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GASTÓN ANDRÉS GARCÍA

Professor  FCE- UNLP
ResearcherCMaLP-CONICET


INFO

I did my undergrad studies (Bachelor + Master) at Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires.

My PhD studies were carried out in three different universities under the supervision of Nicolás Andruskiewitsch together with local supervisors: María Ofelia Ronco at UBA (Buenos Aires, Argentina), Hans-Jürgen Schneider at Mathematisches Institut der Ludwig Maximilians Universität München (Germany), and Nicolás Andruskiewitsch at Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba (Argentina).

I also did some short postdoc stays at different places, among them Mount Allison University (Sackville, Canada), St. Mary's University (Halifax, Canada), Università degli Studi di Padova (Italy), Università degli Studi di Roma "Tor Vergata" (Italy). Partially supported by ICTP-- INDAM (Italy), and ERASMUS (EU).

Nowadays, I am Professor at the Math. Deparment at Universidad Nacional de La Plata and a Researcher of CONICET. My research field is non-commutative algebra. More specifically, Hopf algebras and quantum groups. For more information about the Research Group of Non-commutative algebra and geometry see here


Hopf algebras were introduced in 50´s. From the 60´s they have been studied systematically, first in connection with algebraic groups and later as objects of self interest. Quantum groups, introduced in 1986 by Drinfeld, are a particular class of Hopf algebras that can be considered as deformations of universal enveloping algebras of Lie algebras or deformation of rings of regular functions on a Lie group.


My interest in quantum groups lie in the fact that they are a very important source of examples of Hopf algebras and must be taken into account to study the problem of classifying these algebras. In addition, quantum groups appear naturally codifying the symmetry of braided categories. In such a way they appear in various areas related to conformal field theory, for example, invariants of topological varieties of low dimension, geometric representation theory, and more recently, quantum computing.


Coauthors


N. Andruskiewitsch (UNC, Argentina)
D. Bagio (UFSM, Brazil)
M. Beattie (Mount Allinson University, Canada)
G. Carnovale (Universitá degli studi di Padova, Italy)
F. Fantino (UNC, Argentina)
M. Farinati (UBA, Argentina)
A. García Iglesias (UNC, Argentina)
F. Gavarini (Universitá degli studi di Roma "Tor Vergata", Italy)
J. A. Gutiérrez (Colombia)
J. M. Jury Giraldi (UFRGS, Brazil)
O. Marquez (UFSM, Brazil)
M. Mastnak (St. Mary's University, Canada)
Siu-Hung Ng (Louisiana State University, USA)
J. Roat (SUNY Cortland, USA)
C. Vay (UNC, Argentina)
L. Vendramín (Vrije Universiteit Brussel, Belgium)



Articles


More recent


G. A. García and F. Gavarini . Quantum group deformations and quantum R-(co)matrices vs. Quantum Duality Principle.
Preprint: abstract and file.

N. Andruskiewitsch, G. Carnovale and G. A. García. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(n,q) and PSp(2n,q). J. Algebra, Volume 639, 1 February 2024, 354--397. On-line .
Preprint: abstract and file.

G. A. García and F. Gavarini . Formal multiparameter quantum groups, deformations and specializations. Annales de l'Institut Fourier, to appear
Preprint: abstract and file.

G. A. García and C. Vay. Simple modules of small quantum groups at dihedral groups, DM 29 (2024), no. 1, pp. 1–38.
Open access: here.

G. A. García and M. Mastnak. On Hopf algebras whose coradical is a cocentral abelian cleft extension. Comm. Algebra. DOI: 10.1080/00927872.2024.2316311.
On-line .

Featured publications


G. A. García and F. Gavarini . Formal multiparameter quantum groups, deformations and specializations. Annales de l'Institut Fourier, to appear
Preprint: abstract and file.

G. A. García and C. Vay. Simple modules of small quantum groups at dihedral groups, DM 29 (2024), no. 1, pp. 1–38.
Open access: here.

G. A. García and J. A. Gutiérrez. Quantum subgroups of simple twisted quantum groups at roots of one. Transactions of the AMS, Vol. 370, No. 5, (2018), 3609--3637.
On line here .
Preprint: abstract and file.
Notes.

N. Andruskiewitsch, G. Carnovale and G. A. García. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL(n,q), Rev. Mat. Iberoam. 33 (2017), Nro. 3, 995--1024.
On line.
Preprint: abstract and file.

G. A. García and M. Mastnak. Deformation by cocycles of pointed Hopf algebras over non-abelian groups, Math. Res. Lett. 22 (2015), 59--92.
On line.
Preprint: abstract and file.

G. A. García. Quantum subgroups of GLa,b(n). J. Algebra 324 (2010), 1392–1428.
On line.
Preprint: abstract and file.

N. Andruskiewitsch and G. A. García. Finite subgroups of a simple quantum group. Compositio Math. 145, no. 2 (2009), 476--500.
On line.
Preprint: abstract and file.


By subject: Quantum Groups, Hopf algebras, Representations, Lecture Notes, Others.



Quantum Groups


G. A. García and F. Gavarini . Formal multiparameter quantum groups, deformations and specializations. nnales de l'Institut Fourier, to appear.
Preprint: abstract and file.

G. A. García and F. Gavarini . Twisted deformations vs. cocycle deformations for quantum groups. Comm. Contemp. Math., Vol. 23, No. 08, 2050084 (2021).
On line.
Preprint: abstract and file.
Video
Notes.

M. Farinati and G. A. García. Quantum function algebras from finite-dimensional Nichols algebras. J. Noncommutative Geometry, Volume 14, Issue 3, 2020, pp. 879--911.
On line.
Preprint: abstract and file.

G. A. García and F. Gavarini . Multiparameter quantum groups at roots of unity. J. Noncommut. Geom. 16 (2022), 839--926. DOI: 10.4171/JNCG/471.
Preprint: abstract and file.
Notes.

G. A. García and J. A. Gutiérrez. Quantum subgroups of simple twisted quantum groups at roots of one. Transactions of the AMS, Vol. 370, No. 5, (2018), 3609--3637.
On line here .
Preprint: abstract and file.
Notes.

G. A. García. Multiparameter quantum groups, bosonizations and cocycle deformations. Rev. Un. Mat. Argentina, Vol. 57, no. 2 (2016), 1-23.
Online first.
pdf

G. A. García. Quantum subgroups of GLa,b(n). J. Algebra 324 (2010), 1392–1428.
On line.
Preprint: abstract and file.

N. Andruskiewitsch and G. A. García. Finite subgroups of a simple quantum group. Compositio Math. 145, no. 2 (2009), 476--500.
On line.
Preprint: abstract and file.

N. Andruskiewitsch and G. A. García. Extensions of finite quantum groups by finite groups. Tranformation Groups 14, no. 1 (2009), 1--27.
On line.
Preprint: abstract and file.


Hopf algebras


G. A. García and M. Mastnak. On Hopf algebras whose coradical is a cocentral abelian cleft extension. Comm. Algebra. DOI: 10.1080/00927872.2024.2316311.
On-line .

N. Andruskiewitsch, G. Carnovale and G. A. García. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(n,q) and PSp(2n,q).
Preprint: abstract and file.

D. Bagio, G. A. García, J. M. Jury Giraldi and O. Marquez. Finite-dimensional Nichols algebras over dual Radford algebras. Journal of Algebra and its Applications, Proceedings of the conference "QUANTUM 60 Colloquium on Algebras and Representations", Vol. 20, No. 01, 2140001 (2021).
DOI .
Preprint: abstract and file.
Notes .

N. Andruskiewitsch, G. Carnovale and G. A. García. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups. Manuscripta Mathematica, 166, 605--647 (2021).
On line (free) here.
Preprint: abstract and file.

N. Andruskiewitsch, G. Carnovale and G. A. García. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type IV. Unipotent classes in Chevalley and Steinberg groups. Algebr. Represent. Theory, 23 (2020), 621--655.
On line here.
Preprint: abstract and file.

N. Andruskiewitsch, G. Carnovale and G. A. García. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL(n,q), Rev. Mat. Iberoam. 33 (2017), Nro. 3, 995--1024.
On line.
Preprint: abstract and file.

N. Andruskiewitsch, G. Carnovale and G. A. García. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II. Unipotent classes in symplectic groups. Comm. Contemp. Math., Vol. 18, No. 4 (2016) 1550053 (35 pages)
On-line.
Preprint: abstract and file.
Ktulu (He's so heavy) by Beatallica.

N. Andruskiewitsch, G. Carnovale and G. A. García. Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in PSL(n,q), J. Algebra 442 (2015), 36--65.
On-line
.
Preprint: abstract and file.

The call of cthulhu by Metallica.

F. Fantino, G. A. García and M. Mastnak. On finite-dimensional copointed Hopf algebras over dihedral groups. J. Pure Appl. Algebra 223, Issue 8 (2019), 3611--3634.
On line.
Preprint: abstract and file.

G. A. García and J. M. Jury Giraldi. On Hopf algebras over quantum subgroups. J. Pure Appl. Algebra, Volume 223 (2019), Issue 2, 738--768.
On line.
Preprint: abstract and file.

M. Beattie, G. A. García, Siu-Hung Ng and Jolie Roat. Nonsemisimple Hopf algebras of dimension 8p with the Chevalley property, Contemp. Math., Volume 728 (2019), 49--66.
On line.
Preprint: abstract and file.

G. A. García and M. Mastnak. Deformation by cocycles of pointed Hopf algebras over non-abelian groups, Math. Res. Lett. 22 (2015), 59--92.
On line.
Preprint: abstract and file.

M. Beattie and G. A. García. Classifying Hopf algebras of a given dimension. Contemp. Math. 585 (2013), 125--152.
On line.
Preprint: abstract and file.

M. Beattie and G. A. García. Techniques for classifying Hopf algebras and applications to dimension p^3. Comm. Algebra, 41 (2013), 3108–3129.
On-line.
Preprint: abstract and file.

N. Andruskiewitsch, F. Fantino, G. A. García and L. Vendramín. On Nichols algebras associated to simple racks. Contemp. Math. 537 (2011), 31--56.
Preprint: abstract and file.

F. Fantino and G. A. García. On pointed Hopf algebras over dihedral groups. Pacific J. Math, Vol. 252 (2011), no. 1, 69--91.
On line.
Preprint: abstract and file.

G. A. García and A. García Iglesias. Finite dimensional pointed Hopf algebras over S_4. Israel J. Math. 183 (2011), 417--444.
On line.
Preprint: abstract and file .

N. Andruskiewitsch, F. Fantino, G. A. García and L. Vendramín. On twisted homogeneous racks of type D. Rev. Un. Mat. Argentina 51, no. 2 (2010), 1--16.
On line.
Preprint: abstract and file.

G. A. García and C. Vay. Hopf algebras of dimension 16. Algebr. Represent. Theory 13, no. 4 (2010), 383--405.
On line.
Preprint: abstract and file.

G. A. García. On Hopf algebras of dimension p^3. Tsukuba J. Math. 29 (2005), no. 1, 259--284.
Preprint: abstract and file.


Representations of Hopf algebras


G. A. García and C. Vay. Simple modules of small quantum groups at dihedral groups, DM 29 (2024), no. 1, pp. 1–38.
Open access: here.


Lecture Notes


Short course at VII Jornadas de Álgebra (Spanish): Grupos cuánticos matriciales

Short course at REM/UMA 2012 (Spanish): Números, entre la creación y el descrubrimiento.

Short course at XVIII Coloquio Latinoamericano de Álgebra (English): Quantum Groups and Hopf algebras.

Short course at elENA IV (Spanish): Números p-ádicos.

Short course at virtUMA 2021 (Spanish): Una breve introducción a los grupos cuánticos .


Others


Tesis de doctorado (Spanish): Álgebras de Hopf y grupos cuánticos.

Tesis de licenciatura (Spanish): Representaciones del grupo simétrico.

SOCIAL

ADDRESS
Calle 50 y 115
Departamento de Matemática
Facultad de Ciencias Exactas
Universidad Nacional de La Plata
C.C. 172
1900 La Plata, Pcia. de Buenos Aires
República Argentina

CONTACT
Email: ggarcia@mate.unlp.edu.ar
orcid
Scopus Author ID: 25642709600
ResearchGate
Tel/FAX: +54 221 424-5875
422- 9476/9850