Lie Algebra Of Differential Operators Research Articles

Lie algebras of differential operators for matrix valued Laguerre type polynomials

Lie algebras of differential operators for matrix valued Laguerre type polynomials

A bispinor formalism for spinning Witten diagrams

We develop a new embedding-space formalism for AdS4 and CFT3 that is useful for evaluating Witten diagrams for operators with spin. The basic variables are Killing spinors for the bulk AdS4 and conformal Killing spinors for the boundary CFT3. The more conventional embedding space coordinates XI for the bulk and PI for the boundary are bilinears in these new variables. We write a simple compact form for the general bulk-boundary propagator, and, for boundary operators of spin ℓ ≥ 1, we determine its conservation properties at the unitarity bound. In our CFT3 formalism, we identify an mathfrak{so} (5, 5) Lie algebra of differential operators that includes the basic weight-shifting operators. These operators, together with a set of differential operators in AdS4, can be used to relate Witten diagrams with spinning external legs to Witten diagrams with only scalar external legs. We provide several applications that include Compton scattering and the evaluation of an R4 contact interaction in AdS4. Finally, we derive bispinor formulas for the bulk-to-bulk propagators of massive spinor and vector gauge fields and evaluate a diagram with spinor exchange.

Lie Subalgebras of Differential Operators in One Variable

Let $$\mathrm{Witt}$$ be the Lie algebra generated by the set $$\{L_i\,\vert \, i \in {{\mathbb {Z}}}\}$$ and $$\mathrm{Vir}$$ its universal central extension. Let $$\mathrm{Diff}(V)$$ be the Lie algebra of differential operators on $$V={{\mathbb {C}}}(\!(z)\!)$$, or $$V={{\mathbb {C}}}(z)$$. We explicitly describe all Lie algebra homomorphisms from $$\mathfrak {sl}(2)$$, $$\mathrm{Witt}$$, and $$\mathrm{Vir}$$ to $$\mathrm{Diff}(V)$$, such that $$L_0$$ acts on V as a first-order differential operator.

Algebro-Geometric Solutions of the Generalized Virasoro Constraints

We will describe algebro-geometric solutions of the KdV hierarchy whose $\tau$-functions in addition satisfy a generalization of the Virasoro constraints (and, in particular, a generalization of the string equation). We show that these solutions are closely related to embeddings of the positive half of the Virasoro algebra into the Lie algebra of differential operators on the circle. Our results are tested against the case of Witten-Kontsevich $\tau$-function. As by-products, we exhibit certain links of our methods with double covers of the projective line equipped with a line bundle and with ${\rm Gl}(n)$-opers on the punctured disk.

On an infinite-dimensional Lie algebra of Virasoro-type

In this paper, we study an infinite-dimensional Lie algebra of Virasoro-type which is realized as an affinization of a two-dimensional Novikov algebra. It is a special deformation of the Lie algebra of differential operators on a circle of order at most 1. There is an explicit construction of a vertex algebra associated with the Lie algebra. We determine all derivations of this Lie algebra in terms of some derivations and centroids of the corresponding Novikov algebra. The universal central extension of this Lie algebra is also determined.

Invariant theory and the $\mathcal{W}_{1+\infty}$ algebra with negative integral central charge

The vertex algebra W_{1+\infty,c} with central charge c may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer n\geq 1, it was conjectured in the physics literature that W_{1+\infty,-n} should have a minimal strong generating set consisting of n^2+2n elements. Using a free field realization of W_{1+\infty,-n} due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of GL_n, we prove this conjecture. A consequence is that the irreducible, highest-weight representations of W_{1+\infty,-n} are parametrized by a closed subvariety of C^{n^2+2n}.

The deformed Lie algebra of differential operators of order at most one

In this paper, we study the algebraic properties of the deformed Lie algebra \documentclass[12pt]{minimal}\begin{document}${\mathcal L}$\end{document}L of differential operators of order at most one. We determine all the derivations, central extensions, and automorphism group of \documentclass[12pt]{minimal}\begin{document}${\mathcal L}$\end{document}L. Furthermore, the reducibility criterion for the Verma modules of the universal central extension of \documentclass[12pt]{minimal}\begin{document}${\mathcal L}$\end{document}L is achieved through a study of the Shapovalov form.

Cohomology of Oriented Tree Diagram Lie Algebras

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this article, we use Hodge Laplacian to study the cohomology of these Lie algebras. The “total rank conjecture” and “b 2-conjecture” for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler–Poincaré principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.

Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.

TWISTED VERTEX OPERATORS AND BERNOULLI POLYNOMIALS

Using general principles in the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. We develop new identities and principles in the theory of vertex operator algebras and their twisted modules, and explain the construction by applying general results, including an identity that we call modified weak associativity, to the Heisenberg vertex operator algebra. This paper gives proofs and further explanations of results announced earlier. It is a generalization to twisted vertex operators of work announced by the second author some time ago, and includes as a special case the proof of the main results of that work.

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

Solvability of the rational quantum integrable systems related to exceptional root spaces $G_2, F_4$ is re-examined and for $E_{6,7,8}$ is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form being written in a certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for {\it arbitrary} values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.

Derivations of the Lie algebras of differential operators

This paper encloses a complete and explicit description of the derivations of the Lie algebra D( M)of all linear differential operators of a smooth manifold M, of its Lie subalgebra D 1( M) of all linear first-order differential operators of M, and of the Poisson algebra S( M) = Pol( T* M) of all polynomial functions on T* M, the symbols of the operators in D( M). It turns out that, in terms of the Chevalley cohomology, H 1( D( M), D( M)) = H DR 1( M), H 1 ( D 1( M), D 1( M)) = H DR 1( M) ⊕ R 2, and H 1( S( M), S( M)) = H DR 1 ( M) ⊕ R. The problem of distinguishing those derivations that generate one-parameter groups of automorphisms and describing these one-parameter groups is also solved.

Lie algebraic characterization of manifolds

Abstract Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.

Formal differential operators, vertex operator algebras and zeta-values, II

We introduce certain correlation functions (graded q-traces) associated to vertex operator algebras and superalgebras which we refer to as n-point functions. These naturally arise in the studies of representations of Lie algebras of differential operators on the circle (J. Lepowsky, to appear, J. Lepowsky, In: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Amer. Math. Soc., Providence, RI, 1999, p. 327, A. Milas, Formal Differential Operators, Vertex Operator Algebras and Zeta-values, I, to appear). We investigate their properties and consider the corresponding graded q-traces in parallel with the passage from genus 0 to genus 1 conformal field theory. By using the vertex operator algebra theory we analyze in detail correlation functions in some particular cases. We obtain elliptic transformation properties for q-traces and the corresponding q-difference equations. In particular, our construction leads to certain correlation functions and q-difference equations investigated by Bloch and Okounkov (Adv. Math. 149 (2000), 1).

Twisted modules for vertex operator algebras and Bernoulli polynomials

This work is a continuation of a series of papers of two of the present authors [L3], [L4], [M1]–[M3], stimulated by work of Bloch [Bl]. In those papers we used the general theory of vertex operator algebras to study central extensions of classical Lie algebras and superalgebras of differential operators on the circle in connection with values of ζ–functions at the negative integers. In the present paper, using general principles of the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call “modified weak associativity.” This paper is an announcement. The detailed proofs will appear elsewhere. More specifically, the present goal is to obtain a new general Jacobi identity for twisted operators, and for related iterates of such operators, extending the previous analogous results in the untwisted setting in our papers mentioned above. As a consequence we obtain twisted constructions of certain central extensions of Lie algebras of differential operators on the circle, combining and extending methods from [L3], [L4], [M1]–[M3], [FLM1], [FLM2] and [DL]. In those earlier papers we used vertex operator techniques to analyze untwisted actions of the Lie algebra D, studied in [Bl], on a module for a Heisenberg Lie algebra of a certain standard type, based on a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form. Now consider an arbitrary isometry ν of period say p, that is, with ν = 1. Here we announce that the corresponding ν–twisted modules carry an action of the Lie algebra D in terms of twisted vertex operators, parametrized by certain quadratic vectors in the untwisted module. In particular, we extend a result from [FLM1], [FLM2], [DL] where actions of the Virasoro algebra were constructed using twisted vertex operators. In addition we explicitly compute certain “correction” terms for the generators of the “Cartan subalgebra” of D that naturally appear in any twisted construction. These correction terms are expressed in terms of special values of certain Bernoulli polynomials. They can in principle be generated, in the theory of vertex operator algebras, by the formal operator ex [FLM1], B.D. gratefully acknowledges partial support from an NSERC Postgraduate Scholarship. J.L. and A.M. gratefully acknowledge partial support from NSF grant DMS-0070800.

Prolongations of Vector Fields on Lie Groups and Homogeneous Spaces

We introduce the notion of the \({\mathfrak{g}}{\mathfrak{l}(V)}\)-prolongation of Lie algebras of differential operators on homogeneous spaces. The \({\mathfrak{g}}{\mathfrak{l}(V)}\)-prolongations are topological invariants that coincide with one-dimensional cohomologies of the corresponding Lie algebras in the case where V is a homogeneous space. We apply the obtained results to the spaces S1 (the Virasoro algebra) and \({\mathbb{R}}^1 \).

SOLVABILITY OF THE F4 INTEGRABLE SYSTEM

It is shown that the F4 rational and trigonometric integrable systems are exactly-solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are invariant with respect to the Weyl group of F4 root system and can be obtained by averaging over an orbit of the Weyl group. An alternative way of finding these variables exploiting a property of duality of the F4 model is presented. It is demonstrated that in these variables the Hamiltonian of each model can be expressed as a quadratic polynomial in the generators of some infinite-dimensional Lie algebra of differential operators in a finite-dimensional representation. Both Hamiltonians preserve the same flag of spaces of polynomials and each subspace of the flag coincides with the finite-dimensional representation space of this algebra. Quasi-exactly-solvable generalization of the rational F4 model depending on two continuous and one discrete parameters is found.

Semisimple Lie Algebras of Differential Operators

Starting from the commutation relations in a complex semisimple Lie algebra $${\mathfrak{g}}$$ , one may obtain a space $${\hat {\mathfrak{g}}}$$ of vector fields on Euclidean space such that $${\mathfrak{g}}$$ and $${\hat {\mathfrak{g}}}$$ are isomorphic when $${\hat {\mathfrak{g}}}$$ is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of $${\hat {\mathfrak{g}}}$$ is a Borel subalgebra $${\mathfrak{b}}$$ . Furthermore, one may adjoin to the vector fields in $${\hat {\mathfrak{g}}}$$ multiplication operators to obtain an $$\mathfrak{h}*$$ -parameter family of distinct presentations of $$\mathfrak{g}$$ as spaces of differential operators, where $$\mathfrak{h}*$$ is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of $$\mathfrak{g}$$ can be presented in this way. All of this is carried out explicitly for arbitrary $$\mathfrak{g}$$ . In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.

Cohomologie de l'algèbre de Lie des opérateurs différentiels sur une variété, à coefficients dans les fonctions

The space of local q-cochains of the Lie algebra of differential operators on a manifold, with coefficients in the space of functions, is naturally graded. The homogeneous terms of a cochain are totally ordered and the derivatives may be symbolized by linear forms on ø m. This leads to a method giving the first three cohomology spaces.

Lifting formulas II

In the present paper we generalize the lifting formulas from [Sh] and obtain the formula for the (2n+2l-1)-cocycle on the LIe algebra of differential operators on a n-dimensional space for arbitrary n and l.