Math Sciences Grad Student, Eric Berry's Talk: "On Koszul Duality & Deformation Quantization"
- Wednesday, September 19, 2018 from 4:10pm to 5:00pm
- Wilson Hall, 1-144 - view map
In this talk we will introduce and discuss the ideas of Koszul duality and deformation quantization, both of which deal with deforming algebraic structure. Koszul duality is the idea that deforming a point in an object is controlled by some vector space together with some additional structure on that vector space. Algebraically, deforming a point can be thought of as deforming an augmented algebra, and in the case of associative algebras, it turns out that deformations are controlled by another associative algebra! A classical example is the Koszul duality between the symmetric algebra of a finite dimensional vector space, and the exterior algebra of the dual vector space. Deformation quantization deforms algebraic structure along a different line. Namely, it has to do with deforming the multiplicative structure of algebras. This has a physical interpretation of quantizing the algebra of classical observables of a classical mechanical system into the algebra of quantum observables of the quantized system.
It is natural to wonder if there is any relationship between these two different notions of deforming algebraic structure. Recently, for a particular example, it was shown that deformation quantization in fact respects Koszul duality, in the sense that if we apply deformation quantization to two Koszul dual algebras, then the resulting deformation quantizations are again Koszul dual! This talk will be a tour through the ideas necessary to make the previous statement more precise, and will end with some thoughts and questions regarding extending the previous example to a more general setting.
- Department of Mathematical Sciences