arXiv subjects

Fabio Gavarini

Publications and source records attributed to Fabio Gavarini.

At least 19 recordsLinked to original sources

arXiv2025

Multiparameter quantum general linear supergroup

We introduce uniparametric and multiparametric quantisations of the general linear supergroup, in the form of "quantised function algebras", both in a formal setting - yielding "quantum formal series Hopf superalgebras", a` la Drinfeld - and in a polynomial one - closer to Manin's point of view. In the uniparametric setting, we start from quantised universal enveloping superalgebras over gl(n) - endowed with a super-structure - as in [Ya1] and [Zha]: through a direct approach, we construct their linear dual, thus finding the quantum formal series Hopf superalgebras mentioned above, which are described in detail via an explicit presentation. Starting from the latter, then, we perform a deformation by a well-chosen 2-cocycle, thus getting a multiparametric quantisation, described again by an explicit presentation: this is, in turn, the dual to the multiparametric quantised universal enveloping algebra over gl(n) from [GGP]. We also provide some "polynomial versions" of these quantisations, both for the uniparametric and the multiparametric case. In particular, we compare the latter to Manin's quantum function algebras from [Ma]. Finally, both for the uniparametric and the multiparametric setting, we provide suitable PBW-like theorems, in "formal" and in "polynomial" versions alike.

math.QA↗

arXiv2025

Duality for action bialgebroids

We study the effect of linear duality on action bialgebroids (also known as smash product or scalar extension bialgebroids) and, for those bearing a quantisation nature, the effect of Drinfeld functors underlying the quantum duality principle. By means of various categorical equivalences, it is shown that any braided commutative Yetter-Drinfeld algebra over any bialgebroid is also a braided commutative Yetter-Drinfeld algebra over the respective dual bialgebroid. This implies that the action bialgebroid of the dual exists, which is then proven to be isomorphic, as a bialgebroid, to the dual of the initial action bialgebroid: in short, (linear) duality commutes with the action bialgebroid construction. Similarly, for quantum groupoids to which the Drinfeld duality functors apply and the quantum duality principle holds, these Drinfeld duality functors are shown to commute with the action bialgebroid construction as well.

math.RA↗

arXiv2024

Multiparameter Quantum Supergroups, Deformations and Specializations

In this paper we introduce a multiparameter version of the quantum universal enveloping superalgebras introduced by Yamane in [H. Yamane, "Quantized enveloping algebras associated to simple Lie superalgebras and their universal $R$-matrices", Publ. Res. Inst. Math. Sci. 30 (1994), no. 1, 15-87]. For these objects we consider: - (1) their deformations by twist and by 2-cocycle (both of "toral type"); in particular, we prove that this family is stable under both types of deformations; - (2) their semiclassical limits, which are multiparameter Lie superbialgebras; - (3) the deformations by twist and by 2-cocycle (of "toral type") of these multiparameter Lie superbialgebras: in particular, we prove that this family is stable under these deformations, and that "quantization commutes with deformation".

math.QA↗

arXiv2024

Quantum group deformations and quantum $ R $-(co)matrices vs. Quantum Duality Principle

In this paper we describe the effect on quantum groups -- namely, both QUEA's and QFSHA's -- of deformations by twist and by 2-cocycles, showing how such deformations affect the semiclassical limit. As a second, more important task, we discuss how these deformation procedures can be "stretched" to a new extent, via a formal variation of the original recipes, using "polar twists" and "polar 2-cocycles". These recipes seemingly should make no sense at all, yet we prove that they actually work, thus providing well-defined, more general deformation procedures. Later on, we explain the underlying reason that motivates such a result in light of the "Quantum Duality Principle", through which every "polar twist/2-cocycle" for a given quantum group can be seen as a standard twist/2-cocycle for another quantum group, associated to the original one via the appropriate Drinfeld functor. As a third task, we consider standard constructions involving $R$-(co)matrices in the general theory of Hopf algebras. First we adapt them to quantum groups, then we show that they extend to the case of "polar $R$-(co)matrices", and finally we discuss how these constructions interact with the Quantum Duality Principle. As a byproduct, this yields new special symmetries (isomorphisms) for the underlying pair of dual Poisson (formal) groups that one gets by specialization.

math.QA↗

arXiv2023

Super K\"ahler structures on the complex Abelian Lie supergroups

Let $G$ be a real Abelian Lie supergroup, let $M$ be its complexification. We classify the $G$-invariant super K\"ahler forms on $M$. For the super K\"ahler forms with Hamiltonian actions, we extend the scheme of geometric quantization to the super setting and construct unitary $G$-representations. We show that the irreducible representations that occur in these unitary representations are governed by the image of the moment maps of super K\"ahler forms. As an application, we construct a Gelfand model of $G$, namely a unitary $G$-representation in which every unitary irreducible representation occurs exactly once.

math.DG↗

arXiv2023

Real forms of complex Lie superalgebras and supergroups

We investigate the notion of real form of complex Lie superalgebras and supergroups, both in the standard and graded version. Our functorial approach allows most naturally to go from the superalgebra to the supergroup and retrieve the real forms as fixed points, as in the ordinary setting. We also introduce a more general notion of compact real form for Lie superalgebras and supergroups, and we prove some existence results for Lie superalgebras that are simple contragredient and their associated connected simply connected supergroups.

math.RA↗

arXiv2022

Quantum Duality Principle for quantum continuous Kac-Moody algebras

For the quantized universal enveloping algebra U_h(g_X) associated with a continuous Kac-Moody algebra g_X as in [A. Appel, F. Sala, "Quantization of continuum Kac-Moody algebras", Pure Appl. Math. Q. 16 (2020), no. 3, 439-493], we prove that a suitable formulation of the Quantum Duality Principle holds true, both in a "formal" version - i.e., applying to the original definition of U_h(g_X) as a formal QUEA over the algebra of formal series in h - and in a "polynomial" one - i.e., for a suitable polynomial form of U_h(g_X) over the algebra of Laurent polynomials in q. In both cases, the QDP states that a suitable subalgebra of the given quantization of the Lie bialgebra g_X is in fact a suitable quantization (in formal or in polynomial sense) of a connected Poisson group G^*_X dual to g_X .

math.QA↗

arXiv2022

Formal multiparameter quantum groups, deformations and specializations

We introduce the notion of formal multiparameter quantum universal enveloping algebras - in short FoMpQUEA - as a straightforward generalization of Drinfeld's quantum group. Then we show that the class of FoMpQUEA's is closed under deformations by ("toral") twists and deformations by ("toral") 2-cocycles: as a consequence, all "multiparameter formal QUEA's" considered so far are recovered, as falling within this class. In particular, we prove that any FoMpQUEA is isomorphic to a suitable deformation, by twist or by 2-cocycle, of Drinfeld's standard QUEA. We introduce also multiparameter Lie bialgebras (in short, MpLbA's), and we consider their deformations, by twist and by 2-cocycles. The semiclassical limit of every FoMpQUEA is a suitable MpLbA, and conversely each MpLbA can be quantized to a suitable FoMpQUEA. In the end, we prove that, roughly speaking, the two processes of "specialization" (of FoMpQUEA to a MpLbA) and of "deformation (by toral twist or toral 2-cocycle)" - at the level of FoMpQUEA's or of MpLbA's - do commute with each other.

math.QA↗

arXiv2018

Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$

The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a) $ - are usually considered for "non-singular" values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular values too, giving rise to "singular specializations" that are no longer simple: this extends the family of simple objects of type $D(2,1;a)$ in five different ways. The resulting five families coincide for general values of $a$, but are different at "singular" ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or "degenerations") at singular values of $a$. Although one may work with a single complex parameter $a$, in order to stress the overall $\mathfrak{S}_3$-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter $\boldsymbolσ = (σ_1,σ_2,σ_3)$ ranging in the complex affine plane $σ_1 + σ_2 + σ_3 = 0$.

math.RA↗

arXiv2018

Lie supergroups vs. super Harish-Chandra pairs: a new equivalence

It is known that there exists a natural functor $Φ$ from Lie supergroups to super Harish-Chandra pairs. A functor going backwards, that associates a Lie supergroup with each super Harish-Chandra pair, yielding an equivalence of categories, was found by Koszul [18]; this result was later extended by other authors, to different levels of generality, but always elaborating on Koszul's original idea. In this paper, I provide two new backwards equivalences, i.e. two different functors $Ψ^\circ$ and $Ψ^e$ that construct a Lie supergroup (thought of as a special group-valued functor) out of a given super Harish-Chandra pair, so that any Lie supergroup is recovered from its naturally associated super Harish-Chandra pair; more precisely, both $Ψ^\circ$ and $Ψ^e$ are quasi-inverse to the functor $Φ$.

math.RA↗

arXiv2018

Twisted deformations vs. cocycle deformations for quantum groups

In this paper we study two deformation procedures for quantum groups: deformations by twists, that we call "comultiplication twisting", as they modify the coalgebra structure, while keeping the algebra one -- and deformations by 2-cocycle, that we call "multiplication twisting", as they deform the algebra structure, but save the coalgebra one. We deal with quantum universal enveloping algebras, in short QUEA's, for which we accordingly consider those arising from twisted deformations (in short TwQUEA's) and those arising from 2-cocycle deformations, usually called multiparameter QUEA's (in short MpQUEA's). Up to technicalities, we show that the two deformation methods are equivalent, in that they eventually provide isomorphic outputs, which are deformations (of either kinds) of the "canonical", well-known one-parameter QUEA by Jimbo and Lusztig. It follows that the two notions of TwQUEA's and of MpQUEA's -- which, in Hopf algebra theoretical terms are naturally dual to each other -- actually coincide; thus, that there exists in fact only one type of "pluriparametric deformation" for QUEA's. In particular, the link between the realization of any such QUEA as a MpQUEA and that as a TwQUEA is just a (very simple, and rather explicit) change of presentation.

math.QA↗

arXiv2017

Multiparameter quantum groups at roots of unity

We address the study of multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) $ depending on a matrix of parameters $ \boldsymbol{\rm q} = {\big( q_{ij} \big)}_{i, j \in I} \, $. This is performed via the construction of quantum root vectors and suitable "integral forms" of $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) \, $, a \textsl{restricted one} - generated by quantum divided powers and quantum binomial coefficients - and an \textsl{unrestricted\/} one - where quantum root vectors are suitably renormalized. The specializations at roots of unity of either forms are the "MpQG's at roots of unity" we look for. In particular, we study special subalgebras and quotients of our MpQG's at roots of unity - namely, the multiparameter version of small quantum groups - and suitable associated quantum Frobenius morphisms, that link the MpQG's at roots of 1 with MpQG's at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion - often at the core of our strategy - is that every MpQG is actually a 2-cocycle deformation of the algebra structure of (a lift of) the "canonical" one-parameter quantum group by Jimbo-Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $ \boldsymbol{\rm q} $ our quantum groups yield (through the choice of integral forms and their specialization) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group.

math.QA↗

arXiv2017

The quantum duality principle

The "quantum duality principle" states that the quantization of a Lie bialgebra - via a quantum universal enveloping algebra (QUEA) - provides also a quantization of the dual Lie bialgebra (through its associated formal Poisson group) - via a quantum formal series Hopf algebra (QFSHA) - and, conversely, a QFSHA associated to a Lie bialgebra (via its associated formal Poisson group) yields a QUEA for the dual Lie bialgebra as well; more precisely, there exist functors QUEA --> QFSHA and QFSHA --> QUEA, inverse of each other, such that in either case the Lie bialgebra associated to the target object is the dual of that of the source object. Such a result was claimed true by Drinfeld, but seems to be unproved in literature: we give here a complete detailed proof of it.

math.QA↗

arXiv2017

Tressages des groupe de Poisson formels à dual quasitriangulaire

Let $ \mathfrak{g} $ be a quasitriangular Lie bialgebra over a field $ K $ of characteristic zero, and let $ \mathfrak{g}^* $ be its dual Lie bialgebra. We prove that the formal Poisson group $ K\big[\big[\mathfrak{g}^*\big]\big] $ is a braided Hopf algebra, thus generalizing a result due to Reshetikhin (in the case $ \, \mathfrak{g} = \mathfrak{sl}(2,K) \, $). The proof is via quantum groups, using the existence of a quasitriangular quantization of $ \mathfrak{g}^* $, as well as the fact that this one provides also a quantization of $ K\big[\big[\mathfrak{g}^*\big]\big] \, $.

math.QA↗

arXiv2017

The $ R $--matrix action of untwisted affine quantum groups at roots of 1

Let $\hat{\frak g}$ be an untwisted affine Kac-Moody algebra. The quantum group $U_h(\hat{\frak g})$ (over $\mathbb{C}[[h]]$) is known to be a quasitriangular Hopf algebra: in particular, it has a universal $ R $--matrix, which yields an $ R $--matrix for each pair of representations of $U_h(\hat{\frak g})$. On the other hand, the quantum group $U_q(\hat{\frak g})$ (over $\mathbb{C}(q) $) also has an $ R $--matrix for each pair of representations, but it has not a universal $ R $--matrix so that one cannot say that it is quasitriangular. Following Reshetikin, one introduces the (weaker) notion of braided Hopf algebra: then $ U_q(\hat{\frak g})$ is a braided Hopf algebra. In this work we prove that also the unrestricted specializations of $U_q(\hat{\frak g})$ at roots of 1 are braided: in particular, specializing $q$ at 1 we have that the function algebra $F \big[ \hat{H} \big]$ of the Poisson proalgebraic group $\hat{H}$ dual of $\hat{G}$ (a Kac-Moody group with Lie algebra $\hat{\frak g} \,$) is braided. This is useful because, despite these specialized quantum groups are not quasitriangular, the braiding is enough for applications, mainly for producing knot invariants. As an example, the action of the $ R $--matrix on (tensor products of) Verma modules can be specialized at odd roots of 1.

math.QA↗

arXiv2017

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.

q-alg↗

arXiv2017

A PBW basis for Lusztig's form of untwisted affine quantum groups

Let $ \mathfrak{g} $ be an untwisted affine Kac-Moody algebra over the field $ K \, $, and let $ U_q(\mathfrak{g}) $ be the associated quantum enveloping algebra; let $ \mathfrak{U}_q(g) $ be the Lusztig's integer form of $ U_q(\mathfrak{g}) \, $, generated by $ q $-divided powers of Chevalley generators over a suitable subring $ R $ of $ K(q) \, $. We prove a Poincaré-Birkhoff-Witt like theorem for $ \mathfrak{U}_q(\mathfrak{g}) \, $, yielding a basis over $ R $ made of ordered products of $ q $-divided powers of suitable quantum root vectors.

q-alg↗

arXiv2017

Quantum function algebras as quantum enveloping algebras

Inspired by a result in [Ga], we locate two $ k[q,q^{-1}] $-integer forms of $ F_q[SL(n+1)] $, along with a presentation by generators and relations, and prove that for $ q=1 $ they specialize to $ U({\mathfrak{h}}) $, where $ {\mathfrak{h}} $ is the Lie bialgebra of the Poisson Lie group $ H $ dual of $ SL(n+1) $; moreover, we explain the relation with [loc. cit.]. In sight of this, we prove two PBW-like theorems for $ F_q[SL(n+1)] $, both related to the classical PBW theorem for $ U({\mathfrak{h}}) $.

q-alg↗