Новые подходы к весовым системам, строящимся по алгебрам ЛиNew approaches to Lie algebra weight systems

To a finite type knot invariant, a weight system can be associated, which is a function on chord diagrams satisfying so-called 4-term relations. In particular, a weight system can be associated to any metrized Lie algebra, and any metrized Lie superalgebra. However, computation of  these weight systems is complicated. We define an invariant of permutations taking values in the center of the universal enveloping algebra of gl_N . The restriction of this invariant to involutions without fixed points (such an involution determines a chord diagram) coincides with the value of the gl_N -weight system on this chord diagram. We describe the recursion allowing one to compute the gl_N -invariant of permutations. We extend this construction to the weight system associated to the Lie superalgebra gl(m|n). Then we prove that the gl(m|n)-weight system is equivalent to the gl-one, under the substitution C_0 = m − n.