Lectures on quantum groups

Since its origin about ten years ago, the theory of quantum groups has become one of the most fascinating topics of modern mathematics, with numerous applications to several sometimes rather disparate areas, including low-dimensional topology and mathematical physics. This book is one of the first expositions that is specifically directed to students who have no previous knowledge of the subject. The only prerequisite, in addition to standard linear algebra, is some acquaintance with the classical theory of complex semisimple Lie algebras.Starting with the quantum analog of $SL_2$, the author carefully leads the reader through all the details necessary for full understanding of the subject, particularly emphasizing similarities and differences with the classical theory. The final chapters of the book describe the Kashiwara-Lusztig theory of so-called crystal (or canonical) bases in representations of complex semisimple Lie algebra. The choice of the topics and the style of exposition make Jantzen's book an excellent textbook for a one-semester course on quantum groups.

Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as a Hopf algebra The quantized enveloping algebra $U_q(\mathfrak g)$ Representations of $U_q(\mathfrak g)$ Examples of representations The center and bilinear forms $R$-matrices and $k_q[G]$ Braid group actions and PBW type basis Proof of proposition 8.28 Crystal bases I Crystal bases II Crystal bases III References Notations Index.