An introduction to free stable distributions

Free probability is a fast growing area of research devoted to the study of non-commutative probability spaces. It brings together many different fields of mathematics, for example, operator algebras, random matrices, and combinatorics. Free stable distributions play a prominent role in free probability. In this talk, I firstly introduce the notion of free independence and give some examples. Secondly, I explain some relations between classical infinite divisibility and free infinite divisibility. Finally, I show that free stable distributions are classical infinite divisible if \alpha less than or equal to 1, and symmetric free stable distributions are not classical infinite divisible if \alpha larger than 1. More results are given in my joint work with T. Simon and T. Hasebe.