Mathematics Seminar by Murat Can Aşkaroğulları (İstanbul Technical University)

You are most cordially invited to Yeditepe Mathematics Department Seminars. The details of this week's talk are as follows.

Speaker: Murat Can Aşkaroğulları (İstanbul Technical University)

Title: Leibniz PROP is a crossed presimplicial algebra

Abstract: 

Leibniz algebras, introduced by Loday and Pirashvili [2], are analogues of Lie algebras that are not skew-symmetric. Just as in the Lie case, Leibniz algebras are governed by an operad and can be modeled by an associated PROP [1].

Inspired by the Loday complex of a Leibniz algebra, we define a new set of generators for the Leibniz PROP where specific (1,k)-shuffles are intrinsic to the generators. We prove that the Leibniz PROP is isomorphic as k-linear categories (not as monoidal categories) to the symmetric crossed presimplicial algebra k[(Δ+)opS] where Δ+ is the presimplicial category, but the distributive law between (Δ+)op and the symmetric groups S=⨆n≥1Sn is not the standard one.

In establishing this result, we also extend the standard distributive law between k[(Δ+)op] and k[S] to a distributive law between the nonsymmetric magmatic PROP and Artin's braid monoid k[B] where B=⨆n≥1Bn. Furthermore, our proof yields a description of the boundary maps on the Loday complex as alternating sums of partial boundary maps.

This is a joint work with Atabey Kaygun.

References

1. J.-L. Loday and B. Vallette, Algebraic operads, Grundlehren der mathematischen Wissenschaften, 346, 2012.

2. J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Mathematische Annalen, 296 (1), 139--158, 1993.

3. J. Beck, Distributive laws, In: Sem. on Triples and Categorical Homology Theory, Lecture Notes in Math., No. 80, 119--140, 1969.

Date: Friday, April 3, 2026
Time: 13:00 (Istanbul Time)
Place: Seminar Room