Algebraic approach to q,t-characters

By D. Hernandez. 2002

79 citations recorded by the source corpus.

Abstract Frenkel and Reshetikhin (in: Recent Developments in Quantum Affine Algebras and Related Topics, Contemporary Mathematics, Vol. 248, 1999, pp. 163–205) introduced q-characters to study finite dimensional representations of the quantum affine algebra U q ( g ) . In the simply laced case Nakajima (in: Physics and Combinatorics, Proceedings of the Nagoya 2000 International Workshop, World Scientific, Singapore, 2001, pp. 181–212; Preprint arXiv:math.QA/0105173) defined deformations of q-characters called q,t-characters. The definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case. In this article we propose an algebraic general (non-necessarily simply laced) new approach to q,t-characters motivated by the deformed screening operators (Internat. Math. Res. Not. 2003 (8) (2003) 451). The t-deformations are naturally deduced from the structure of U q ( g ) : the parameter t is analog to the central charge c∈ U q ( g ) . The q,t-characters lead to the construction of a quantization of the Grothendieck ring and to general analogues of Kazhdan–Lusztig polynomials in the same spirit as Nakajima did for the simply laced case.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9065fe9c35719a766ba0caa533cdab24e919917e

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Citations in the corpus

Yangians, quantum loop algebras and abelian difference equations (2013, methodology): “We follow the presentation in [31,32].”
Cluster algebras and quantum affine algebras (2009, unknown): “M, Th. 5.9] as follows: a sufﬁcient condition for having FM(m)= χq(L(m)) is that FM(m) contains all the dominant monomials of χq(L(m)). The proof is essentially the same as in [FM, Th. 5.9]. (iii) In [H1], a polynomial F(m) deﬁned by formulas similar to (20), (21), (22), has been deﬁned for any m ∈M”
The Kirillov-Reshetikhin conjecture and solutions of T-systems (2005, unknown): “Although q, t-characters can be defined in general (see [12]), this result in [31] has not been proved for non simply-laced cases (see the conjectures of [12]).”
Quantum Grothendieck rings and derived Hall algebras (2011, background): “rate a commutativeC-algebra isomorphic to the complexiﬁed Grothendieck ring of the category of ﬁnite-dimensional irreducible Uq(Lg)-modules. Nakajima [N3], Varagnolo and Vasserot [VV1], and Hernandez [He2], have studied t-deformations of the qcharacters of the standard modules and of the simple modu”
Quantum toroidal algebras and their representations (2008, methodology): “In particular, as for the applications developed in [Nak5, Nak2], it should be possible to interpret geometrically corresponding analogs of Kazhdan{Lusztig polynomials in order to get information on the fusion product f . Candidates for these polynomials are constructed in [ He1 , He2] where precise”
Simple tensor products (2009, background): “∈ Kˆt ∩ Y i,a 1 +Vi,aq + X V <Vi,aq Z[t±1]V . Li,a is a t-analog of χq(L(Yi,a)). The existence of Li,a is non trivial and is proved in [N1] (it can also be proved purely algebraically [H1]). We deﬁne t-analogs of q-characters of tensor products of fundamental representations (or Weyl modules”
Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, methodology): “denote the t-analog of the q-characters (q,t-character for short) χq,t : Rt →Y= Z[Y± (39) i,a](i,a)∈I×Z. χq,t is an injective ring homomorphism with respect to the twisted product ∗, cf. [Nak04][VV03][Her04]. We refer the reader to [Qin14] for its precise deﬁnition in our convention. In particular, ”
Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras (2013, methodology): “ces a ring isomorphism φFe: K(TJ)/(q−1)K(TJ)−→∼K(CJ). Therefore TJ may be regarded as a Z-graded lifting of the rigid tensor category CJ. Recall that the ring K(CJ) admits interesting t-deformations ([37, 33, 11]). In [13], Hernandez and Leclerc gave a presentation of Kt, one of those t-deformations”
Representations of Quantum Affinizations and Fusion Product (2003, background): “In [H3] we extended the denition of the algorithm for generalized Cartan matrix and showed that it is well-dened if i 6= j ) Ci;jCj;i 3 (see also [ H2 ] for the detailed description of this algorithm at t = 1). Theorem 5.15 allows us to prove two results announced in [H3] : the algorithm is well den”
On Minimal Affinizations of Representations of Quantum Groups (2006, methodology): “Observe that this result is false for not simply-laced g (for example there is a thin fundamental representation for type G2 which is not simple as aUq (g)-module, see [ H1 , Sect.”