Vol 52, No 2 (2018)

Finite-order invariants (Vassiliev invariants) of knots are expressed in terms of weight systems, that is, functions on chord diagrams (embedded graphs with a single vertex) satisfying the four-term relations. Weight systems have graph analogues, the so-called 4-invariants of graphs, i.e., functions on graphs that satisfy the four-term relations for graphs. Each 4-invariant determines a weight system.

The notion of a weight system is naturally generalized to the case of embedded graphs with an arbitrary number of vertices. Such embedded graphs correspond to links; to each component of a link there corresponds a vertex of an embedded graph. Recently, two approaches have been suggested to extend the notion of 4-invariants of graphs to the case of combinatorial structures corresponding to embedded graphs with an arbitrary number of vertices. The first approach is due to V. Kleptsyn and E. Smirnov, who considered functions on Lagrangian subspaces in a 2n-dimensional space over F2 endowed with a standard symplectic form and introduced four-term relations for them. The second approach, due to V. Zhukov and S. Lando, gives four-term relations for functions on binary delta-matroids.

In this paper, these two approaches are proved to be equivalent.

Let G be a connected reductive algebraic group over ℂ, and let Λ_G⁺ be the monoid of dominant weights of G. We construct integrable crystals B^G(λ), λ ∈ Λ_G⁺, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group of G. We also construct tensor product maps \(P{\lambda _1},{\lambda _2}:{B^G}({\lambda _2}) \to {B^G}({\lambda _1} + {\lambda _2}) \cup \{ 0\} \) in terms of multiplication in generalized transversal slices. Let L ⊂ G be a Levi subgroup of G. We describe the functor Res_L^G: Rep(G) → Rep(L) of restriction to L in terms of the hyperbolic localization functors for generalized transversal slices.

Let G be a finite Abelian group acting (linearly) on space ℝⁿ and, therefore, on its complexification ℂⁿ, and let W be the real part of the quotient ℂⁿ/G (in the general case, W ≠ ℝⁿ/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the residue bilinear form on the G-invariant part of the quotient of the space of germs of n-forms on (ℝⁿ, 0) by the subspace of forms divisible by the 1-form under consideration.

Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected but the connected components are not distinguished by the linking numbers with the connected components of the curve.

We propose and study elements of potential theory for the sub-Laplacian on homogeneous Carnot groups. In particular, we show the continuity of the single-layer potential and establish Plemelj-type jump relations for the double-layer potential. As a consequence, we derive a formula for the trace on smooth surfaces of the Newton potential for the sub-Laplacian. Using this, we construct a sub-Laplacian version of Kac’s boundary value problem.