An introduction to L∞-algebras and their homotopy theory for the working mathematician
Abstract
In this paper, we give a detailed introduction to the theory of (curved) [Formula: see text]-algebras and [Formula: see text]-morphisms, avoiding the concept of operads and providing explicit formulas. In particular, we recall the notion of (curved) Maurer-Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of [Formula: see text]-algebras and [Formula: see text]-modules. In particular, one can interpret [Formula: see text]-morphisms and morphisms of [Formula: see text]-modules as Maurer-Cartan elements in certain [Formula: see text]-algebras, and we show that twisting the morphisms with equivalent Maurer-Cartan elements yields homotopic morphisms. We hope that these notes provide an accessible entry point to the theory of [Formula: see text]-algebras.
- Publication:
-
Reviews in Mathematical Physics
- Pub Date:
- February 2024
- DOI:
- 10.1142/S0129055X23300066
- arXiv:
- arXiv:2207.01861
- Bibcode:
- 2024RvMaP..3630006K
- Keywords:
-
- Mathematics - Quantum Algebra;
- 17B55;
- 16E45
- E-Print:
- 62 pages, comments are welcome!