# Technical report 7/2006

## {\bf SuperLie}. A package for Lie algebra and Super Algebra computations in {\it Mathematica} software system

### Pavel Grozman and Dimitry A. Leites

**Contact the author:** Please use for correspondence this email.**Submission date: **23. Aug. 2006**Pages: 170****paper prepared for:** Springer**MSC-Numbers: **68-04, 68W30**Keywords and phrases: **Lie algebra, Lie superalgebra, scientific computing**Download full preprint:** PDF (1246 kB)**Abstract:**

A *Mathematica*-based package **SuperLie** for
the study of Lie (super)algebras and their cohomology is offered.
Among applications we find:

(1) calculation of the analogs of the Riemannian curvature tensors
needed to write Supergravity Equations for any *N*-extended Minkowski
superspace and to classify possible models for these superspaces;

(2) description of the analogue of the curvature tensor for nonlinear nonholonomic constraints and the fields of solids or their surfaces, e.g., cones, as in optimal control;

(3) a new method for the study of formal integrability of differential equations (numerical methods can provide with individual solutions but are unable, generally, to study qualitative behavior, e.g., stability).

All the above problems can be expressed in terms of Lie algebra
cohomology and **SuperLie** makes it possible to

(1) determine Lie algebras as subalgebras over integers as well as over various fields of any characteristic, via defining relations, from Cartan matrix, realize via vector fields, as polynomials with Poisson or contact (Legendre) bracket, and so on,

(2) determine various modules over these Lie algebras (tensors; modules with a vacuum vector; submodules, quotients, duals, and so on), and calculate the bracket as well as the action of the Lie algebra on its module,

(3) solve systems of equations with scalar and (linear) vector unknowns; find the kernel and the image of a linear operator;

(4) calculates the exterior differential in various complexes;

(5) lists of ``natural'' operators, i.e., the operators between sections of tensor fields (or jets) invariant with respect to the group of diffeomorphism or its subgroup;

(6) list central extensions and deformations of Lie algebras and Lie suepralgebras.

A partial list of open problems, currently under study, is offered (separately): the students in search of a topic for a M.S. or a Ph.D. thesis, as well as other interested, are invited to join forces.