We begin by discussing motivation for our consideration of a structure called a Lie 2-algebra, in particular an important class of Lie 2-algebras are the Courant Algebroids introduced in 1990 by Courant. We wish to attach some natural definitions from operad theory, mainly the notion of a module over an algebra, to Lie 2-algebras and hence to Courant algebroids. To this end our goal is to show that Lie 2-algebras can be described as what are called \emph{homotopy algebras over an operad}. Describing Lie 2-algebras using operads also solves the problem of showing that the equations defining a Lie 2-algebra are consistent.
Our technical discussion begins by introducing some notions from operad theory, which is a generalization of the theory of operations on a set and their compositions. We define the idea of a quadratic operad and a homotopy algebra over a quadratic operad. We then proceed to describe Lie 2-algebras as homotopy algebras over a given quadratic operad using a theorem of Ginzburg and Kapranov.
Next we briefly discuss the structure of a braided monoidal category. Following this, motivated by our discussion of braided monoidal categories, a new structure is introduced, which we call a commutative 2-algebra. As with the Lie 2-algebra case we show how a commutative 2-algebra can be seen as a homotopy algebra over a particular quadratic operad.
Finally some technical results used in previous theorems are mentioned. In discussing these technical results we apply some ideas about distributive laws and Koszul operads.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/31946 |
| Date | 11 January 2012 |
| Creators | Squires, Travis |
| Contributors | Arkhipov, Sergey |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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