Lectures on differentials, generalized differentials and on some examples related to theoretical physics

Lectures on graded differential algebras and noncommutative geometry

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.

Lectures on Noncommutative Geometry

These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results, in particular, noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative differential forms and double-tangent bundles.

A lecture on Kac--Moody Lie algebras of the arithmetic type

We name an indecomposable symmetrizable generalized Cartan matrix $A$ and the corresponding Kac--Moody Lie algebra ${\goth g} ^\prime (A)$ {\it of the arithmetic type} if for any $β\in Q$ with $(β| β)<0$ there exist $n(β)\in {\Bbb N}$ and an imaginary root $α\in Δ^{im}$ such that $n(β)β\equiv α\mod Ker\ (.|.)$ on $Q$. Here $Q$ is the root lattice. This generalizes "symmetrizable hyperbolic" type of Kac and Moody. We show that generalized Cartan matrices of the arithmetic type are divided in $4$ types: (a) finite, (b) affine, (c) rank two, and (d) arithmetic hyperbolic type. The last type is ve…

覆水难收：关于不可逆对称性的 TASI 讲义

What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries

本文综述了不同时空维度中不可逆对称性的最新进展，从伊辛模型的不可逆 Kramers-Wannier 对偶构造出发，最终给出中性π介子衰变的新解释。

An introduction to quantum groups and non-commutative differential calculus

An introduction to quantum groups and non-commutative differential calculus (Lecture at the III Workshop on Differential Geometry, Granada, September 1994)

Differential and integral forms on non-commutative algebras

An extended summary of the lecture course given at the V School on Geometry and Physics, Białoweża 2016, in which an algebraic approach to differentiation and integration that is characteristic for non-commutative geometry is described.