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Distributive laws and Koszulness

Martin Markl — 1996

Annales de l'institut Fourier

Distributive law is a way to compose two algebraic structures, say 𝒰 and 𝒱 , into a more complex algebraic structure 𝒲 . The aim of this paper is to understand distributive laws in terms of operads. The central result says that if the operads corresponding respectively to 𝒰 and 𝒱 are Koszul, then the operad corresponding to 𝒲 is Koszul as well. An application to the cohomology of configuration spaces is given.

G L n -Invariant tensors and graphs

Martin Markl — 2008

Archivum Mathematicum

We describe a correspondence between GL n -invariant tensors and graphs. We then show how this correspondence accommodates various types of symmetries and orientations.

Cohomology operations and the Deligne conjecture

Martin Markl — 2007

Czechoslovak Mathematical Journal

The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.

Homotopy Lie algebras and fundamental groups via deformation theory

Martin MarklStefan Papadima — 1992

Annales de l'institut Fourier

We formulate first results of our larger project based on first fixing some easily accessible invariants of topological spaces (typically the cup product structure in low dimensions) and then studying the variations of more complex invariants such as π * Ω S (the homotopy Lie algebra) or gr * π 1 S (the graded Lie algebra associated to the lower central series of the fundamental group). We prove basic rigidity results and give also an application in low-dimensional topology.

Combinatorial differential geometry and ideal Bianchi–Ricci identities II – the torsion case

Josef JanyškaMartin Markl — 2012

Archivum Mathematicum

This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.

Operads for n -ary algebras – calculations and conjectures

Martin MarklElisabeth Remm — 2011

Archivum Mathematicum

In [8] we studied Koszulity of a family t 𝒜 𝑠𝑠 d n of operads depending on a natural number n and on the degree d of the generating operation. While we proved that, for n 7 , the operad t 𝒜 𝑠𝑠 d n is Koszul if and only if d is even, and while it follows from [4] that t 𝒜 𝑠𝑠 d n is Koszul for d even and arbitrary n , the (non)Koszulity of t 𝒜 𝑠𝑠 d n for d odd and n 8 remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.

Homotopy algebras via resolutions of operads

Markl, Martin — 2000

Proceedings of the 19th Winter School "Geometry and Physics"

Summary: All algebraic objects in this note will be considered over a fixed field k of characteristic zero. If not stated otherwise, all operads live in the category of differential graded vector spaces over k . For standard terminology concerning operads, algebras over operads, etc., see either the original paper by [“The geometry of iterated loop spaces”, Lect. Notes Math. 271 (1972; Zbl 0244.55009)], or an overview [, “La renaissance des opérads”, Sémin. Bourbaki 1994/95, Exp. No. 792, Asterisque...

Free loop spaces and cyclohedra

Markl, Martin — 2003

Proceedings of the 22nd Winter School "Geometry and Physics"

It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an A -operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let 𝒫 be an operad, M a 𝒫 -module and U a 𝒫 -algebra. An M -trace over U consists of a space V and a module homomorphism T : M End U , V over the operad homomorphism 𝒫 End U given by the algebra structure on U . Let 𝒞 1 be the little 1-cubes...

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