Jack Morava
The group of diffeomorphisms of a circle is not an infinite-dimensional algebraic group, though in many ways it behaves as if it were. Here we construct an algebraic model for this object, and discuss some of its representations, which appear in the Kontsevich-Witten theory of two-dimensional topological gravity through the homotopy theory of moduli spaces.
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The main aim of this article is to prove the one-dimensionality of the third algebraic cohomology of the Virasoro algebra with values in the adjoint module. We announced this result in a previous publication with only a sketch of the proof. The detailed proof is provided in the present article. We also show that the third algebraic cohomology of the Witt and the Virasoro algebra with values in the trivial module is one-dimensional. We consider purely algebraic cohomology, i.e. our results are independent of any topology chosen. The vanishing of the third algebraic cohomology of the Witt algebr…
Graded anti-pre-Lie algebraic structures on Witt and Virasoro algebras
本文通过分类Witt代数的不可分解权表示，给出了其上的分次反pre-Lie代数结构及同构分类，并证明了Virasoro代数上不存在满足自然条件的此类结构。
In this paper, we complete the classification of the {\bf Z}-graded modules of the intermediate series over the $q$-analog Virasoro-like algebra $L$. We first construct four classes of irreducible {\bf Z}-graded $L$-modules of the intermediate series. Then we prove that any {\bf Z}-graded $L$-modules of the intermediate series must be the direct sum of some trivial $L$-modules or one of the modules constructed by us.
An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra $U_q(\hat{sl}_2)$. A similar construction is proposed for the elliptic algebra $A_{q,p}(\hat{sl}_2)$.
We give an algebraic proof of the formula on the norm of logarithmic primary of Virasoro algebra, which was proposed by Al. Zamolodchikov. This formula appears in the recursion formula for the norm of Gaiotto state, which guarantees the AGT relation for the four-dimensional SU(2) pure gauge theory.
We study the Madsen-Tillmann spectrum $\C P^\infty_{-1}$ as a quotient of the Mahowald pro-object $\C P^{\infty}_{-\infty}$, which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten.
We identify the algebra of matrix elements of big projective modules in category O with the regular functions on the big Bruhat cell of G. Analogous extensions of the regular representations of the affine Lie and Virasoro algebras yield vertex operator algebras, equipped with two commuting actions with special values of the total central charge. In the generic case, we identify the structure of these VOAs, compute their semi-infinite cohomology. Its superalgebra structure encodes the fusion rules for the associated tensor categories. Full details are provided for the affine sl(2) and Virasoro …
We study the fused currents of the deformed Virasoro algebra (DVA). By constructing a homotopy operator we show that for special values of the parameter of the algebra fused currents pairwise coincide on the cohomologies of the Felder resolution. Within the algebraic approach to lattice models these currents are known to describe neutral excitations of the solid-on-solid (SOS) models in the transfer-matrix picture. It allows us to prove the closeness of the system of excitations for a special nonunitary series of restricted SOS (RSOS) models. Though the results of the algebraic approach to lat…
We study an analog of the AGT relation in five dimensions. We conjecture that the instanton partition function of 5D N=1 pure SU(2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.
Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via $q_R$-conformal symmetries in the Verma modules over the Lie algebra $sl(2,C)$ is found. The analysis and the unraveling of the algebraic structure of these families of hidden symmetries are performed.
A proof of the vanishing of the third cohomology group of the Witt algebra with values in the adjoint module is given. Moreover, we provide a sketch of the proof of the one-dimensionality of the third cohomology group of the Virasoro algebra with values in the adjoint module. The proofs given in the present article are completely algebraic and independent of any underlying topology. They are a generalization of the ones provided by Schlichenmaier, who proved the vanishing of the second cohomology group of the Witt and the Virasoro algebra by using purely algebraic methods. In the case of the t…
By elementary and direct calculations the vanishing of the (algebraic) second Lie algebra cohomology of the Witt and the Virasoro algebra with values in the adjoint module is shown. This yields infinitesimal and formal rigidity or these algebras. The first (and up to now only) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler.
We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL(2). The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL(2), and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in q-alg/9505025. We also consider a discrete analogue of this Poisson algebr…
To a given algebraic curve we assign an infinite family of quantum curves (Schrödinger equations), which are in one-to-one correspondence with, and have the structure of, Virasoro singular vectors. For a spectral curve of a matrix model we build such quantum curves out of an appropriate representation of the Virasoro algebra, encoded in the structure of the $α/β$-deformed matrix integral and its loop equation. We generalize this construction to a large class of algebraic curves by means of a refined topological recursion. We also specialize this construction to various specific matrix models w…
We study algebraic aspects of Kontsevich integrals as generating functions for intersection theory over moduli space and review the derivation of Virasoro and KdV constraints. 1. Intersection numbers 2. The Kontsevich integral 2.1. The main theorem 2.2 Expansion of Z on characters and Schur functions 2.3 Proof of the first part of the Theorem 3. From Grassmannians to KdV 4. Matrix Airy equation and Virasoro highest weight conditions 5. Genus expansion 6. Singular behaviour and Painlev'e equation. 7. Generalization to higher degree potentials
We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices are associated to (the conjugacy classes of) elements s of the Weyl group W of g. For such slices we prove an analogue of the Kostant cross-section theorem for the action of a unipotent group.
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