Untwisted Quantum Affine Algebra Research Articles

Crystal bases for reduced imaginary Verma modules of untwisted quantum affine algebras

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras Uq(gˆ) and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra Kq constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category Ored,imq containing the reduced imaginary Verma modules for Uq(gˆ).

The (q, t)-Cartan matrix specialized at $$q=1$$ and its applications

The $(q,t)$-Cartan matrix specialized at $t=1$, usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of kew-symmetric type. In this paper, we study the $(q,t)$-Cartan matrix specialized at $q=1$, called the $t$-quantized Cartan matrix, and investigate the relations with (ii') its corresponding quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.

The q-characters of minimal affinizations of type G 2 arising from cluster algebras

Hernandez and Leclerc proved that the Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional modules of any untwisted quantum affine algebra has a cluster algebra structure. This makes it possible to compute -characters of real prime simple modules associated to cluster variables as a result of a sequence of cluster mutations. In this paper, for minimal affinizations of type G 2, we describe a cluster algebra algorithm to compute their -characters and prove the geometric -character formula conjectured by Hernandez and Leclerc.

Combinatorics of the q-characters of Hernandez-Leclerc modules

Let g be a complex simple Lie algebra and Uq(gˆ) be the corresponding untwisted quantum affine algebra. In this paper, we give a path description of the q-characters of Hernandez-Leclerc modules, show that up to spectral parameter shift, the equivalent classes of Hernandez-Leclerc modules in the Grothendieck ring of the category of finite-dimensional Uq(sln+1ˆ)-modules are cluster variables in the cluster algebra introduced by Hernandez and Leclerc, and finally prove that the geometric q-character formula conjecture is true for Hernandez-Leclerc modules.

Stable maps, Q-operators and category 𝒪

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category O \mathcal {O} of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new R R -matrices in the category O \mathcal {O} and we establish that a large family of simple modules, including the prefundamental representations associated to Q Q -operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified Q Q ∗ QQ^* -systems in terms of the R R -matrices we construct.

Q-data and Representation Theory of Untwisted Quantum Affine Algebras

For a complex finite-dimensional simple Lie algebra $\mathfrak{g}$, we introduce the notion of Q-datum, which generalizes the notion of a Dynkin quiver with a height function from the viewpoint of Weyl group combinatorics. Using this notion, we develop a unified theory describing the twisted Auslander-Reiten quivers and the twisted adapted classes introduced in [O.-Suh, J. Algebra, 2019] with an appropriate notion of the generalized Coxeter elements. As a consequence, we obtain a combinatorial formula expressing the inverse of the quantum Cartan matrix of $\mathfrak{g}$, which generalizes the result of [Hernandez-Leclerc, J. Reine Angew. Math., 2015] in the simply-laced case. We also find several applications of our combinatorial theory of Q-data to the finite-dimensional representation theory of the untwisted quantum affine algebra of $\mathfrak{g}$. In particular, in terms of Q-data and the inverse of the quantum Cartan matrix, (i) we give an alternative description of the block decomposition results due to [Chari-Moura, Int. Math. Res. Not., 2005] and [Kashiwara-Kim-O.-Park, arXiv:2003.03265], (ii) we present a unified (partially conjectural) formula of the denominators of the normalized R-matrices between all the Kirillov-Reshetikhin modules, and (iii) we compute the invariants $\Lambda(V,W)$ and $\Lambda^\infty(V, W)$ introduced in [Kashiwara-Kim-O.-Park, Compos. Math., 2020] for each pair of simple modules $V$ and $W$.

Spectra of Quantum KdV Hamiltonians, Langlands Duality, and Affine Opers

We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was shown that solutions of this system can be attached to certain affine opers for the Langlands dual affine Kac-Moody algebra of g^, introduced in [FF5]. Together with the results of [BLZ3, BHK], which enable one to associate quantum g^-KdV Hamiltonians to representations from the category O, this provides strong evidence for the conjecture of [FF5] linking the spectra of quantum g^-KdV Hamiltonians and affine opers for the Langlands dual affine algebra. As a bonus, we obtain a direct and uniform proof of the Bethe Ansatz equations for a large class of quantum integrable models associated to arbitrary untwisted quantum affine algebras, under a mild genericity condition. We also conjecture analogues of these results for the twisted quantum affine algebras and elucidate the notion of opers for twisted affine algebras, making a connection to twisted opers introduced in [FG].

TRACES OF INTERTWINERS FOR QUANTUM AFFINE ALGEBRAS AND DIFFERENCE EQUATIONS (AFTER ETINGOF–SCHIFFMANN–VARCHENKO)

We modify and give complete proofs for the results of Etingof-Schiffmann-Varchenko on traces of intertwiners of untwisted quantum affine algebras in the opposite coproduct and the standard grading. More precisely, we show that certain normalized generalized traces for $U_q(\hat{\mathfrak{g}})$ solve four commuting systems of q-difference equations: the Macdonald-Ruijsenaars, dual Macdonald-Ruijsenaars, q-KZB, and dual q-KZB equations. In addition, we show a symmetry property for these renormalized trace functions. Our modifications are motivated by their appearance in recent work of the author.

A cluster algebra approach to $q$-characters of Kirillov–Reshetikhin modules

We describe a cluster algebra algorithm for calculating q -characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra U_q(\widehat{\mathfrak{g}}) . This yields a geometric q -character formula for tensor products of Kirillov–Reshetikhin modules. When \mathfrak g is of type A, D, E , this formula extends Nakajima's formula for q -characters of standard modules in terms of homology of graded quiver varieties.

Baxter’s relations and spectra of quantum integrable models

Generalized Baxter's relations on the transfer-matrices (also known as Baxter's TQ relations) are constructed and proved for an arbitrary untwisted quantum affine algebra. Moreover, we interpret them as relations in the Grothendieck ring of the category O introduced by Jimbo and the second author in arXiv:1104.1891 involving infinite-dimensional representations constructed in arXiv:1104.1891, which we call here "prefundamental". We define the transfer-matrices associated to the prefundamental representations and prove that their eigenvalues on any finite-dimensional representation are polynomials up to a universal factor. These polynomials are the analogues of the celebrated Baxter polynomials. Combining these two results, we express the spectra of the transfer-matrices in the general quantum integrable systems associated to an arbitrary untwisted quantum affine algebra in terms of our generalized Baxter polynomials. This proves a conjecture of Reshetikhin and the first author formulated in 1998 (arXiv:math/9810055). We also obtain generalized Bethe Ansatz equations for all untwisted quantum affine algebras.

Reprint of: Minimal affinizations as projective objects

We prove that the specialization to q = 1 of a Kirillov–Reshetikhin module for an untwisted quantum affine algebra of classical type is projective in a suitable category. This yields a uniform character formula for the Kirillov–Reshetikhin modules. We conjecture that these results hold for specializations of minimal affinization with some restriction on the corresponding highest weight. We discuss the connection with the conjecture of Nakai and Nakanishi on q -characters of minimal affinizations. We establish this conjecture in some special cases. This also leads us to conjecture an alternating sum formula for Jacobi–Trudi determinants.

Minimal affinizations as projective objects

We prove that the specialization to q = 1 of a Kirillov–Reshetikhin module for an untwisted quantum affine algebra of classical type is projective in a suitable category. This yields a uniform character formula for the Kirillov–Reshetikhin modules. We conjecture that these results hold for specializations of minimal affinization with some restriction on the corresponding highest weight. We discuss the connection with the conjecture of Nakai and Nakanishi on q -characters of minimal affinizations. We establish this conjecture in some special cases. This also leads us to conjecture an alternating sum formula for Jacobi–Trudi determinants.

Weight Functions and Drinfeld Currents

A universal weight function for a quantum affine algebra is a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. In representations of the quantum affine algebra it gives off-shell Bethe vectors and is used in the construction of solutions of the qKZ equations. We construct a universal weight function for each untwisted quantum affine algebra, using projections onto the intersection of Borel subalgebras of different types, and study its functional properties.

The Kirillov-Reshetikhin conjecture and solutions of T-systems

We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras: we prove that the characters of Kirillov-Reshetikhin modules solve the Q-system and we give an explicit formula for the character of their tensor products. In the proof we show that Kirillov-Reshetikhin modules are special in the sense of monomials and that their q-characters solve the T-system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the T-system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by Nakajima in [H. Nakajima, Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.], [H. Nakajima, t-analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, Represent. Th. 7 (2003), 259–274.] with geometric arguments (main result of [H. Nakajima, Quiver Varieties and t-Analogs of q-Characters of Quantum Affine Algebras, Ann. Math. 160 (2004), 1057–1097.]) which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.

Quiver varieties and t-analogs of q-characters of quantum affine algebras

We consider a specialization of an untwisted quantum affine algebra of type ADE at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ?computable? polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ?compute? q-characters for all simple modules. The result is based on ?computations? of Betti numbers of graded/cyclic quiver varieties. (The reason why we use ? ? will be explained at the end of the introduction.)

The t-analogs of q-characters at roots of unity for quantum affine algebras and beyond

The q-characters were introduced by Frenkel and Reshetikhin [The q-characters of representations of quantum affine algebras and deformations of W-algebras, in: Recent Developments in Quantum Affine Algebras and Related Topics, in: Contemp. Math., vol. 248, Amer. Math. Soc., Providence, RI, 1999, pp. 163–205] to study finite dimensional representations of the untwisted quantum affine algebra U q( g ̂ ) for q generic. The ε-characters at roots of unity were constructed by Frenkel and Mukhin [The q-characters at roots of unity, Adv. Math. 171 (1) (2002) 139–167] to study finite dimensional representations of various specializations of U q( g ̂ ) at q s =1. In the finite simply laced case Nakajima [ t-analogue of the q-characters of finite dimensional representations of quantum affine algebras, in: Physics and Combinatorics, Proc. Nagoya 2000 Internat. Workshop, World Scientific, Singapore, 2001, pp. 181–212; Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math., in press; preprint arXiv: math.QA/0105173] defined deformations of q-characters called q, t-characters for q generic and also at roots of unity. 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 [Algebraic approach to q, t-characters, Adv. Math., in press, preprint arXiv: math.QA/0212257] we proposed an algebraic general (non-necessarily simply laced) new approach to q, t-characters for q generic. In this paper we propose two developments of this approach: we treat the root of unity case and the case of a larger class of generalized Cartan matrices (including finite and affine cases except A 1 (1), A 2 (2)). In particular, we generalize the construction of analogs of Kazhdan–Lusztig polynomials at roots of unity of [H. Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math., in press, preprint arXiv: math.QA/0105173] to those cases. We also study properties of various objects used in this article: deformed screening operators at roots of unity, t-deformed polynomial algebras, bicharacters arising from symmetrizable Cartan matrices, deformation of the Frenkel–Mukhin's algorithm.

The q-Characters at Roots of Unity

We consider various specializations of the untwisted quantum affine algebras at roots of unity. We define and study the q-characters of their finite-dimensional representations.