David Hernandez

Publications grouped by projects

My research interests concern the representation theory of quantum groups and infinite-dimensional Lie algebras and their applications, from various perspectives involving algebraic geometry, combinatorics and mathematical physics. Here are the main projects in my research work, with the corresponding publications grouped by projects.

Monoidal categorification of cluster algebras

Cluster algebras and quantum affine algebras (with B. Leclerc) : PDF , Duke Mathematical Journal 154 (2010), no. 2, 265--341

Simple tensor products : PDF , Inventiones Mathematicae 181 (2010), no. 3, 649--675

Monoidal categorifications of cluster algebras of type A and D (with B. Leclerc) : PDF , in Symmetries, Integrable systems and Representations, Springer Proceedings in Mathematics and Statistics 40 (2013), 175--193

Cluster algebras and category O for representations of Borel subalgebras of quantum affine algebras (with B. Leclerc) : PDF , Algebra and Number Theory 10 (2016), no. 9, 2015--2052.

Quantum affine algebras and cluster algebras (with B. Leclerc) : PDF, in Progress in Mathematics 337 (2021), vol. in Honor of V. Chari.

Toroidal Grothendieck rings and cluster algebras (with L. Fedele) : PDF , Mathematische Zeitschrift 300 (2022), no. 1, 377--420.

Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case (with C. Geiss and B. Leclerc) : PDF, Preprint arXiv:2401.04616.

Quantum integrable systems and Baxter Q-operators

Asymptotic representations and Drinfeld rational fractions (with M. Jimbo) : PDF , Compositio Mathematica 148 (2012), no. 5, 1593--1623

Spectre des systèmes intégrables quantiques représentations linéaires : PDF , Gazette des Mathématiciens 141 (2014), 34--47.

Baxter's Relations and Spectra of Quantum Integrable Models (with E. Frenkel) : PDF , Duke Mathematical Journal 164 (2015), no. 12, 2407-2460.

Folded quantum integrable models and deformed W-algebras (with E. Frenkel and N. Reshetikhin) : PDF , Letters in Mathematical Physics 112 (2022), no. 4, 80.

Weyl group symmetry of q-characters (with E. Frenkel) : PDF , Preprint arXiv:2211.09779.

Extended Baxter relations and QQ-systems for quantum affine algebras (with E. Frenkel) : PDF, Preprint arXiv:2312.13256.

Kazhdan-Lusztig polynomials and quantum Grothendieck rings

t-analogues des opérateurs d'écrantage associés aux q-caractères : PDF , International Mathematics Research Notices 2003, no. 8, 451--475

Algebraic Approach to q,t-Characters : PDF , Advances in Mathematics 187 (2004), no. 1, 1--52

Monomials of q and q,t-characters for non simply-laced quantum affinizations : PDF , Mathematische Zeitschrift 250 (2005), no. 2, 443--473

Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm (with H. Oya) : PDF , Advances in Mathematics 347 (2019), 192--272 .

Isomorphisms among quantum Grothendieck rings and propagation of positivity (with R. Fujita, S-J. Oh and H. Oya) : PDF , Crelle - Journal für die reine und angewandte Mathematik 785 (2022), 117--185.

Isomorphisms among quantum Grothendieck rings and cluster algebras (with R. Fujita, S-J. Oh and H. Oya) : PDF, Preprint arXiv:2304.02562.

[57] Monoidal Jantzen filtrations (with R. Fujita) : PDF, Preprint arXiv:2402.13544.

Shifted quantum groups associated to Coulomb branches

Representations of shifted quantum affine algebras : PDF , International Mathematical Research Notices (2023), no. 13, 11035–11126..

Shifted Yangians and polynomial R-matrices (with H. Zhang) : PDF , to appear in Publications of the Research Institute for Mathematical Sciences .

Quiver varieties

Smallness problem for quantum affine algebras and quiver varieties : PDF , Annales Scientifiques de l'Ecole Normale Supérieure (4) 41 (2008) , no. 2, 271-306

Quantum Grothendieck rings and derived Hall algebras (with B. Leclerc) : PDF ,Crelle - Journal für die reine und angewandte Mathematik 701 (2015), 77--126

A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules (with B. Leclerc) : PDF , Journal of the European Mathematical Society 18 (2016), no. 5, 1113--1159

Oper and Langlands duality for quantum groups

Langlands duality for representations of quantum groups (with E. Frenkel) : PDF , Mathematische Annalen 349 (2011), no. 3, 705--746

Langlands duality for finite-dimensional representations of quantum affine algebras (with E. Frenkel) : PDF , Letters in Mathematical Physics 96 (2011), no. 1-3, 217--261

Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers (with E. Frenkel) : PDF , Communications in Mathematical Physics 362 (2018), no. 2, 361--414.

Stable maps and R-matrices

Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...) : PDF (translation) , Sém. Bourbaki 1129, Astérisque 407 (2019), 297--331.

Cyclicity and R-matrices : PDF , Selecta Mathematica (N.S.) 25 (2019), no.2, 25:19..

Stable maps, Q-operators and category O : PDF, Representation Theory 26 (2022), 179--210.

Kirillov-Reshetikhin modules, crystals and T-systems

The Kirillov-Reshetikhin conjecture and solutions of T-systems : PDF , Crelle - Journal für die reine und angewandte Mathematik 596 (2006), 63--87

Level 0 monomial crystals (with H. Nakajima) : PDF , Lusztig's issue of Nagoya Mathematical Journal 184 (2006), 85--153

On minimal affinizations of representations of quantum groups : PDF , Communications in Mathematical Physics 277 (2007), no. 1, 221--259

Kirillov-Reshetikhin conjecture : the general case : PDF, International Mathematics Research Notices 2010, no. 1, 149--193

Beyond Kirillov-Reshetikhin modules (with V. Chari) : PDF , Contemporary Mathematics 506 (2010), 49--81

Schur positivity and Kirillov-Reshetikhin modules (with G. Fourier) : PDF , Special issue of SIGMA "New Directions in Lie Theory" 10 (2014), 058

Quantum periodicity and Kirillov-Reshetikhin modules : PDF , in Progress in Mathematics 340 (2021), vol. in Honor of N. Reshetikhin.

Geometry of loop groups and quantum groups at roots of unity

The t-Analogs of q-Characters at Roots of Unity for Quantum Affine Algebras and Beyond : PDF , Journal of Algebra 279 (2004), no. 2 , 514--557

Geometry of the analytic loop group (with C. De Concini and N. Reshetikhin) : PDF , Advances in Mathematics 238 (2013), 290--321

Toroidal algebras, Drinfeld coproduct and fusion product

Representations of Quantum Affinizations and Fusion Product : PDF , Transformation Groups 10 (2005), no. 2, 163--200

Drinfeld coproduct, quantum fusion tensor category and applications : PDF , Proceedings of the London Mathematical Society (3) 95 (2007), no. 3, 567--608

Quantum toroidal algebras and their representations : PDF , Selecta Mathematica (N.S.) 14 (2009), no. 3-4, 701--725

Double affine Lie algebras and finite groups (with N. Guay and S. Loktev) : PDF , Pacific Journal of Mathematics 243 (2009), no. 1, 1--41

The algebra $U_q(\hat{sl}_\infty)$ and applications : PDF , De Concini's issue of Journal of Algebra 329 (2011), no. 1, 147--162