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\centerline{\bfb Calculation of central generators}
\centerline{\bfb of the universal enveloping algebras}
\centerline{\bfb and Vassiliev--Kontsevich weight systems}
\bigskip
\centerline{\slb S.~Duzhin, A.~Kaishev}
\bigskip
\centerline{\ss Laboratory of problems of small dimension, PSI}
\bigskip

The set of Vassiliev knot invariants [1] forms a filtered Lie
algebra. The elements of the associated graded algebra
referred to as weight systems, admit a combinatorial description
as functions on the set of chord diagrams that satisfy 1 and 4 term
relations.
\smallskip

   With each positive integer $n$ and any semi-simple Lie algebra $\g$
we can associate [2] a weight system $T_n^{\g}$ of degree $n$
with values in the universal enveloping algebra $U(\g)$.
The image of $T_n^{\g}$ is always contained in the centre of $U(\g)$,
whose elements can be represented as polynomials in a finite
number of commuting generators. Among these generators,
there is a distinguished element $C$ of degree 2 (Casimir element),
and the initial conjecture was that the mapping $T_n^{\g}$
can always be expressed in terms of $C$ --- a conjecture
confirmed by manual calculations for small values of the degree.
\smallskip

    Using FORM [3], we have calculated the tables of polynomials
corresponding to chord diagrams of degree up to 7 for $\g=\spl_2$
and up to 6 for $\g=\spl_3$ which showed this conjecture to be
false.
\smallskip

    In the process of these computations, we have found the
systems of generators of the center of $U(\g)$ for the simple
Lie algebras of small dimension $A_1$, $A_2$, $A_3$, $A_4$,
$B_2$, $B_3$, $C_3$ and transformed them to the lexicographic
canonical form. We hope that similar results for the algebras
$A_5$, $D_4$, $G_2$ will be ready in a week or two.
\smallskip

    The work was organized as follows.
\smallskip

    Step 1. Find a basis $x_1,\dots,x_m$ of the given Lie algebra $\g$,
the commutator relations between them, and the adjoint basis
$x_1^*,\dots,x_m^*$ with respect to the invariant bilinear form
$(x,y)\rightarrow \tr(xy)$.
This was done by means of Maple programs {\tt fndbas*.ms} [4].
\smallskip

    Step 2. Find a system of generators in the algebra of invariant
polynomials on the space $\g$ and the corresponding system of generators
in the center of the universal enveloping algebra $U(\g)$: Maple programs
{\tt sysgen*.ms} [4].
\smallskip

    Step 3. Check that the generators found in step 2 really belong to
the center of $U(\g)$: FORM program {\tt chkcnt*.frm} [4].
\smallskip

    Step 4. Starting from the list of chord diagrams of given order [5],
generate the list of elements of $U(\g)$ that correspond to these
chord diagrams by the construction of Kontsevich: Pascal programs
{\tt ueainv*.pas} [4].
\smallskip

    Step 5. Express the polynomials found in step 4 through
the generators of the center found in step 2: FORM program
{\tt exprcg*.frm} [4].
\medskip

\centerline{\bf References}
\medskip

[1] S.~Chmutov, S.~Duzhin, S.~Lando. Vassiliev Knot Invariants, I, II, III.
    Adv. in Soviet Math. 21 (1994), p. 117--126.
\smallskip

[2] M.~Kontsevich, Vassiliev's knot invariants, Adv. in Soviet Math.,
    vol.~16, Part 2, pp. 137--150, 1993.
\smallskip

[3] Jos Vermaseren. Computer algebraic system FORM. User's manual.
\smallskip

[4] S.~Duzhin, A.~Kaishev. Programs and data on UEA. ---
    {\tt ftp://math.botik.ru/pub/uea/}.
\smallskip

[5] D.~Bar-Natan. Programs and data on Vassiliev invariants. ---
    {\tt ftp://math.harvard.edu/dror/}.
\bigskip

\rightline{\tt duzhin@math.botik.yaroslavl.su}
\rightline{\tt AnIKa@math.botik.yaroslavl.su}

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