## Tuesday Seminar on Topology

Seminar information archive ～10/09｜Next seminar｜Future seminars 10/10～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

### 2018/03/30

15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)

**Matteo Felder**(University of Geneva)Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)

[ Abstract ]

The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.

The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.