Language: C

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DWSGCoeff - computation of DeWitt-Seeley-Gilky coefficients;
FPLSA - computation of finitely resented Lie algebras and superalgebras;
LieCohomology - computation of cohomologies of Lie algebras and superalgebras.

DWSGCoeff - COMPUTATION OF DEWITT-SEELEY-GILKY COEFFICIENTS

The DWSGCoeff program is intended for calculation of asymptotic spectral invariants (heat kernel coefficients) of elliptic differential operators acting on compact closed curved manifolds with torsion and gauge connection.

The program (file DWSG.c), the instruction on compilation and use, examples of input files can be received from the author.

Method:

The algorithm of calculation is based on the Vidom's covariant generalization of pseudodifferential calculus.
The C text is compiled in two executed files. One of them calculates the coincidence limits of covariant derivatives of the phase and transport functions constituting the basis of Widom's approach. These limits, being universal (i.e., not dependent on type of the operator) geometrical characteristics of manifold, are written to a disk and then they are used by another executed file for computing the DWSG coefficients of particular operators.

References:

1. V.P. Gusynin, V.V. Kornyak. Symbolic Computation of DeWitt Seeley Gilkey Coefficients on Curved Manifolds. Journal of Symbolic Computation. 1994. V. 17, No.3. p.283-294.
2. V.P. Gusynin, V.V. Kornyak.
   DeWitt-Seeley-Gilkey coefficients for nonminimal operators in curved space. Fundamental and Applied Mathematics. 1999. V. 5, No. 3. p.649-674. (in Russian)

FPLSA - COMPUTATION OF FINITELY PRESENTED LIE ALGEBRAS
AND SUPERALGEBRAS

Authors: V.P.Gerdt, V.V. Kornyak

The FPLSA program is intended for constructing the complete system of relations (Groebner basis), basis elements and commutator table for Lie algebras and superalgebras defined by finite set of generators satisfying to finite set of relations. The program outputs also the Hilbert series of the constructed algebra and, in the case the input data contain arbitrary parameters, the list of parametric expressions which zero values may cause branching in the structure of the algebra.

The program (file FPLSA4.c) and auxiliary files (initiating file - FPLSA4.ini, message file - FPLSA4.msg, examples of input data, the instruction on compilation and use) are available from the author.

Method:

Computation of noncommutative and nonassociative Groebner basis of ideal in free Lie (super)algebra. As a basis of free Lie (super)algebra the Hall regular monomials are used.

References:

1. V.P. Gerdt, V.V. Kornyak. Construction of Finitely Presented Lie Algebras and Superalgebras. Journal of Symbolic Computation. 1996. V. 21, No 3. p.337-349.
2. V.P. Gerdt, V.V. Kornyak. Program for Constructing a Complete System of Relations, Basis Elements, and Commutator Table for Finitely Presented Lie Algebras and Superalgebras. Programming and Computer Software, Vol.23, No.3, 1997, pp.164-172.

LieCohomology - COMPUTATION OF COHOMOLOGIES OF LIE ALGEBRAS AND
SUPERALGEBRAS

Author: V.V. Kornyak

The LieCohomology program is intended for computing nontrivial cohomology classes of finitedimensional and graded infinitedimensional Lie algebras and superalgebras with coefficients in the trivial, adjoint and coadjoint modules. The algebra can be defined via basis elements and commutator table. For some Lie (super)algebras of vector fields (general W(n|m) and special S(n|m) vectorial algebras; Poisson algebra Po(2n|m), Hamilton H(2n|m) and special Hamilton algebra SH(0|m); contact algebra K(2n+1|m); Buttin B(n), Leites Le(n) algebras and their special forms SB(n) and SLe(n); odd contact algebra M(n) and its special form SM(n)) the program constructs basis elements and their commutators automatically.

The program (file LieCoho1.c) and auxiliary files (initiating file - LieCoho1.ini, message file - LieCoho1.msg, examples of input data, the instruction on compilation and use) are available from the author.

Method:

Construction of the part of cochain complex, corresponding to given cohomological dimension (cochain degree) and grade, and computation of basis elements of the quotient space of the cocycle space with respect to subspace of coboundaries.

References:

1. V.V. Kornyak. Computation of Cohomologies of Lie Superalgebras: Algorithm and Implementation. Programming and Computer Software, Vol. 27, No. 3, 2001, p.142-145.
2. V.V. Kornyak. Computation of Cohomology of Lie Superalgebras of Vector Fields. International Journal of Modern Physics C. 2000. V.11, No.2. p.397-414.

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