DWSGCoeff,               Library  "JINRLIB"                      
    FPLSA,LieCohomology
                                                                     You are   
    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:
    -------
    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. 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.