2022/12/07 by Oskar Frost, Frost, Oskar
Biochemistry, Genetics and Molecular Biology · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Sphingolipid Metabolism and Signaling
paper · pdf · doi:10.48550/arxiv.2212.03739
openalex publication_date 2022/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well-known that the Lie algebra of homotopy non-trivial degree zero derivations of the properad of strongly homotopy Lie bialgebras Holieb can be identified with the Grothendieck-Teichmuller Lie algebra \mathfrakgrt. We study in this paper the derivation complex of the wheeled closure Holieb^\circlearrowleft (and of its degree shifted version Holiebp,q^\circlearrowleft, ∀ p,q∈ℤ) and establishing a quasi-isomorphism to a version of the Kontsevich graph complex. This result leads us to a surprising conclusion that the Lie algebra of homotopy non-trivial derivations of the wheeled properad Holieb\circlearrowleft can be identified with the direct sum of two copies of \mathfrakgrt. As an illustrative example, we describe explicitly how the famous tetrahedron class in \mathfrakgrt acts as a derivation of Holieb\circlearrowleft in two homotopy inequivalent ways.