Deformations of algebras in noncommutative geometry

Abstract

These are significantly expanded lecture notes for the author's minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of the deformation theory of associative algebras in terms of the Hochschild cochain complex as well as quantization of Poisson structures, and Kontsevich's formality theorem in the smooth setting. We then discuss quantization and deformation via Calabi-Yau algebras and potentials. Examples discussed include Weyl algebras, enveloping algebras of Lie algebras, symplectic reflection algebras, quasihomogeneous isolated hypersurface singularities (including du Val singularities), and Calabi-Yau algebras.

A few minor corrections after publication (e.g., continuity of star products)

References in corpus (3)

• Etingof conjecture for quantized quiver varieties
• On global deformation quantization in the algebraic case
• Graded Calabi Yau Algebras of dimension 3

Cited by in corpus (6)

• Graded twisted Calabi-Yau algebras are generalized Artin-Schelter regular
• Quantizations of local surfaces and rebel instantons
• Deformations of path algebras of quivers with relations
• Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties
• On Lie algebroid over algebraic spaces
• Simply-laced quantum connections generalising KZ