Lie Conformal Algebras Research Articles

Universal central extensions of Lie conformal algebras, Part 2: Supercase

In the previous part of this study we considered universal central extensions of Lie conformal algebras and conditions for existence of such extensions. The aim of this paper is some refinement of previous results and extension of these results to Lie conformal superalgebras.

Lie Conformal Algebra Cohomology and the Variational Complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a g-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.

The exceptional Lie algebra E7(−25): multiplets and invariant differential operators

In the present paper, we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional algebra E7(−25). Our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras, which we call ‘conformal Lie algebras’, which have very similar properties to the conformal algebras of n-dimensional Minkowski spacetime. This class of algebras is identified and summarized in a table. Another motivation is related to the AdS/CFT correspondence. We give the multiplets of indecomposable elementary representations, including the necessary data for all relevant invariant differential operators.

Differential Invariants of the 2D Conformal Lie Algebra Action

We consider the Lie algebra that corresponds to the Lie pseudogroup of all conformal transformations on the plane. This conformal Lie algebra is canonically represented as the Lie algebra of holomorphic vector fields in ℝ2≃ℂ. We describe all representations of \(\mathfrak{g}\) via vector fields in J0ℝ2=ℝ3(x,y,u), which project to the canonical representation, and find their algebra of scalar differential invariants.

The $so(d + 2, 2)$ minimal representation and ambient tractors: the conformal geometry of momentum space

Tractor Calculus is a powerful tool for analyzing Weyl invariance; although fundamentally linked to the Cartan connection, it may also be arrived at geometrically by viewing a conformal manifold as the space of null rays in a Lorentzian ambient space. For dimension d conformally flat manifolds we show that the (d+2)-dimensional Fefferman--Graham ambient space corresponds to the momentum space of a massless scalar field. Hence on the one hand the null cone parameterizes physical momentum excitations, while on the other hand, null rays correspond to points in the underlying conformal manifold. This allows us to identify a fundamental set of tractor operators with the generators of conformal symmetries of a scalar field theory in a momentum representation. Moreover, these constitute the minimal representation of the non-compact conformal Lie symmetry algebra of the scalar field with Kostant--Kirillov dimension d+1. Relaxing the conformally flat requirement, we find that while the conformal Lie algebra of tractor operators is deformed by curvature corrections, higher relations in the enveloping algebra corresponding to the minimal representation persist. We also discuss potential applications of these results to physics and conformal geometry.

Non-linear Lie conformal algebras with three generators

We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sl_2. In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras R_{-1}^d and the corresponding freely generated vertex algebras V_{-1}^d, which includes for d=1 the affine vertex algebra of sl_2 at the critical level k=-2. We construct free-field realizations of the algebras V_{-1}^d extending the Wakimoto realization of the affine vertex algebra of sl_2 at the critical level, and we compute their Zhu algebras.

Varieties of dialgebras and conformal algebras

We introduce and study the concept of a variety of dialgebras which is closely related to the concept of a variety of conformal algebras: Each dialgebra of a given variety embeds into an appropriate conformal algebra of the same variety. In particular, the Leibniz algebras are exactly Lie dialgebras, and each Leibniz algebra embeds into a conformal Lie algebra.

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

We extend classical results of Kostant et al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.

Universally defined representations of Lie conformal superalgebras

We distinguish a class of irreducible finite representations of the conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra L is completely determined by commutation relations of L and by the requirement of associative locality of generators. We describe such representations for conformal superalgebras W n , n ≥ 0 , with respect to a natural set of generators. We also consider the problem for superalgebras K n . In particular, we find a universally defined representation for the Neveu–Schwartz conformal superalgebra K 1 and show that the analogues of this representation for n ≥ 2 are not universally defined.

A remark on simplicity of vertex algebras and Lie conformal algebras

I give a short proof of the following algebraic statement: in a simple vertex algebra, the underlying Lie conformal algebra is either abelian, or it is an irreducible central extension of a simple Lie conformal algebra. This provides many examples of non-finite simple Lie conformal algebras, and should prove useful for classification purposes.

Finite vertex algebras and nilpotence

I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (= without nilpotent elements) finite vertex algebra is nilpotent.

Steinberg unitary Lie conformal algebras

In this paper, we study Steinberg unitary Lie conformal algebras, which are universal central extensions of unitary Lie conformal algebras. We describe the kernels of these extensions by means of skew-dihedral homology.

Supersymmetric Vertex Algebras

We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields.

Lie symmetries of semi-linear Schrödinger equations and applications

Conditional Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied in case the mass (or the diffusion constant) is considered as an additional variable and/or where the couplings of the non-linear part have a non-vanishing scaling dimension. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to certain subalgebras of a three-dimensional conformal Lie algebra (conf3)C. The representations of these subalgebras are classified and the complete list of conditionally invariant semi-linear Schrödinger equations is obtained. Applications to the phase-ordering kinetics of simple magnets and to simple particle-reaction models are briefly discussed.

Finite vs affine W-algebras

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra ZhuΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established.

Higher-Dimensional Generalizations of Affine Kac-Moody and Virasoro Conformal Lie Algebras

We discuss the generalizations of the notion of Conformal Algebra and Local Distribution Lie algebras for multi-dimensional bases. We replace the algebra of Laurent polynomials on Open image in new window by an infinite-dimensional representation (with some additional structures) of a simple finite-dimensional Lie algebra Open image in new window in the space of regular functions on the corresponding Grassmann variety Open image in new window that can be described as a ``right'' higher-dimensional generalization of Open image in new window from the point of view of a corresponding group action. For Open image in new window it gives us the usual Vertex Algebra notion. We construct the higher dimensional generalizations of the Virasoro and the Affine Kac-Moody Conformal Lie algebras explicitly and in terms of the Operator Product Expansion.

Irreducible modules over finite simple Lie pseudoalgebras I. Primitive pseudoalgebras of type [formula omitted] and [formula omitted

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C [ ∂ ] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra. The finite (i.e., finitely generated over H ) simple Lie pseudoalgebras were classified in our previous work [B.Bakalov, A.D’Andrea, V.G. Kac, Theory of finite pseudoalgebras, Adv. Math. 162 (2001) 1–140]. In a series of papers, starting with the present one, we classify all irreducible finite modules over finite simple Lie pseudoalgebras.

Dynamical symmetries of semi-linear Schrödinger and diffusion equations

Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra ( conf 3 ) C . We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of ( conf 3 ) C are classified and the complete list of conditionally invariant semi-linear Schrödinger equations is obtained. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed.

Steinberg conformal algebras

The structure of Steinberg conformal algebras is studied; these are analogues of Steinberg groups (algebras, superalgebras). A Steinberg conformal algebra is defined as an abstract algebra by a system of generators and relations between the generators. It is proved that a Steinberg conformal algebra is the universal central extension of the corresponding conformal Lie algebra; the kernel of this extension is calculated.

IFFT-equivariant quantizations

The existence and uniqueness of quantizations that are equivariant with respect to conformal and projective Lie algebras of vector fields were recently obtained by Duval, Lecomte and Ovsienko. In order to do so, they computed spectra of some Casimir operators. We give an explicit formula for those spectra in the general framework of I FFT-algebras classified by Kobayashi and Nagano. We also define t ree-like subsets of eigenspaces of those operators in which eigenvalues can be compared to show the existence of IFFT-equivariant quantizations. We apply our results to prove the existence and uniqueness of quantizations that are equivariant with respect to the infinitesimal action of the symplectic (resp. pseudo-orhogonal) group on the symplectic (resp. pseudo-orthogonal) Grassmann manifold.