Conformal Lie superalgebras and moduli spaces

A conformal Lie superalgebra is a superextension of the centerless Virasoro algebra W—the Lie algebra of complex vector fields on the circle. The algebras of Ramond and Neveu-Schwarz are not the only examples of such superalgebras. All known superconformal algebras can be obtained as comlexifications of Lie superalgebras of vector fields on a supercircle with an additional structure. For every such superalgebra

a class of geometric objects—complex

— is defined. For the superalgebras of Neveu-Schwarz and Ramond they are super Riemann surfaces with punctures of different kinds. We construct moduli superspaces for compact

, and show that the superalgebra

acts infinitesimally on the corresponding moduli space.     

Semiclassical Dynamics¶with Exponentially Small Error Estimates

We construct approximate solutions to the time–dependent Schr?dinger?equation

Chaotic Expansions of Elements of the Universal Enveloping Superalgebra Associated with a Z2-graded Quantum Stochastic Calculus

Given a polynomial function f of classical stochastic integrator processes whose differentials satisfy a closed Ito multiplication table, we can express the stochastic derivative of f as

Block averaging renormalization group for lattice and continuum euclidean fermions: Expected and unexpected results

Simple relations, given by the trace and supertrace, between expectation values of tensor operators connected with a Lie superalgebra g are shown to be generated by the Weyl group of a Lie algebra 0 acting on a Lie superalgebra.     

Canonical Basis for Quantum {\mathfrak{osp}(1|2)}

We introduce a modified quantum enveloping algebra as well as a (modified) covering quantum algebra for the ortho-symplectic Lie superalgebra ${\mathfrak{osp}(1|2)}$ . Then we formulate and compute the corresponding canonical bases, and relate them to the counterpart for ${\mathfrak{sl}(2)}$ . This provides a first example of canonical basis for quantum superalgebras.     

Freely Generated Vertex Algebras and Non–Linear Lie Conformal Algebras

We introduce the notion of a non–linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique, up to isomorphism, non–linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras.Acknowledgement. We would like to thank M. Artin, B. Bakalov, A. Dandrea and P. Etingof for useful discussions. This research was conducted by A. De Sole for the Clay Mathematics Institute. The paper was partially supported by the NSF grant DMS0201017.     

Superconformal Structures on Generalized Calabi-Yau Metric Manifolds

We construct an embedding of two commuting copies of the N = 2 superconformal vertex algebra in the space of global sections of the twisted chiral-anti-chiral de Rham complex of a generalized Calabi-Yau metric manifold, including the case when there is a non-trivial H-flux and non-vanishing dilaton. The 4 corresponding BRST charges are well defined on any generalized Kähler manifold. This allows one to consider the half-twisted model defining thus the chiral de Rham complex of a generalized Kähler manifold. The classical limit of this result allows one to recover the celebrated generalized Kähler identities as the degree zero part of an infinite dimensional Lie superalgebra attached to any generalized Kähler manifold. As a byproduct of our study we investigate the properties of generalized Calabi-Yau metric manifolds in the Lie algebroid setting.     

Number Operator Algebras and Deformations of ε-Poisson Algebras

It is well known that the Lie algebra structure on quantum algebras gives rise to a Poisson algebra structure on classical algebras as the Planck constant goes to 0. We show that this correspondence still holds in the generalization of superalgebra introduced by Scheunert, called -algebra. We illustrate this with the example of Number Operator Algebras, a new kind of object that we have defined and classified under some assumptions.     

Pure E 0-Semigroups and Absorbing States

An E ₀-semigroup acting on is called pure if its tail von Neumann algebra is trivial in the sense that

Fluid Dynamical Profiles and Constants of Motion¶from d-Branes

Various fluid mechanical systems enjoy a hidden, higher-dimensional dynamical Poincaré symmetry, which arises owing to their descent from a Nambu–Goto action. Also, for the same reason, there are equivalence transformations between different models. These interconnections, summarized by the diagram below, are discussed in our paper.

and as Vertex Operator Extensions¶of Dual Affine Algebras

We discover a realisation of the affine Lie superalgebra and of the exceptional affine superalgebra as vertex operator extensions of two algebras with “dual” levels (and an auxiliary level-1 algebra). The duality relation between the levels is . We construct the representation of on a sum of tensor products of , , and modules and decompose it into a direct sum over the spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to is traced to the properties of embeddings into and their relation with the dual pairs. Conversely, we show how the representations are constructed from representations. Received: 29 July 1999 / Accepted: 6 February 2000     

Lie Superalgebra Structures in <Emphasis Type="Italic">H</Emphasis><Superscript>⋅</Superscript> (<Emphasis Type="Italic">g</Emphasis>;<Emphasis Type="Italic">g</Emphasis>)

Let g=vect(M) be the Lie (super)algebra of vector fields on any connected (super)manifold M; let - be the change of parity functor, C ⁱ and H ⁱ the space of i-chains and i-cohomology. The Nijenhuis bracket makes into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in L ^* and for any Lie superalgebra g. We use a Mathematica-based package SuperLie (already proven useful in various problems) to explicitly describe the algebras l ^* for some simple finite dimensional Lie superalgebras g and their relatives - the nontrivial central extensions or derivation algebras of the considered simple ones.     

Differential geometry of the quantum supergroupGL q (1/1)

We construct a right-invariant differential calculus on the quantum supergroupGL _q (1/1) and we show that the quantum Lie algebra generators satisfy the undeformed Lie superalgebra. The deformation becomes apparent when one studies the comultiplication for these generators. We bring the algebra into the standard Drinfeld-Jimbo form by performing a suitable change of variables, and we check the consistency of the map with the induced comultiplication.     

Nonlinear Reformulation of Heisenberg's Dynamics

A structural similarity between Classical Mechanics (CM) and Quantum Mechanics (QM) was revealed by P.A.M. Dirac in terms of Lie Algebras: while in CM the dynamics is determined by the Lie algebra of Poisson brackets on the manifold of scalar fields for classical position/momentum observables Q/P, QM evolves (in Heisenberg's picture) according to the formally similar Lie algebra of commutator brackets of the corresponding operators:

On the Road Map of Vogel’s Plane

We define “population” of Vogel’s plane as points for which universal character of adjoint representation is regular in the finite plane of its argument. It is shown that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. We find all their solutions: classical series of simple Lie algebras (including an “odd symplectic” one), \({D_{2,1,\lambda}}\) superalgebra, the line of sl(2) algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely \({knm=4k+4n+2m+12,}\) contains all simple Lie algebras, except so\({(2N+1).}\) Among isolated solutions are, besides exceptional simple Lie algebras, so called \({\mathfrak{e}_{7\frac{1}{2}}}\) algebra and also two other similar unidentified objects with positive dimensions. In addition, there are 47 isolated solutions in “unphysical semiplane” with negative dimensions. Isolated solutions mainly belong to the few lines in Vogel plane, including some rows of Freudenthal magic square. Universal dimension formulae have an integer values on all these solutions at least for first three symmetric powers of adjoint representation.     

A new quantum group associated with a ‘nonstandard’ braid group representation

A new quantum group is derived from a nonstandard braid group representation by employing the Faddeev-Reshetikhin-Takhtajan constructive method. The classical limit is not a Lie superalgebra, despite relations like x ²–y ²=0. We classify all finite-dimensional irreducible representations of the new Hopf algebra and find only one- and two-dimensional ones.     

Markovian Anderson Model: Bounds for the Rate of Propagation

We consider the Anderson model in with potentials whose values at any site of the lattice are Markovian independent random functions of time. For solutions to the time-dependent Schr?dinger equation we show under some conditions that with probability 1