Affine Algebras Research Articles

On the primality of totally ordered q-factorization graphs

AbstractWe introduce the combinatorial notion of a q-factorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with three decorations: a coloring and a weight map on vertices, and an exponent map on arrows (the exponent map can be seen as a weight map on arrows). Such graphs do not contain oriented cycles and, hence, the set of arrows induces a partial order on the set of vertices. In this first paper on the topic, beside setting the theoretical base of the concept, we establish several criteria for deciding whether or not a tensor product of two simple modules is a highest- $\ell $ -weight module and use such criteria to prove, for type A, that a simple module whose q-factorization graph has a totally ordered vertex set is prime.

PBW theory for quantum affine algebras

Let U_q'(\mathfrak{g}) be a quantum affine algebra of arbitrary type and let \mathscr{C}_\mathfrak{g}^0 be Hernandez-Leclerc’s category. We can associate the quantum affine Schur–Weyl duality functor \mathcal{F}_\mathcal{D} to a duality datum \mathcal{D} in \mathscr{C}_\mathfrak{g}^0 . In this paper, we introduce the notion of a strong (complete) duality datum \mathcal{D} and prove that, when \mathcal{D} is strong, the induced duality functor \mathcal{F}_\mathcal{D} sends simple modules to simple modules and preserves the invariants \Lambda , \tilde\Lambda and \Lambda^\infty introduced by the authors. We next define the reflections \mathscr{S}_k and \mathscr{S}^{-1}_k acting on strong duality data \mathcal{D} . We prove that if \mathcal{D} is a strong (resp.\ complete) duality datum, then \mathscr{S}_k(\mathcal{D}) and \mathscr{S}_k^{-1}(\mathcal{D}) are also strong (resp. complete) duality data. This allows us to make new strong (resp. complete) duality data by applying the reflections \mathscr{S}_k and \mathscr{S}^{-1}_k from known strong (resp. complete) duality data. We finally introduce the notion of affine cuspidal modules in \mathscr{C}_\mathfrak{g}^0 by using the duality functor \mathcal{F}_\mathcal{D} , and develop the cuspidal module theory for quantum affine algebras similar to the quiver Hecke algebra case. When \mathcal{D} is complete, we show that all simple modules in \mathscr{C}_\mathfrak{g}^0 can be constructed as the heads of ordered tensor products of affine cuspidal modules. We further prove that the ordered tensor products of affine cuspidal modules have the unitriangularity property. This generalizes the classical simple module construction using ordered tensor products of fundamental modules.

On string functions and double-sum formulas

String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certain symmetry relations. We evaluate the series using special double-sum formulas that express Hecke-type double-sums in terms of Appell--Lerch functions and theta functions, where we point out that Appell--Lerch functions are the building blocks of Ramanujan's classical mock theta functions.

Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems

We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras in the spirit of Feigin and Odesskii (Funktsional. Anal. Prilozhen. 31(3):57–70, 1997), thus generalizing its rational version of Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) and the type A construction of Finkelberg and Tsymbaliuk (Arnold Math. J. 5(2–3):197–283, 2019). As an application, this allows us to construct large families of commuting and q-commuting difference operators, in particular, providing a convenient approach to the Q-systems where it proves a conjecture of Di Francesco and Kedem (Commun. Math. Phys. 369(3):867–928, 2019).

Cosets of Free Field Algebras via Arc Spaces

AbstractUsing the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra ${{\mathcal {V}}}$, we have a surjective homomorphism of differential algebras $\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$; equivalently, the singular support of ${{\mathcal {V}}}$ is a closed subscheme of the arc space of the associated scheme $X_{{{\mathcal {V}}}}$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$ for all positive integers $n$ and $k$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular ${{\mathcal {W}}}$-algebra of ${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.

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.

Affine Kac-Moody groups and Lie algebras in the language of SGA3

In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.

Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks

We give a mathematical definition of irregular conformal blocks in the genus-zero WZNW model for any simple Lie algebra, using coinvariants of modules for affine Lie algebras whose parameters match up with those of moduli spaces of irregular meromorphic connections: the open de Rham spaces. The Segal–Sugawara representation of the Virasoro algebra is used to show that the spaces of irregular conformal blocks assemble into a flat vector bundle over the space of isomonodromy times à la Klarès, and we provide a universal version of the resulting flat connection generalising the irregular KZ connection of Reshetikhin and the dynamical KZ connection of Felder–Markov–Tarasov–Varchenko.

New partition identities from \(C^{(1)}_\ell\)-modules

In this paper we conjecture combinatorial Rogers-Ramanu­jan type colored partition identities related to standard representations of the affine Lie algebra of type \(C^{(1)}_\ell\), \(\ell\geq2\), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras.

$$Q{\widetilde{Q}}$$-Systems for Twisted Quantum Affine Algebras

We establish the \(Q {\widetilde{Q}}\)-systems for the twisted quantum affine algebras that were conjectured in Frenkel and Hernandez (Commun Math Phys 362(2):361–414, 2018). We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their prefundamental representations. We also propose a general conjecture on the relations between twisted and non-twisted types. We prove this conjecture for some particular classes of representations, including prefundamental representations.

Bases for upper cluster algebras and tropical points

It is known that many (upper) cluster algebras possess different kinds of good bases which contain the cluster monomials and are parametrized by the tropical points of cluster Poisson varieties. For a large class of upper cluster algebras (injective-reachable ones with full rank coefficients), we describe all of their bases with these properties. Moreover, we show the existence of the generic basis for them. In addition, we prove that Bridgeland’s representation-theoretic formula is effective for their theta functions (weak genteelness). Our results apply to (almost) all known cluster algebras arising from representation theory or higher Teichmüller theory, including quantum affine algebras, unipotent cells, double Bruhat cells, skein algebras over surfaces, where we change the coefficients if necessary so that the full rank assumption holds.

Affine Lie algebra representations induced from Whittaker modules

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary affine Kac-Moody algebras. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the affine Lie algebra modules. Both functors preserve irreducibility when the central charge is nonzero.

An Ising-type formulation of the six-vertex model

We show that the celebrated six-vertex model of statistical mechanics (along with its multistate generalizations) can be reformulated as an Ising-type model with only a two-spin interaction. Such a reformulation unravels remarkable factorization properties for row to row transfer matrices, allowing one to uniformly derive all functional relations for their eigenvalues and present the coordinate Bethe ansatz for the eigenvectors for all higher spin generalizations of the six-vertex model. The possibility of the Ising-type formulation of these models raises questions about the precedence of the traditional quantum group description of the vertex models. Indeed, the role of a primary integrability condition is now played by the star-triangle relation, which is not entirely natural in the standard quantum group setting, but implies the vertex-type Yang-Baxter equation and commutativity of transfer matrices as simple corollaries. As a mathematical identity the emerging star-triangle relation is equivalent to the Pfaff-Saalschütz-Jackson summation formula, originally discovered by J. F. Pfaff in 1797. Plausibly, all vertex models associated with quantized affine Lie algebras and superalgebras can be reformulated as Ising-type models with only two-spin interactions.

NONRECURSIVE CANONICAL BASIS COMPUTATIONS FOR LOW RANK KASHIWARA CRYSTALS OF TYPE A

We give an improved algorithm for the action of the divided power of a Chevalley basis element of an affine Lie algebra of type A on canonical basis elements satisfying an easily checked uniformity condition and compare calculation times for our algorithm against the standard algorithm. For symmetric Kashiwara crystals of affine type A and rank e=2, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a nonrecursive manner. In particular, for a symmetric crystal with Λ=aΛ0+aΛ1, we give formulae for the canonical basis elements for all the e-regular multipartitions with defects either k(a−k) or k(a−k)+2a, for 0≤k≤a.

Strip bundle realization of the crystals over Uq(D4(3))

We introduce new combinatorial models called strip bundles for the quantum affine algebra Uq(D4(3)) and give new realizations of the crystal B(∞) and the highest weight crystals B(λ) over Uq(D4(3)) in terms of strip bundles. In these strip bundle realizations, the highest weight crystal B(Λ1) of the natural representation V(Λ1) over the quantum finite algebra Uq(G2) plays a crucial role.

Unimodular rows over affine algebras over algebraic closure of a finite field

In this paper, we prove that if [Formula: see text] is an affine algebra of dimension [Formula: see text] over [Formula: see text] and [Formula: see text] then any unimodular row over [Formula: see text] of length [Formula: see text] can be mapped to a factorial row by elementary transformations.

Equivariant multiplicities via representations of quantum affine algebras

For any simply-laced type simple Lie algebra mathfrak {g} and any height function xi adapted to an orientation Q of the Dynkin diagram of mathfrak {g}, Hernandez–Leclerc introduced a certain category mathcal {C}^{le xi } of representations of the quantum affine algebra U_q(widehat{mathfrak {g}}), as well as a subcategory mathcal {C}_Q of mathcal {C}^{le xi } whose complexified Grothendieck ring is isomorphic to the coordinate ring mathbb {C}[textbf{N}] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism {widetilde{D}}_{xi } on a torus mathcal {Y}^{le xi } containing the image of K_0(mathcal {C}^{le xi }) under the truncated q-character morphism. We prove that the restriction of {widetilde{D}}_{xi } to K_0(mathcal {C}_Q) coincides with the morphism overline{D} recently introduced by Baumann–Kamnitzer–Knutson in their study of equivariant multiplicities of Mirković–Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov–Reshetikhin modules in mathcal {C}_Q, as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of overline{D} allows us to prove a conjecture by the first author on the distinguished values of overline{D} on the flag minors of mathbb {C}[textbf{N}]. We also provide applications of our results from the perspective of Kang–Kashiwara–Kim–Oh’s generalized Schur–Weyl duality. Finally, we use Kashiwara–Kim–Oh–Park’s recent constructions to define a cluster algebra overline{mathcal {A}}_Q as a subquotient of K_0(mathcal {C}^{le xi }) naturally containing mathbb {C}[textbf{N}], and suggest the existence of an analogue of the Mirković–Vilonen basis in overline{mathcal {A}}_Q on which the values of {widetilde{D}}_{xi } may be interpreted as certain equivariant multiplicities.

Vertex algebraic construction of modules for twisted affine Lie algebras of type [formula omitted

Let g˜ be the affine Lie algebra of type A2l(2). The integrable highest weight g˜-module L(kΛ0) called the standard g˜-module is realized by a tensor product of the twisted module VLT for the lattice vertex operator algebra VL. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in [15].

Classification of minimal locally affine Lie algebras

Minimal locally affine Lie algebras, minimal LALAs for short, are classified. In particular, we will have a condition for a minimal LALA to be isomorphic to a standard minimal LALA containing the so-called degree derivation. In fact, we will see that there are many kinds of non-standard minimal LALAs.

Tableau models for semi-infinite Bruhat order and level-zero representations of quantum affine algebras

We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application, we give new tableau models for the crystal bases of a level-zero fundamental representation and a level-zero extremal weight module over a quantum affine algebra of classical untwisted type, which we call quantum Kashiwara-Nakashima columns and semi-infinite Kashiwara-Nakashima tableaux. We give an explicit description of the crystal isomorphisms among three different realizations of the crystal basis of a level-zero fundamental representation by quantum Lakshmibai-Seshadri paths, quantum Kashiwara-Nakashima columns, and (ordinary) Kashiwara-Nakashima columns.