The defining properties of the Kontsevich unoriented graph complex

Abstract

Consider the real vector space of formal sums of non-empty, finite unoriented graphs without multiple edges and loops. Let the vertices of graphs be unlabelled but let every graph γ be endowed with an ordered set of edges E(γ). Denote by Gra the vector space of formal sums of graphs modulo the relation (γ 1 , E(γ 1 )) − sign(σ)(γ 2 , E(γ 2 )) = 0 for topologically equal graphs γ 1 and γ 2 whose edge orderings differ by a permutation σ. The zero class in Gra is represented by sums of graphs that cancel via the above relation. The Lie bracket of graphs with ordered edge sets is defined using the insertion of a graph into vertices of the other one. We give an explicit proof of the theorems which state that the space Gra is a well-defined differential graded Lie algebra: both the Lie bracket [·, ·] and the vertex-expanding differential d = [•−•, ·] respect the calculus modulo zero graphs.

The chromospheric component of coronal bright points

“…Chromospheric low-temperature features appear dark in these channels (but also in other coronal channels). This is caused by the extinction of photons emitted from the hot coronal plasma in overlying neutral H i or He i atoms in cool plasma (Rutten 1999). The AIA EUV data have a 12 s cadence and 0.6 × 0.6 pixel size.…”

The orientation morphism: from graph cocycles to deformations of Poisson structures

“…Having studied the natural differential graded Lie algebra (dgLa) structure on the space of graded skew-symmetric endomorphisms End * , * skew (T poly (M) [1]), we observe that its construction goes in parallel with the dgLa structure on the vector space k Gra i edge i #Vert=:k 1 S k of finite non-oriented graphs with wedge ordering of edges (and without leaves). Referring to [8,9,10,15] (and references therein), as well as to [6,7,14] with explicit examples of calculations in the graph complex, we summarize the set of analogous objects and structures in Table 1 below. Table 1.…”

“…The precise right-hand side is all but written in[9]; still to the best of our knowledge, the exact formula is presented here and on p. 7 below for the first time. -The same applies to Jacobi identity (2) for the Lie bracket of graphs (cf [14]…”

Open problems in the Kontsevich graph construction of Poisson bracket symmetries

“…In the fundamental work [29], see also [27], Willwacher related the unoriented graph complex to generators of the Grothendieck-Teichmüller Lie algebra grt. In what follows, we expect some familiarity with the subject, e.g., on the basis of [22,24] and [2,4,9,19,28] or the lectures [18] available on-line from the Steklov MI RAS; the notation is standard.…”

“…Are the graph cocycles truly graphs ? In the papers [21] (see also [22] and [4,8,28] for a pedagogical review), Kontsevich introduced the graph complex -one of the many -with parity-even vertices, with a wedge ordering of parity-odd edges, and the differential d = [•−•, ·] produced by the graded commutator of graph insertions into vertices. This direction of research was furthered by Willwacher et al [11,14,29]: in particular, in [30] a generating function counts the numbers of nonzero (i.e.…”