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DWSGCoeff, Library "JINRLIB" 
FPLSA,LieCohomology
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Author: V.V.Kornyak 
Language: C visitor here.
PROGRAMS FOR STUDY OF NONCOMMUTATIVE AND NONASSOCIATIVE
STRUCTURES IN PROBLEMS OF MATHEMATICAL PHYSICS
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:
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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:
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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:
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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:
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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:
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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:
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1. 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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