Affine Algebras Research Articles

ODE/IM correspondence and supersymmetric affine Toda field equations

We study the linear differential system associated with the supersymmetric affine Toda field equations for affine Lie superalgebras, which has a purely odd simple root system. For an affine Lie algebra, the linear problem modified by conformal transformation leads to an ordinary differential equation (ODE) that provides the functional relations in the integrable models. This is known as the ODE/IM correspondence. For the affine Lie superalgebras, the linear equations modified by a superconformal transformation are shown to reduce to a couple of ODEs for each bosonic subalgebra. In particular, for osp(2,2)(2), the corresponding ODE becomes the second-order ODE with squared potential, which is related to the N=1 supersymmetric minimal model via the ODE/IM correspondence. We also find ODEs for classical affine Lie superalgebras with purely odd simple root systems.

Quadratic Relations of the Deformed W-Algebra for the Twisted Affine Lie Algebra of Type A<sup>(2)</sup><sub>2N</sub>

We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.

Restricted Gröbner fans and re-embeddings of affine algebras

In this paper we continue the study of good re-embeddings of affine K-algebras started in Kreuzer et al. (J Algebra Appl, 2021. https://doi.org/10.1142/S0219498822501882 ). The idea is to use special linear projections to find isomorphisms between a given affine K-algebra K[X]/I, where $$X=(x_1,\dots ,x_n)$$ , and K-algebras having fewer generators. These projections are induced by particular tuples of indeterminates Z and by term orderings $$\sigma $$ which realize Z as leading terms of a tuple F of polynomials in I. In order to efficiently find such tuples, we provide two major new tools: an algorithm which reduces the check whether a given tuple F is Z-separating to an LP feasibility problem, and an isomorphism between the part of the Gröbner fan of I consisting of marked reduced Gröbner bases which contain a Z-separating tuple and the Gröbner fan of $$I\cap K[X\setminus Z]$$ . We also indicate a possible generalization to tuples Z which consist of terms. All results are illustrated by explicit examples.

On Hopf algebraic structures of quantum toroidal algebras

ABSTRACT We define an algebra using a simplified set of generators for the quantum toroidal algebra and show that there exists an epimorphism from to . We derive a closed formula of the comultiplication on the generators of that extends that of the quantum affine algebra . As a consequence, we show that is a Hopf algebra for n = 1, 2 and give conjectural formulas in the general case. We further show that is isomorphic to a double algebra.

Subregular W-algebras of type A

Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the [Formula: see text] subregular W-algebra can be realized in terms of the [Formula: see text] regular W-algebra and the half lattice vertex algebra [Formula: see text]. This generalizes the realizations found for [Formula: see text] and [Formula: see text] in [D. Adamović, Realizations of simple affine vertex algebras and their modules: The cases [Formula: see text] and [Formula: see text], Comm. Math. Phys. 366 (2019) 1025–1067, arXiv:1711.11342 [math.QA]; D. Adamović, K. Kawasetsu and D. Ridout, A realization of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys., 111 (2021) 1–30, arXiv:2007.00396 [math.QA]] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamović. We use this realization to explore the representation theory of [Formula: see text] subregular W-algebras. Much of the structure encountered for [Formula: see text] and [Formula: see text] is also present for [Formula: see text]. Particularly, the simple [Formula: see text] subregular W-algebra at nondegenerate admissible levels can be realized purely in terms of the [Formula: see text] minimal model vertex algebra and [Formula: see text].

Cancellation of vector bundles of rank 3 with trivial Chern classes on smooth affine fourfolds

If n ≡ 0 , 1 m o d 4 , we prove a sum formula V θ 0 ( a 0 , a R n ) = n ⋅ V θ 0 ( a 0 , a R ) for the generalized Vaserstein symbol whenever R is a smooth affine algebra over a perfect field k with c h a r ( k ) ≠ 2 such that − 1 ∈ k × 2 . This enables us to generalize a result of Fasel-Rao-Swan on transformations of unimodular rows via elementary matrices over normal affine algebras of dimension d ≥ 4 over algebraically closed fields of characteristic ≠2. As a consequence, we prove that any projective module of rank 3 with trivial Chern classes over a smooth affine algebra of dimension 4 over an algebraically closed field k with c h a r ( k ) ≠ 2 , 3 is cancellative.

Diagram automorphisms and canonical bases for quantum affine algebras, II

Let Uq− be the negative part of the quantum enveloping algebra, and σ the algebra automorphism on Uq− induced from a diagram automorphism. Let U_q− be the quantum algebra obtained from σ, and B˜ (resp. B_˜) the canonical signed basis of Uq− (resp. U_q−). Assume that Uq− is simply-laced of finite or affine type. In our previous papers [10], [11], we have proved by an elementary method, that there exists a natural bijection B˜σ≃B_˜ in the case where σ is admissible. In this paper, we show that such a bijection exists even if σ is not admissible, possibly except some small rank cases.

Irreducible modules of toroidal Lie algebras arising from ϕϵ-coordinated modules of vertex algebras

In this paper, for every ϵ∈Z, we introduce an extension of the 2-toroidal Lie algebra by certain derivations. Based on the ϕϵ-coordinated modules theory for vertex algebras, we give an explicit realization of a class of irreducible highest weight modules for this extended toroidal Lie algebra. When ϵ=1, this affords a realization of certain irreducible modules for the toroidal extended affine Lie algebras first constructed by Billig.

Snowflake modules and Enright functor for Kac–Moody superalgebras

We introduce a class of modules over Kac-Moody superalgebras; we call these modules snowflake. These modules are characterized by invariance property of their characters with respect to a certain subgroup of the Weyl group. Examples of snowflake modules appear as admissible modules in representation theory of affine vertex algebras and in classification of bounded weight modules. Using these modules we prove Arakawa's Theorem for the Lie superalgebra osp(1|2n).

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.

Generalisations of Capparelli's and Primc's identities, I: Coloured Frobenius partitions and combinatorial proofs

The partition identities of Capparelli and Primc were originally discovered via representation theoretic techniques, and have since then been studied and refined combinatorially, but the question of giving a very broad generalisation remained open. In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra An−1(1).In this first paper, we focus on combinatorial aspects. We give a n2-coloured generalisation of Primc's identity by constructing a n2×n2 matrix of difference conditions, Primc's original identities corresponding to n=2 and n=3. While most coloured partition identities in the literature connect partitions with difference conditions to partitions with congruence conditions, in our case, the natural way to generalise these identities is to relate partitions with difference conditions to coloured Frobenius partitions. This gives a very simple expression for the generating function. With a particular specialisation of the colour variables, our generalisation also yields a partition identity with congruence conditions.Then, using a bijection from our new generalisation of Primc's identity, we deduce a large family of identities on (n2−1)-coloured partitions which generalise Capparelli's identity, also in terms of coloured Frobenius partitions. The particular case n=2 is Capparelli's identity and one of the cases where n=3 recovers an identity of Meurman and Primc.In the second paper, we will focus on crystal theoretic aspects. We will show that the difference conditions we defined in our n2-coloured generalisation of Primc's identity are actually energy functions for certain An−1(1) crystals. We will then use this result to retrieve the Kac-Peterson character formula and derive a new character formula as a sum of infinite products for all the irreducible highest weight An−1(1)-modules of level 1.

The crossing multiplier for solvable lattice models

We study the large class of solvable lattice models, based on the data of conformal field theory. These models are constructed from any conformal field theory. We consider the lattice models based on affine algebras described by Jimbo et al, for the algebras ABCD and by Kuniba et al for G 2. We find a general formula for the crossing multipliers of these models. It is shown that these crossing multipliers are also given by the principally specialized characters of the model in question. Therefore we conjecture that the crossing multipliers in this large class of solvable interaction round the face lattice models are given by the characters of the conformal field theory on which they are based. We use this result to study the local state probabilities of these models and show that they are given by the branching rule, in regime III.

Folded quantum integrable models and deformed W-algebras

We propose a novel quantum integrable model for every non-simply laced simple Lie algebra \({{\mathfrak {g}}}\), which we call the folded integrable model. Its spectra correspond to solutions of the Bethe Ansatz equations obtained by folding the Bethe Ansatz equations of the standard integrable model associated with the quantum affine algebra \(U_q(\widehat{{{\mathfrak {g}}}'})\) of the simply laced Lie algebra \({{\mathfrak {g}}}'\) corresponding to \({{\mathfrak {g}}}\). Our construction is motivated by the analysis of the second classical limit of the deformed W-algebra of \({{\mathfrak {g}}}\), which we interpret as a “folding” of the Grothendieck ring of finite-dimensional representations of \(U_q(\widehat{{{\mathfrak {g}}}'})\). We conjecture, and verify in a number of cases, that the spaces of states of the folded integrable model can be identified with finite-dimensional representations of \(U_q({}^L{\widehat{{{\mathfrak {g}}}}})\), where \(^L{\widehat{{{\mathfrak {g}}}}}\) is the (twisted) affine Kac–Moody algebra Langlands dual to \({\widehat{{{\mathfrak {g}}}}}\). We discuss the analogous structures in the Gaudin model which appears in the limit \(q \rightarrow 1\). Finally, we describe a conjectural construction of the simple \({{\mathfrak {g}}}\)-crystals in terms of the folded q-characters.

Root components for tensor product of affine Kac-Moody Lie algebra modules

Let g \mathfrak {g} be an affine Kac-Moody Lie algebra and let λ , μ \lambda , \mu be two dominant integral weights for g \mathfrak {g} . We prove that under some mild restriction, for any positive root β \beta , V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) contains V ( λ + μ − β ) V(\lambda +\mu -\beta ) as a component, where V ( λ ) V(\lambda ) denotes the integrable highest weight (irreducible) g \mathfrak {g} -module with highest weight λ \lambda . This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) . Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G \mathcal {G} -Schubert varieties in the product partial flag variety G / P × G / P \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of G \mathcal {G} -sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.

Correlation Functions of Quantum Toroidal gln Algebra

In this paper, we study the correlation functions of the quantum toroidal \mathfrak{gl}_n algebra. Their first key properties are established in analogy to those of the correlation functions of quantum affine algebras U_q\mathfrak{n}_+. The core of the paper is the proof of the vanishing of those correlation functions at length 3 wheel conditions, which is done separately for n = 1, n = 2, n≥ 3 cases with different ``Master Equalities" of formal series (the n = 1 case has been previously discussed by the author in (Cui, 2021)). Another important contribution is the discovery of cubic Serre relations for the quantum toroidal.

Representations of twisted affine Lie superalgebras–A recognition theorem

Over the past three decades, representation theory of affine Lie (super)algebras has been a research spotlight in both areas Mathematics and Physics. The even parts of almost all affine Lie superalgebras L contain two affine Lie algebras; say L0(1) and L0(2). Irreducible finite weight modules (f.w.m's for short) are the first interesting class of modules over such an L. The structure of these modules strongly depends on whether of root vectors corresponding to real roots of L0(1) and L0(2) act locally nilpotently or not. We know that there is no nonzero irreducible f.w.m. of nonzero level over L for which root vectors corresponding to nonzero real roots of both L0(1) and L0(2) act locally nilpotently. This leads us to define quasi-integrable modules and study the case that root vectors corresponding to nonzero real roots of one of L0(1) and L0(2) act locally nilpotently. We give a recognition theorem for quasi-integrable modules over twisted affine Lie superalgebras.

Parafermionic Bases of Standard Modules for Twisted Affine Lie Algebras of Type $A_{2l-1}^{(2)}$, $D_{l+1}^{(2)}$, $E_{6}^{(2)}$ and $D_{4}^{(3)}$

Using the bases of principal subspaces for twisted affine Lie algebras except \(A_{2l}^{(2)}\) by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight kΛ0 and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in Hatayama et al. (2001).

Formula omitted] at level [formula omitted

Let Ll=L(sl2l+1,−l−12) be the simple vertex operator algebra based on the affine Lie algebra slˆ2l+1 at boundary admissible level −l−12.We consider a lift ν of the Dynkin diagram involution of A2l=sl2l+1 to an involution of Ll. The ν-twisted Ll-modules are A2l(2)-modules of level −l−12 with an anti-homogeneous realization. We classify simple ν-twisted highest-weight (weak) Ll-modules using twisted Zhu algebras and singular vectors for slˆ2l+1 at level −l−12 obtained by Perše.We find that there are finitely many such modules up to isomorphism, and the ν-twisted (weak) Ll-modules that are in category O for A2l(2) are semi-simple.

A note on relative Vaserstein symbol

In an unpublished work of Fasel–Rao–Swan the notion of the relative Witt group [Formula: see text] is defined. In this paper, we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol [Formula: see text]. We establish injectivity of this symbol if [Formula: see text] is a non-singular affine algebra of dimension 3 over a perfect [Formula: see text]-field and [Formula: see text] is a local complete intersection ideal of [Formula: see text]. It is natural to ask whether the Vaserstein symbol is injective for a singular 3-dimensional affine algebra. At the end of the paper, we will give an example of a singular 3-dimensional algebra over a perfect [Formula: see text]-field for which the Vaserstein symbol is injective.

Principal subspaces for the affine Lie algebras in types D, E and F

We consider the principal subspaces of certain level $k\geqslant 1$ integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types $D$, $E$ and $F$. Generalizing the approach of G. Georgiev we construct their quasi-particle bases. We use the bases to derive presentations of the principal subspaces, calculate their character formulae and find some new combinatorial identities.