Feng Xu
We prove a conjecture about mininmal index of certain representations of Coset Algebraic Conformal Field Theories under general conditions as formulated previously by us. As a by-product, the Kac-Wakimoto Conjecture (KWC) which is related to the asympotics of the coset characters is true under the same conditions. The same idea in the proof also proves a recent conjecture related to subfactors from conformal inclusions.
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Character formulas for Lie superalgebras have been shown to have important applications to number theory and combinatorics. We prove the Kac-Wakimoto character formula for the general linear Lie superalgebra gl(m|n). This formula specializes to the well-known Kac-Weyl character formula when the modules are typical and to the Weyl denominator identity when the module is trivial. We also prove a determinantal character formula for KW-modules using the Kac-Wakimoto character formula.
For the Kac-Wakimoto hierarchy constructed from the principal vertex operator realization of the basic representation of the affine Lie algebra $D_n^{(1)}$, we compute the coefficients of the corresponding Hirota bilinear equations, and verify the coincidence of these bilinear equations with the ones that are satisfied by Givental's total descendant potential of the $D_n$ singularity, as conjectured by Givental and Milanov in \cite{GM}
Coset constructions and Kac-Wakimoto Hypothesis
本文通过范畴化陪集构造证明了伪酉模张量范畴中的Kac-Wakimoto假设，并给出场识别，进而应用于顶点算子代数理论。
We classify finite-dimensional tame modules over the ortho-symplectic Lie superalgebras. For these modules we show that their characters are given by the Kac-Wakimoto character formula, thus establishing the Kac-Wakimoto conjecture for the ortho-symplectic Lie superalgebras. We further relate the Kac-Wakimoto formula to the super Jacobi polynomials of Sergeev and Veselov, and show that these polynomials, up to a sign, give the super characters for these tame modules.
We prove the Kac-Wakimoto conjecture for the periplectic Lie superalgebra $\mathfrak{p}(n)$, stating that any simple module lying in a block of non-maximal atypicality has superdimension zero.
Atypicality is a fundamental combinatorial invariant for simple supermodules of a basic Lie superalgebra. Boe, Nakano, and the author gave a conjectural geometric interpretation of atypicality via support varieties. Inspired by low dimensional topology, Geer, Patureau-Mirand, and the author gave a generalization of the Kac-Wakimoto atypicality conjecture. We prove both of these conjectures for the Lie superalgebra osp(m|2n).
We study some non-highest weight modules over an affine Kac-Moody algebra at non-critical level. Roughly speaking, these modules are non-commutative localizations of some non-highest weight "vacuum" modules. Using free field realization, we embed some rings of differential operators in endomorphism rings of our modules. These rings of differential operators act on a localization of the space of coinvariants of any module over the Kac-Moody algebra with respect to a certain level subalgebra. In a particular case this action is identified with the Casimir connection.
We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we describe explicitly geometric realizations of the Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group.
First, we establish the relation between the associated varieties of modules over Kac-Moody algebras \hat{g} and those over affine W-algebras. Second, we prove the Feigin-Frenkel conjecture on the singular supports of G-integrable admissible representations. In fact we show that the associated variates of G-integrable admissible representations are irreducible G-invariant subvarieties of the nullcone of g, by determining them explicitly. Third, we prove the C_2-cofiniteness of a large number of simple W-algebras, including all minimal series principal W-algebras and the exceptional W-algebras …
Representation theory of W-algebras, II: Ramond twisted representations
本文研究了 f 容许良好偶分次时仿射 W-代数 W^k(g,f) 的 Ramond 扭曲表示，确立了 BRST 上同调的消没性与几乎不可约性，证实了 Kac 和 Wakimoto 的猜想，并在 A 型中给出了不可约表示的特征标及证明了模不变表示的存在性。
We prove the conjecture of Frenkel, Kac and Wakimoto on the existence of two-sided BGG resolutions of G-integrable admissible representations of affine Kac-Moody algebras at fractional levels. As an application we establish the semi-infintie analogue of the generalized Borel-Weil theorem for mimimal parabolic subalgebras which enables an inductive study of admissible representations.
We prove the conjecture of Kac-Wakimoto on the rationality of exceptional W-algebras for the first non-trivial series, namely, for the Bershadsky-Polyakov vertex algebras $W_3^{(2)}$ at level $k=p/2-3$ with $p=3,5,7,...$. This gives new examples of rational conformal field theories.
In this paper we discuss an analogue of the Kac-Weisfeiler conjecture for a certain class of almost commutative algebras. In particular, we prove the Kac-Weisfeiler type statement for rational Cherednik algebras.
In this paper we prove a conjecture by De Concini, Kac and Procesi \cite{CP} (Corollary \ref{conj}): The dimension of any $M\in U_q-\mood^χ$ is divisible by $l^{codim_\mathcal{B}\mathcal{B}_χ}$.
In this paper we prove the Kazhdan-Lusztig type character formula for irreducible highest weight modules with positive rational highest weights over symmetrizable Kac-Moody Lie algebras.
Let Q be an affine quiver and let $\mathfrak{n}$ be the positive part of the affine Lie algebra associated to Q. We provide a construction of $\mathfrak{n}$ using the semistable irreducible components in the Lusztig nilpotent variety associated to Q. This confirms a conjecture of Frenkel, Malkin, and Vybornov on defining the so-called algebra of BPS states on the minimal resolution of a Kleinian singularity. Using the results of Crawley-Boevey and Van den Bergh, we show that our construction is closely connected to Kac's constant term conjecture in the case of an affine quiver.
The exceptional infinite-dimensional linearly compact simple Lie superalgebra ${E}(5,10)$, which Kac believes, is the algebra of symmetries of the ${SU}_{5}$ Grand Unified Model. In this paper, we give a proof of Kac and Rudakov's conjecture about the classification of all the degenerate generalized Verma module over ${E}(5,10)$. Also, we work out all the nontrivial singular vectors degree by degree. It is a potential that the representation theory of ${E}(5,10)$ will shed new light on various features of the the ${SU}_{5}$ Grand unified model.
We give a criterion for the Kac conjecture asserting that the free term of the polynomial counting the absolutely indecomposable representations of a quiver over a finite field of given dimension coincides with the corresponding root multiplicity of the associated Kac-Moody algebra. Our criterion suits very well for computer tests.
We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.
Given a Lie algebra $\mathfrak{g}$ of an algebraic group over a ring $S,$ we show that the first Kac-Weisfeiler conjecture holds for reductions of $\mathfrak{g} \mod p$ for large enough primes $p,$ reproving a recent result of Martin, Stewart and Topley. As a byproduct of our proof, we show that the center of the skew field of fractions of the the enveloping algebra $\mathfrak{U}\mathfrak{g}_{k}$ for a field $k$ of characteristic $p>>0$ is generated by the $p$-center and by the reduction $\mod p$ of the center of the fraction skew field of $\mathfrak{U}\mathfrak{g}.$
We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.
We prove locality of superconformal algebras: every pluperfect superconformal algebra is spanned by coefficients of a finite family of mutually local distributions. We also introduce quasi-Poisson algebras and show that they can be used to construct all known simple superconformal algebras.
On odd-spin $A_{1}^{(1)}$-string functions, cross-spin identities, and mock theta conjecture-like identities
本文研究了奇自旋A₁^(1)-字符串函数的极有限分解，并发现新的类似假theta猜想的恒等式。
In 1971 Kac and Weisfeiler made two important conjectures regarding the representation theory of restricted Lie algebras over fields of positive characteristic. The first of these predicts the maximal dimension of the simple modules, and can be stated without the hypothesis that the Lie algebra is restricted. In this short article we construct the first example of a non-restricted Lie algebra for which the prediction of the first Kac--Weisfeiler conjecture fails. Our method is to present pairs of Lie algebras which have isomorphic enveloping algebras but distinct indexes.
We provide a new proof of the Kac positivity conjecture for an arbitrary quiver $Q$. The ingredients are the cohomological integrality theorem in Donaldson-Thomas theory, dimensional reduction, and an easy purity result. These facts imply the purity of the cohomological Donaldson-Thomas invariants for a quiver with potential $(\tilde{Q},W)$ associated to $Q$, which in turn implies positivity of the Kac polynomials for $Q$.
Q-W-algebras, Zhelobenko operators and a proof of De Concini-Kac-Procesi conjecture
本文通过q-W代数与泽洛布金诺操作符的描述，证明了量子群在根数处不可约模的维度猜想。
We propose in this thesis a new deformation process of Kac-Moody algebras and their representations. The direction of deformation is given by a collection of numbers, called a colouring. The natural numbers lead for example to the classical algebras, while the quantum numbers lead to the associated quantum algebras. We first establish sufficient and necessary conditions on colourings to allow the process depend polynomially on a formal parameter and to provide the generalised quantum enveloping (GQE) algebras. We then lift the restrictions and show that the process still exists via the coloure…
In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $\mathfrak{g}$. The first predicts the maximal dimension of simple $\mathfrak{g}$-modules and in this paper we apply the Lefschetz principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of $\mathfrak{gl}_n(k)$ whenever $k$ is an algebraically closed field of characteristic $p \gg n$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebrai…
Irreducible modules of modular Lie superalgebras and super version of the first Kac-Weisfeiler conjecture
本文针对特征p>2的代数闭域上有限维限制李超代数，提出第一Kac-Weisfeiler猜想的超推广猜想，证明其对基本典型李超代数和完全可解限制李超代数成立，并研究了可解李超代数的不可约表示。
Modular representations of strange classical Lie superalgebras and the first super Kac-Weisfeiler conjecture
本文初步研究了periplectic李超代数的模表示，并验证了关于其不可约模最大维数的第一个超Kac-Weisfeiler猜想。
Asymptotic behaviour of Kac polynomials
本文研究箭图Kac多项式的渐近行为，提出仅依赖顶点环数的赋值猜想，证明重整化Kac多项式随箭数增加在幂级数环中线性收敛，并进一步猜想其系数的整体行为。
We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac co…
We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Sch…
In this paper, we prove that the ring of polynomial invariants of the Weyl group for an indecomposable and indefinite Kac-Moody Lie algebra is generated by invariant symmetric bilinear form or is trivial depending on $A$ is symmetrizable or not. The result was conjectured by Moody and assumed by Kac. As applications we discuss the rational homotopy types of Kac-Moody groups and their flag manifolds.
We study the restricted category O for an affine Kac--Moody algebra at the critical level. In particular, we prove the first part of the Feigin-Frenkel conjecture: the linkage principle for restricted Verma modules. Moreover, we prove a version of the BGGH-reciprocity principle and we determine the block decomposition of the restricted category O. For the proofs we need a deformed version of the classical structures, so we mostly work in a relative setting.
New polar-finite forms of generalized Euler identities for $A_{1}^{(1)}$-string functions and mock theta conjecture-like identities
本文研究了$A_{1}^{(1)}$-弦函数的极有限形式，利用准周期性方法推导出与拟theta猜想类似的恒等式，并引入了mock theta函数进行计算。
This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the affine Grassmannian for an affine Kac-Moody group. We formulate a conjecture describing the (local) intersection cohomology of these varieties in terms of integrable representations of the Langlands dual affine Kac-Moody group and check this conjecture in a number of cases. Roughly speaking the above singularities are constructed by looking at the Uhlenbeck sp…
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