Representations Of Affine Lie Algebras Research Articles

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.

Screening operators and parabolic inductions for affine [formula omitted]-algebras

(Affine) W-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra g over C, a nilpotent element f in [g,g], a good grading Γ and a symmetric invariant bilinear form κ on g. We introduce free field realizations of W-algebras by using Wakimoto representations of affine Lie algebras, where W-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free fields realizations of W-algebras. As applications, under certain conditions that are valid in all cases of type A, we construct parabolic inductions for W-algebras, which we expect to induce the parabolic inductions of finite W-algebras defined by Premet and Losev. In type A, we show that our parabolic inductions are a chiralization of the coproducts for finite W-algebras defined by Brundan-Kleshchev. In type BCD, we are able to obtain some generalizations of the coproducts in some special cases. This paper also contains an appendix by Shigenori Nakatsuka on the compatibility of screening operators with Miura maps.

Chern-Simons theory on Seifert manifold and matrix model

Chern-Simons (CS) theories with rank $N$ and level $k$ on Seifert manifold are discussed. The partition functions of such theories can be written as a function of modular transformation matrices summed over different integrable representations of affine Lie algebra $u(N{)}_{k}$ associated with the boundary Wess-Zumino-Witten model. Using properties of modular transform matrices we express the partition functions of these theories as a unitary matrix model. We show that the eigenvalues of unitary matrices are discrete and proportional to hook lengths of the corresponding integrable Young diagram. As a result, in the large $N$ limit, the eigenvalue density develops an upper cap. We consider CS theory on ${S}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ coupled with fundamental matters and express the partition functions in terms of modular transformation matrices. Solving this model at large $N$ we find the dominant integrable representations and show how large $N$ representations are related to each other by transposition of Young diagrams as a result of level rank duality. Next we consider $U(N)$ CS theory on ${S}^{3}$ and observed that in Seifert framing the dominant representation is no longer an integrable representation after a critical value of 't Hooft coupling. We also show that CS on ${S}^{3}$ admits multiple (two-gap phase) large $N$ phases with the same free energy.

Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras

We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over sl ^ 2 . In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights.

Modular Representation Theory of Affine and Cyclotomic Yokonuma-Hecke Algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type A (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic p. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type A.

Local martingales associated with Schramm-Loewner evolutions with internal symmetry

We consider Schramm-Loewner evolutions (SLEs) with internal degrees of freedom that are associated with representations of affine Lie algebras, following group theoretical formulation of SLEs. We reconstruct the SLEs considered by Bettelheim et al. [Phys. Rev. Lett. 95, 251601 (2005)] and Alekseev et al. [Lett. Math. Phys. 97, 243–261 (2011)] in correlation function formulation. We also explicitly formulate stochastic differential equations on internal degrees of freedom for Heisenberg algebras and the affine sl2. Our formulation enables us to find several local martingales associated with SLEs with internal degrees of freedom from computation on a representation of an affine Lie algebra. Indeed, we formulate local martingales associated with SLEs with internal degrees of freedom described by Heisenberg algebras and the affine sl2. We also find an affine sl2 symmetry of a space of SLE local martingales for the affine sl2.

A uniform realization of the combinatorial R-matrix for column shape Kirillov–Reshetikhin crystals

Kirillov–Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape Kirillov–Reshetikhin crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the so-called quantum alcove model. We enhance this model by using it to give a uniform realization of the corresponding combinatorial R-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of column shape KR crystals. In other words, we are generalizing to all Lie types Schützenberger's sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial R-matrix in type A. Our construction is in terms of certain combinatorial moves, called quantum Yang–Baxter moves, which are explicitly described by reduction to the rank 2 root systems. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves joining the fundamental one to a translation of it.

Pattern avoidance seen in multiplicities of maximal weights of affine Lie algebra representations

We prove that the multiplicities of certain maximal weights of $\mathfrak{g}(A^{(1)}_{n})$-modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Misra-Rebecca. We also prove similar phenomena in types $A^{(2)}_{2n}$ and $D^{(2)}_{n+1}$. Both proofs are applications of Kashiwara's crystal theory.

A uniform realization of the combinatorial $R$-matrix

Kirillov-Reshetikhin (KR) crystals are colored directed graphs encoding the structure of certain finite-dimensional representations of affine Lie algebras. A tensor product of column shape KR crystals has recently been realized in a uniform way, for all untwisted affine types, in terms of the quantum alcove model. We enhance this model by using it to give a uniform realization of the combinatorial $R$-matrix, i.e., the unique affine crystal isomorphism permuting factors in a tensor product of KR crystals. In other words, we are generalizing to all Lie types Schützenberger’s sliding game (jeu de taquin) for Young tableaux, which realizes the combinatorial $R$-matrix in type $A$. We also show that the quantum alcove model does not depend on the choice of a sequence of alcoves Les cristaux de Kirillov–Reshetikhin (KR) sont des graphes orientés avec des arêtes étiquetées qui encodent certaines représentations de dimension finie des algèbres de Lie affines. Les produits tensoriels des cristaux KR de type colonne ont été récemment réalisés de manière uniforme, pour tous les types affines symétriques, en termes du modèle des alcôves quantique. Nous enrichissons ce modèle en l’utilisant pour donner une réalisation uniforme de la $R$-matrice combinatoire, c’est à dire, l’isomorphisme des cristaux affines unique quit permute les facteurs dans un produit tensoriel des cristaux KR. En d’autres termes, nous généralisons pour tous les types de Lie le jeu de taquin de Schützenberger sur les tableaux de Young, qui réalise la $R$-matrice combinatoire dans le type $A$. Nous montrons aussi que le modèle des alcôves quantique ne dépend pas du choix d’une suite d’alcôves.

Curve counting, instantons and McKay correspondences

We survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kähler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson–Thomas theory for ideal sheaves on Calabi–Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson–Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch–Jung spaces.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

In this paper we present Affine.m—a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Program summaryProgram title: Affine.mCatalogue identifier: AENA_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, UKLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 24844No. of bytes in distributed program, including test data, etc.: 1045908Distribution format: tar.gzProgramming language: Mathematica.Computer: i386–i686, x86_64.Operating system: Linux, Windows, Mac OS, Solaris.RAM: 5–500 MbClassification: 4.2, 5.Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems.Solution method:We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases.Restrictions:Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical.Unusual features:We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface.Running time:From seconds to days depending on the rank of the algebra and the complexity of the representation.

Vertex operator algebras associated to modified regular representations of affine Lie algebras

Let G be a simply-connected complex Lie group with simple Lie algebra g and let g ˆ be its affine Lie algebra. We use intertwining operators and Knizhnik–Zamolodchikov equations to construct a family of N -graded vertex operator algebras (VOAs) associated to g . These vertex operator algebras contain the algebra of regular functions on G as the conformal weight 0 subspaces and are g ˆ ⊕ g ˆ -modules of dual levels k , k ¯ ∉ Q in the sense that k + k ¯ = − 2 h ∨ , where h ∨ is the dual Coxeter number of g . This family of VOAs was previously studied by Arkhipov–Gaitsgory and Gorbounov–Malikov–Schechtman from different points of view. We show that when k is irrational, the vertex envelope of the vertex algebroid associated to G and the level k is isomorphic to the vertex operator algebra we constructed above. The case of rational levels is also discussed.

Graded level zero integrable representations of affine Lie algebras

We study the structure of the category of integrable level zero representations with finite dimensional weight spaces of affine Lie algebras. We show that this category possesses a weaker version of the finite length property, namely that an indecomposable object has finitely many simple constituents which are non-trivial as modules over the corresponding loop algebra. Moreover, any object in this category is a direct sum of indecomposables only finitely many of which are non-trivial. We obtain a parametrization of blocks in this category.

A construction of some ideals in affine vertex algebras

We study ideals generated by singular vectors in vertex operator algebras associated with representations of affine Lie algebras of types A and C. We find new explicit formulas for singular vectors in these vertex operator algebras at integer and half‐integer levels. These formulas generalize the expressions for singular vectors from Adamović (1994). As a consequence, we obtain a new family of vertex operator algebras for which we identify the associated Zhu′s algebras. A connection with the representation theory of Weyl algebras is also discussed.

A BASIS OF THE BASIC $\mathfrak{sl} ({\bf 3}, {\mathbb C})^~$-MODULE

J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers–Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard [Formula: see text]-modules. We obtained several infinite series of new combinatorial identities through the construction of all standard [Formula: see text]-modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic [Formula: see text]-module and, by using the principal specialization of the Weyl–Kac character formula, we obtain a Rogers–Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky–Wilson's approach for affine Lie algebras of higher ranks, say for [Formula: see text], n ≥ 2, in a way parallel to the next level of complexity seen when passing from the Rogers–Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli ≥7.

Generating function identities for untwisted standard modules of affine Lie algebras

The present work generalizes to affine Lie algebras of type A,D or E the generating function identities used to construct bases of untwisted representations of the affine Lie algebra . For this, we apply multi-operator extensions of the Jacobi identity for generalized vertex algebras to the relative vertex operators associated to a dijrect sum of copies of the weight lattice of a Lie algebra of type A D or E. Then, using the properties of the δ-function, we construct mtalti-operator extensions of the Jacobi identity for relative untwisted vertex operators, and we obtain generating function identities for standard representations of untwisted affine Lie algebras as combinations of coefficients of these multi-operator Jacobi identities.

Unitary representations of some infinite-dimensional Lie algebras motivated by string theory on AdS 3

We consider some unitary representations of infinite-dimensional Lie algebras motivated by string theory on AdS 3. These include examples of two kinds: the A,D,E type affine Lie algebras and the N=4 superconformal algebra. The first presents a new construction for free field representations of affine Lie algebras. The second is of a particular physical interest because it provides some hints that a hybrid of the NSR and GS formulations for string theory on AdS 3 exists.

Sugawara construction for higher genus Riemann surfaces

By the classical genus zero Sugawara construction one obtains representations of the Virasoro algebra from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type). In this lecture, the classical construction is recalled first. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in a joint work with O. K. Sheinman. We were able to show that also in the higher genus, multi-point situation one obtains (from representations of the global algebras of affine type) representations of a centrally extended algebra of meromorphic vector fields on Riemann surfaces. The latter algebra is a generalization of the Virasoro algebra to higher genus.

On inner product in modular tensor categories II: Inner product on conformal blocks and affine inner product identities

This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space $\Hom (\bigoplus X_i\otimes X^*_i, U)$ is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group $U_q \sln$ at roots of unity and $U$ being a symmetric power of the fundamental representation, this inner product coincides with so-called Macdonald's inner product on symmetric polynomials. In this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras. In this case our construction immediately gives a hermitian form on the spaces of conformal blocks, and this form is modular invariant (Warning: we cannot prove that it is positive definite). We show that this form can be rewritten in terms of asymptotics of KZ equations, and calculate it for $sl_2$, in which case the formula is a natural affine analogue of Macdonald's inner product identities. We also formulate as a conjecture similar formula for $sl_n$.

Regularity of Rational Vertex Operator Algebras

Rational vertex operator algebras, which play a fundamental role in rational conformal field theory (see [BPZ] and [MS]), single out an important class of vertex operator algebras. Most vertex operator algebras which have been studied so far are rational vertex operator algebras. Familiar examples include the moonshine module V ♮ ([B], [FLM], [D2]), the vertex operator algebras VL associated with positive definite even lattices L ([B], [FLM], [D1]), the vertex operator algebras L(l, 0) associated with integrable representations of affine Lie algebras [FZ] and the vertex operator algebras L(cp,q, 0) associated with irreducible highest weight representations for the discrete series of the Virasoro algebra ([DMZ] and [W]). A rational vertex operator algebra as studied in this paper is a vertex operator algebra such that any admissible module is a direct sum of simple ordinary modules (see Section 2). It is natural to ask if such complete reducibility holds for an arbitrary weak module (defined in Section 2). A rational vertex operator algebra with this property is called a regular vertex operator algebra. One motivation for studying such vertex operator algebras arises in trying to understand the appearance of negative fusion rules (which are computed by the Verlinde formula) for vertex operator algebras L(l, 0) for certain rational l (cf. [KS] and [MW]). In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the rational vertex operator algebras V , L(l, 0) for positive integers l, L(cp,q, 0) and VL for positive definite even lattices L are regular. Our result for L(l, 0) implies that any restricted integrable module of level l for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. This result is expected to be useful in comparing the construction of tensor product of modules for L(l, 0) in [F] based on Kazhdan-Lusztig’s approach [KL] with the construction of tensor product of modules [HL] in this special case. We should remark that VL in general is a vertex algebra in the sense of [DL] if L is not positive definite. In this case we establish the complete reducibility of any weak module. Since the definition of vertex operator algebra is by now well-known, we do not define vertex operator algebra in this paper. We refer the reader to [FLM] and [FHL] for their elementary properties. The reader can find the details of the constructions of V ♮ and VL in [FLM], and L(l, 0) and L(cp,q, 0) in [DMZ], [DL], [FLM], [FZ], [L1] and [W].