantizations of bialgebroids: here the notion of bialgebroid is a suitable generalization of that of bialgebra, we deal with quantization in a formal sense and, at the semiclassical level, Lie (bi)algebras are replaced by LieRinehart (bi)algebras  sometimes called just "Lie (bi)algebroids". In particular, we develop a convenient form of QDP for these objects (also showing this "at work" in a specific example).
RMAT:
Rmatrices and braidings are the core of the third trend. The notion of
Rmatrix for a QUEA is the quantization of the notion of classical
rmatrix for a Lie bialgebra, which corresponds to consider those Lie bialgebras whose Lie cobracket be a particular coboundary. More in general, the Hopf algebras endowed with an
Rmatrix correspond, via TannakaKrein duality and associated reconstruction theorems, to the braided monoidal categories, i.e. those endowed with an analogue of the tensor product and of the twist ("exchange of factors") automorphism for it. It springs out of this the interest of this algebras in quantum and conformal field theories, as well as in topology for the construction of link invariants and of 3variety invariants. One gets a weaker notion by substituting the
Rmatrix with a suitable automorphism of the Hopf algebra, which is called
braiding.
In my work upon this topic I applied the QDP (see above) to those QUEA endowed with an
Rmatrix (the socalled "quasitriangular" ones): the main result is that, given a Lie bialgebra endowed with a classical rmatrix (also called "quasitriangular"), one finds a geometrical counterpart of such an
rmatrix for
dual formal Poisson group, thus explaining the relationship between
rmatrices and Poisson duality (among Poisson groups).
In
[
1],
given a QUEA over a semisimple Lie algebra, and its standard
Rmatrix, sorting out of the QUEA a QFA (after the QDP, like in
[
E1]
or in
[
E2]
or in
[
22])
I prove that the adjoint action of the
Rmatrix does specialize to a distinguished automorphism on this QFA. Moreover, the semiclassical limit of the latter is a birational automorphism of the dual Poisson group, and more precisely a braiding, in geometrical sense; thus I widely generalize a result proved by Reshetikhin for
SL(2). All this I later extend to the case of KacMoody algebras in
[
8].
In
[
10]
we perform a further generalization, proving that a similar result does hold for any quasitriangular QUEA: here we apply the QDP as formulated in
[
12],
i.e. for topological quantum groups directly using the general definition of Drinfeld's functor from the QUEAs to the QFAs rather than an explicit description as one does in
[
1]
and
[
8].
In
[
13]
we compare the results of
[
1]
with those of Weinstein and Xu, who make an analogous braiding on the dual of a quasitriangular Poisson group by means of purely geometrical methods. Our first result is that both braidings are "infinitesimally trivial". The second is that in the case of
SL(2) these braidings do coincide: the proof comes out of an explicit description of both of them, via direct computation.
Finally in
[
14]
we show that, given a formal quasitriangular Poisson group
G, with classical
rmatrix
r, a braiding associated to it on the dual formal Poisson group
G^{*} is
unique: in particular the one in
[
13]
and that in Weinstein and Xu always coincide. Furthermore we make precise the nature of such a braiding, proving that it is Hamiltonian, corresponding to some function
f_{r} on
G^{*}, which in turn is a "lifting" of
r from the cotangent Lie bialgebra
of
G^{*} to the function algebra of
G^{*} itself. We provide two independent construction of such a lifting
f_{r}: in the first one,
f_{r} is given as semiclassical limit of the "logarithm" of a (rescaled) quantum
Rmatrix which quantizes
r; in the second one, we make out
f_{r} directly as a lifting of
r by iterated approximations, where the possibility of performing the
nth step is proved by cohomological means.
Hopf Algebras and related structures: The theory of Hopf algebras is a classical topic which gained new interest in the last twenty years, mainly for its interplay with such diverse fields as quantum groups, lowdimensional topology, tensor categories, supergeometry, etc.
My contributions in this area divide into three main trends.
CDP: The
crystal duality principle, or CDP in short, is an important corollary of the QDP, which is obtained as an application of the latter to Hopf algebras defined over a field and for which scalars are extended to polynomials over that field. Nevertheless, such a result can be achieved almost entirely by means of techniques and tools of the "classical", i.e. "non quantum", theory of Hopf algebras over a field: in this way one gives life to a new chapter of the "standard" theory, in which to any Hopf algebra (which can be thought of as a generalized symmetry) one associates Poisson groups and Lie bialgebras (which are geometrical symmetries). This approach of "classical" type is realized in
[
17].
A short version of such a work is
[
15]
(proceedings of a conference). Instead,
[
16]
is the explicit study in detail of an important example, a Hopf algebra built out of the Nottingham group of formal series of degree 1, with the composition product. This is just but one among several examples of Hopf algebras built upon combinatorial data (graphs, trees, Feynman diagrams, etc.) which naturally show up in (co)homology, noncommutative geometry and quantum physics; so it is highly instructive as a "toy model" of more general situations.
Quasitriangular structures (and generalizations): A very special class of Hopf algebras is that for which  roughly speaking  the lack of cocommutativity is somehow "under control". This idea is encoded in the notion of
quasitriangular Hopf algebras and in its various generalizations. I studied this topic in a series of paper 
[
1],
[
8],
[
10],
[
13]
and
[
14]
 in which the Hopf algebras under exam are always quantum groups: for further details, see section
RMAT in "Quantum Groups" above.
Generalizations (quasiHopf algebras, Hopf superalgebras, etc.): Hopf algebras have been generalized in several ways: among these, I consider the cases of
quasiHopf algebras and of
Hopf superalgebras. In the first case, one is weakening the coassociativity axiom; in the second, one is considering Hopf algebras in the category of
super (i.e., Z_{2}graded) vector spaces  or supermodules over a ring  so that tensor products must be handled in a different way.
The study of quasiHopf algebras became very important due to Drinfeld's works in the second half of the '80s of last century. The main ingredient in this study is the notion of "associator": roughly speaking, it measures the lack of coassociativity in the quasiHopf algebra. Moreover, associators also turn useful in other contexts as well: for instance, to solve the general problem of the quantization of Lie bialgebras. As a matter of fact, to date the sole associator which is really known is the socalled
KZ associator, obtained as solution to the KnizhnikZamolodchikov differential equation (with respect to the same name connection on
C^{n}, for which it was known  explicitly  an additive formula only. In
[
20]
we provide instead an explicit formula for the
logarithm of this associator (as a special application of a more general result), in terms of multiple
Zfunctions.
As to Hopf superalgebras, the commutative ones (in a "super sense") have a geometrical meaning: namely, their spectra are the socalled affine algebraic
supergroups, just as classically the affine algebraic groups are the spectra of commutative Hopf algebras. My main contributions to this topic are in
[
29],
[
30],
[
31],
[
32],
[
33],
[
36],
[
37],
[
39],
[
41],
and
[
43],
whose content is explained in detail above  see the section
Supergroups in "Algebraic Groups, Representations and related topics".
Finally, an important extension of the notion of Hopf algebra (actually, of bialgebra indeed) is that of
bialgebroid: this has shown increasing importance in several contexts, e.g. in noncommutative geometry. I start investigating this in
[
34],
which is devoted to study quantizations (in formal sense) of bialgebroids. In particular, here we study the functors providing linear duality among (quantum) bialgebroids  something more or less already known  and we introduce new, suitable functors "à la Drinfeld" which establish a QDP for quantum bialgebroids  the (main) original contribution of this paper (see also the section
QDP in "Quantum Groups" here above). Still in this context one has also
[
38],
that is specifically devoted to the study of duality for bialgebroids with suitable additional structure.
Works in progress

[43]
R. Fioresi, F. Gavarini, "Real forms of complex Lie supergroups"
Preprints

[42]
G. A. García, F. Gavarini, "Twisted deformations vs. cocycle deformations for quantum groups", preprint 1807.01149 [math.QA] (2018), 39 pages  see
http://arxiv.org/abs/1807.01149

[41]
K. Iohara, F. Gavarini, "Singular degenerations of Lie supergroups of type D(2,1;a)", preprint arXiv:1709.04717 [math.RA] (2017), 36 pages  see
http://arxiv.org/abs/1709.04717

[40]
G. A. García, F. Gavarini, "Multiparameter quantum groups at roots of unity", preprint arXiv:1708.05760 [math.QA] (2017), 84 pages  see
http://arxiv.org/abs/1708.05760

[39]
F. Gavarini, "Lie supergroups vs. super HarishChandra pairs: a new equivalence", preprint arXiv:1609.02844 [math.RA] (2016), 42 pages  see
http://arxiv.org/abs/1609.02844
Journal articles, Proceedings, etc.

[38]
S. Chemla, F. Gavarini, N. Kowalzig, "Duality features of left Hopf algebroids", Algebras and Representation Theory 19 (2016), no. 4, 913941

DOI: 10.1007/s1046801696049

[37]
F. Gavarini, "Corrigendum to Algebraic supergroups of Cartan type", Forum Mathematicum 28 (2016), no. 5, 10051009

DOI: 10.1515/forum20150044  complement to [31] here above

[36]
F. Gavarini, "Global splittings and super HarishChandra pairs for affine supergroups", Transactions of the American Mathematical Society 368 (2016), 39734026

DOI: 10.1090/tran/6456

[35]
N. Ciccoli, F. Gavarini, "A global quantum duality principle for
subgroups and homogeneous spaces", Documenta Mathematica 19 (2014), 333380

[34]
S. Chemla, F. Gavarini, "Duality functors for quantum groupoids", Journal of Noncommutative Geometry (2015), no. 2, 287358

DOI: 10.4171/JNCG/194

[33]
F. Gavarini, "Chevalley Supergroups of type D(2,1;a)",
Proceedings of the Edinburgh Mathematical Society (2) 57 (2014), no. 2, 465491

DOI: 10.1017/S0013091513000503

[32]
R. Fioresi, F. Gavarini, "Algebraic supergroups with Lie superalgebras of classical type", Journal of Lie Theory 23 (2013), no. 1, 143158

[31]
F. Gavarini, "Algebraic supergroups of Cartan type", Forum Mathematicum 26 (2014), no. 5, 14731564 
DOI: 10.1515/forum20110144

[30]
R. Fioresi, F. Gavarini, "On the construction of Chevalley supergroups", in: S. Ferrara, R. Fioresi, V. S. Varadarajan (eds.), Supersymmetry in Mathematics and Physics, UCLA Los Angeles, U.S.A. 2010, Lecture Notes in Mathematics 2027, SpringerVerlag, BerlinHeidelberg, 2011, pp. 101123 
DOI: 10.1007/9783642217449_5

[29]
R. Fioresi, F. Gavarini, "Chevalley supergroups", Memoirs of the American Mathematical Society 215, no. 1014 (2012), p. 177 
DOI: 10.1090/S006592662011006337

[28]
R. Fioresi, F. Gavarini, "Quantum Duality Principle for Quantum
Grassmannians", in: M. Marcolli, D. Parashar (eds.), Quantum Groups and Noncommutative Spaces. Perspectives on Quantum Geometry, 8095, Aspects of Mathematics E41, Vieweg+Teubner, Wiesbaden, 2011 
DOI: 10.1007/9783834898319_4

[27]
N. Ciccoli, R. Fioresi, F. Gavarini, "Quantization of Projective Homogeneous Spaces and Duality Principle", Journal of Noncommutative Geometry 2 (2008), no. 4, 449496  DOI: 10.4171/JNCG/26

[26]
F. Gavarini, "On the radical of Brauer algebras", Mathematische Zeitschrift 260 (2008), 673697 
DOI: 10.1007/s002090070296z

[25]
F. Gavarini, "PBW theorems and Frobenius structures for quantum
matrices", Glasgow Mathematical Journal 49
(2007), no. 3, 479488 
DOI: 10.1017/S0017089507003813

[24]
F. Gavarini, Z. Rakic, "F_{q}[M_{n}] ,
F_{q}[GL_{n}] and
F_{q}[SL_{n}] as
quantized hyperalgebras", Journal of Algebra
315 (2007), no. 2, 761800 
DOI: 10.1016/j.jalgebra.2007.03.040

[23]
F. Gavarini, Z. Rakic, "F_{q}[M_{2}],
F_{q}[GL_{2}] and
F_{q}[SL_{2}] as
quantized hyperalgebras", Communications in Algebra
37 (2009), no. 1, 95119 
DOI: 10.1080/00927870802241238

[22]
F. Gavarini, "The global quantum duality principle",
Journal für die reine und angewandte Mathematik 612 (2007),
1733 
DOI: 10.1515/CRELLE.2007.082

[21]
N. Ciccoli, F. Gavarini, "Quantum duality principle for coisotropic
subgroups and Poisson quotients", in: N. Bokan, M. Djoric,
A. T. Fomenko, Z. Rakic, B. Wegner, J. Wess (eds.), Contemporary
Geometry and Related Topics, Proceedings of the Workshop
(Belgrade, June 26July 2, 2005), EMIS ed., 2006, pp. 99118
 see also
http://www.emis.de/proceedings/CGRT2005

[20]
B. Enriquez, F. Gavarini, "A formula for the logarithm of the KZ
associator", Symmetry, Integrability and Geometry: Methods
and Applications 2, Vadim Kuznetsov Memorial
Issue "Integrable Systems and Related Topics" (2006), Paper 080, 3 pages

DOI: 10.3842/SIGMA.2006.080

see also
http://www.emis.de/journals/SIGMA/2006/Paper080

[19]
F. Gavarini, "Presentation by Borel subalgebras and
Chevalley generators for quantum enveloping algebras",
Proceedings of the Edinburgh Mathematical Society (2)
49 (2006), no. 2, 291308 
DOI: 10.1017/S0013091504000689

[18]
N. Ciccoli, F. Gavarini, "A quantum duality principle
for coisotropic subgroups and Poisson quotients",
Advances in Mathematics 199
(2006), no. 1, 104135 
DOI: 10.1016/j.aim.2005.01.009

[17]
F. Gavarini, "The crystal duality principle: from Hopf
algebras to geometrical symmetries", Journal of Algebra
285 (2005), no. 1, 399437 
DOI: 10.1016/j.jalgebra.2004.12.003

[16]
F. Gavarini, "Poisson geometrical symmetries associated
to noncommutative formal diffeomorphisms", Communications in
Mathematical Physics 253 (2005), no. 1, 121155 
DOI: 10.1007/s0022000411757

[15]
F. Gavarini, "The crystal duality principle: from general
symmetries to geometrical symmetries", in: N. Bokan, M. Djoric,
Z. Rakic, A. T. Fomenko, J. Wess (eds.), Contemporary Geometry and
Related Topics, Proceedings of the Workshop (Belgrade, 1521 May
2002), World Scientific, 2004, pp. 223249

[14]
B. Enriquez, F. Gavarini, G. Halbout, "Uniqueness of
braidings of quasitriangular Lie bialgebras and lifts of classical
rmatrices", International Mathematics Research
Notices 46 (2003), 24612486 
DOI: 10.1155/S1073792803208138

[13]
F. Gavarini, G. Halbout, "Braiding structures on formal
Poisson groups and classical solutions of the QYBE", Journal
of Geometry and Physics 46 (2003), no. 34, 255282 
DOI: 10.1016/S03930440(02)00147X

[12]
F. Gavarini, "The quantum duality principle",
Annales de l'Institut Fourier 52
(2002), no. 3, 809834 
DOI: 10.5802/aif.1902

[11]
F. Gavarini, "On the global quantum duality principle",
in: Zoran Kadelburg (ed.), Proceedings of the 10th Congress
of Yugoslav Mathematicians (2124 January 2001; Belgrade, Yugoslavia),
Vedes, Belgrade, 2001, pp. 161168

[10]
F. Gavarini, G. Halbout, "Tressages des groupes de
Poisson formels à dual quasitriangulaire", Journal
of Pure and Applied Algebra 161 (2001), no. 2, 295307 
DOI: 10.1016/S00224049(00)000992 
(hereafter also available in English version)

[9]
F. Gavarini, "A global version of the quantum duality
principle", Czechoslovak Journal of Physics
51 (2001), no. 12, 13301335 
DOI: 10.1023/A:1013322103870

[8]
F. Gavarini, "The Rmatrix action of untwisted
affine quantum groups at roots of 1", Journal of Pure and
Applied Algebra 155 (2001), no. 1, 4152 
DOI: 10.1016/S00224049(99)001176

[7]
F. Gavarini, "Dual affine quantum groups",
Mathematische Zeitschrift 234 (2000), no. 1, 952 
DOI: 10.1007/s002090050502

[6]
F. Gavarini, "A PBW basis for Lusztig's form of untwisted
affine quantum groups", Communications in Algebra
27 (1999), no. 2, 903918 
DOI: 10.1080/00927879908826468

[5]
F. Gavarini, "A Brauer algebra theoretic proof of
Littlewood's restriction rules", Journal of Algebra
212 (1999), no. 1, 240271 
DOI: 10.1006/jabr.1998.7536

[4]
F. Gavarini, "Quantum function algebras as quantum
enveloping algebras", Communications in Algebra
26 (1998), no. 6, 17951818 
DOI: 10.1080/00927879808826240

[3]
F. Gavarini, "Quantization of Poisson groups",
Pacific Journal of Mathematics 186 (1998), no. 2, 217266 
DOI: 10.2140/pjm.1998.186.217

[2]
F. Gavarini, P. Papi, "Representations of the Brauer algebra
and Littlewood's restriction rules", Journal of Algebra
194 (1997), no. 1, 275298 
DOI: 10.1006/jabr.1996.7003

[1]
F. Gavarini, "Geometrical Meaning of Rmatrix action for
Quantum Groups at Roots of 1", Communications in Mathematical Physics 184 (1997), no. 1, 95117 
DOI: 10.1007/s002200050054

[0]
F. Gavarini, "Quantizzazione di gruppi di Poisson",
Ph. D. Thesis, Università degli Studi di Roma "La Sapienza" (Roma, ITALY), 1996
Unpublished essays

[E2]
F. Gavarini, "The global quantum duality principle:
a survey through examples", in: Proceedings des "Rencontres
Mathématiques de Glanon"  6éme édition (15/7/2002;
Glanon, France), 2004, 60 pages  see
http://arxiv.org/abs/1109.3729

[E1]
F. Gavarini, "The global quantum duality principle:
theory, examples, and applications", 120 pages, see
http://arxiv.org/abs/math.QA/0303019
(2003)
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