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Fabio Gavarini

Publications and source records attributed to Fabio Gavarini.

47 records · Page 3Linked to original sources
arXiv2007

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.

math.RT↗
arXiv2007

F_q[M_n], F_q[GL_n] and F_q[SL_n] as quantized hyperalgebras

The quantized universal enveloping algebra U_q(gl(n)) has two integral forms - over Z[q,q^{-1}] - the restricted (by Lusztig) and the unrestricted (by De Concini and Procesi) one. Dually, the quantum function algebra F_q[GL(n)] has two integral forms, namely those of all elements - of F_q[GL(n)] - which take values in Z[q,q^{-1}] when paired respectively with the restricted or the unrestricted form of U_q(gl(n)). The first one is the well-known form generated over Z[q,q^{-1}] by the entries of a q-matrix and the inverse of its quantum determinant. In this paper instead we study the second integral form, say F'_q[GL(n)], i.e. that of all elements which are Z[q,q^{-1}]-valued over the unrestricted form of U_q(gl(n)). In particular we yield a presentation of it by generators and relations, and a PBW-like theorem: in short, it is an algebra of "quantum divided powers" and "quantum binomial coefficients". Moreover, we give a direct proof that F'_q[GL(n)] is a Hopf subalgebra of F_q[GL(n)], and that its specialization at q=1 is the Z-hyperalgebra over gl(n)^*, the Lie bialgebra dual to gl(n). In addition, we describe explicitly the specializations of F'_q[GL(n)] at roots of 1, and the associated quantum Frobenius (epi)morphism. The same analysis is done for F'_q[SL(n)] and (as a key step) F'_q[Mat(n)]: in fact, for the latter the strongest results are obtained. This work extends to general n>2 the results for n=2, already treated in math.QA/0411440.

math.QA↗
arXiv2007

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.

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arXiv2007

PBW theorems and Frobenius structures for quantum matrices

Let G be either of Mat(n), GL(n) or SL(n), let O_q(G) be the quantum function algebra - over Z[q,q^{-1}] - associated to G, and let O_e(G) be the specialisation of O_q(G) at a root of unity, of odd order l. Then O_e(G) is a module over the corresponding classical function algebra O(G) via the quantum Frobenius morphism, which embeds O(G) as a central subbialgebra of O_e(G). In this note we prove a PBW-like theorem for O_q(G) - more or less known in literature, but not in this form (to the best of the author's knowledge) - and we show that it yields explicit bases of O_e(G) over O(G) when G is Mat(n) or GL(n): in particular, O_e(G) is free of rank l^{dim(G)}. Also, we apply the latter result to prove that O_e(G) is a free Frobenius extensions over O(G), and to compute explicitly the corresponding Nakayama automorphism, again for G being Mat(n) or GL(n) . This extends previous results by Brown, Gordon and Stroppel (see [BG], [BGS2]).

math.QA↗
arXiv2006

A Formula for the Logarithm of the KZ Associator

We prove that the logarithm of a group-like element in a free algebra coincides with its image by a certain linear map. We use this result and the formula of Le and Murakami for the Knizhnik-Zamolodchikov (KZ) associator $Φ$ to derive a formula for $\log(Φ)$ in terms of MZV's (multiple zeta values).

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arXiv2006

A quantum duality principle for coisotropic subgroups and Poisson quotients

We develop a quantum duality principle for coisotropic subgroups of a (formal) Poisson group and its dual. Namely, starting from a quantum coisotropic subgroup (for a quantization of a given Poisson group) we provide functorial recipes to produce quantizations of the dual coisotropic subgroup (in the dual formal Poisson group). By the natural link between subgroups and homogeneous spaces, we argue a quantum duality principle for Poisson homogeneous spaces which are Poisson quotients, i.e. have at least one zero-dimensional symplectic leaf. As an application, we provide an explicit quantization of the homogeneous SL(n)^*-space of Stokes matrices, with the Poisson structure given by Dubrovin and Ugaglia.

math.QA↗
arXiv2005

The Crystal Duality Principle: from Hopf Algebras to Geometrical Symmetries

We give functorial recipes to get, out of any Hopf algebra over a field, two pairs of Hopf algebras which have some geometrical content. When the ground field has characteristic zero, the first pair is made by a function algebra over a connected Poisson group and a universal enveloping algebra over a Lie bialgebra. In addition, the Poisson group as a variety is an affine space, and the Lie bialgebra as a Lie algebra is graded. Forgetting these last details, the second pair is of the same type. When the Hopf algebra H we start from is already of geometric type the result involves Poisson duality: the first Lie bialgebra associated to H = F[G] is g^* (with g := Lie(G)), and the first Poisson group H = U(g) is of type G^*, i.e. it has g as cotangent Lie bialgebra. If the ground field has positive characteristic, then the same recipes give similar results, but for the fact that the Poisson groups obtained have dimension 0 and height 1, and restricted universal enveloping algebras are obtained. We show how these "geometrical" Hopf algebras are linked to the original one via 1-parameter deformations, and explain how these results follow from quantum group theory. The cases of hyperalgebras and group algebras are examined in some detail (thus recovering some well-known, classical construction), along with some relevant examples.

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arXiv2004

Poisson geometrical symmetries associated to non-commutative formal diffeomorphisms

Let G be the group of all formal power series starting with x with coefficients in a field k of zero characteristic (with the composition product), and let F[G] be its function algebra. C. Brouder and A. Frabetti introduced a non-commutative, non-cocommutative graded Hopf algebra H, via a direct process of ``disabelianisation'' of F[G], i.e. taking the like presentation of the latter as an algebra but dropping the commutativity constraint. In this paper we apply a general method to provide four one-parameters deformations of H, which are quantum groups whose semiclassical limits are Poisson geometrical symmetries such as Poisson groups or Lie bialgebras, namely two quantum function algebras and two quantum universal enveloping algebras. In particular the two Poisson groups are extensions of G, isomorphic as proalgebraic Poisson varieties but not as proalgebraic groups. This analysis easily extends to a hudge family of Hopf algebras of similar nature, thus yielding a method to associate to such "generalized symmetries" some classical geometrical symmetries (such as Poisson groups and Lie bialgebras) in a natural way: the present case then stands as a simplest, toy model for the general situation.

math.QA↗
arXiv2002

Braiding structures on formal Poisson groups and classical solutions of the QYBE

If g is a quasitriangular Lie bialgebra, one can asks what is the geometrical meaning of its r-matrix. A first answer was given in a paper by Weinstein and Xu, using purely geometrical means: roughly, one has that the formal Poisson group F[[g^*]] is endowed with a "braiding", i.e. a distinguished operator on its tensor square which satisfy quasitriangularity conditions (in particular, it is a solution of the QYBE). Independently, the authors also found, by means of quantum groups, that F[[g^*]] has a braiding. In this paper we compare these two approaches and their outcomes. First, we show that the two braidings obtained in the two processes do share several similar properties (in particular, the construction is functorial). Second, in the simplest case (G = SL_2) we prove that the two braidings do coincide. The question then rises of whether they are always the same: this problem is addressed and solved in math.QA/0207235, in a much broader context, in which unicity of braidings is proved.

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arXiv1999

Braidings of Poisson groups with quasitriangular dual (Tressages des groupes de Poisson à dual quasitriangulaire)

Let g be a quasitriangular Lie bialgebra over a field k of characteristic zero, and let g^* be its dual Lie bialgebra. We prove that the formal Poisson group F[[g^*]] is a braided Hopf algebra. More generally, we prove that if (U_h,R) is any quasitriangular QUEA, then (U_h', Ad(R)|_{U_h' \otimes U_h'}) --- where U_h' is defined by Drinfeld --- is a braided QFSHA. The first result is then just a consequence of the existence of a quasitriangular quantization (U_h,R) of U(g) and of the fact that U_h' is a quantization of F[[g^*]]. ----- Soit g une bigèbre de Lie quasitriangulaire sur un corps k de characteristique zero, et soit g^* sa bigèbre de Lie duale. Nous prouvons que le groupe de Poisson formel F[[g^*]] est une algebre de Hopf tressée. Plus en général, nous prouvons que, si (U_h,R) est une QUEA quasitriangulaire, alors (U_h', Ad(R)|_{U_h' \otimes U_h'}) --- où U_h' est definie par Drinfeld --- est une QFSHA tressée. Le premier résultat est alors une consequence de l'existence d'une quantification quasitriangulaire (U_h,R) de U(g) et du fait que U_h' est une quantification de F[[g^*]].

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