Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The defining properties of the Kontsevich unoriented graph complex   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 $\gamma$ be endowed with an ordered set of edges
$\mathsf{E}(\gamma)$. Denote by Gra the vector space of formal sums of graphs
modulo the relation $(\gamma_1,\mathsf{E}(\gamma_1))-\text{sign}(\sigma)
(\gamma_2,\mathsf{E}(\gamma_2)) = 0$ for topologically equal graphs $\gamma_1$
and $\gamma_2$ whose edge orderings differ by a permutation $\sigma$. 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 $[{\cdot},{\cdot}]$ and
the vertex\/-\/expanding differential ${\mathrm
d}=[{\bullet}\!{-}\!{\bullet},{\cdot}]$ respect the calculus modulo zero
graphs.

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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: cs
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg