Algebra, geometry and physics. (Q2702032)

In this work, the author reviews some of the connections that have recently appeared between Algebra, Geometry and Physics. Unlike classical Physics that mainly uses mathematics developed time ago, modern Physics requires new mathematical tools. This situation has originated new ways for treating geometrical and algebraic problems, arising some surprising results. Noncommutative Geometry is one of the most important fields that has emerged from this new relation. The starting idea is to express geometrical results in a language that permits its extension to situations that exceed usual geometry. Some intuitive notions disappear in this step but geometrical information subsists.NEWLINENEWLINEAlgebraic language has been the most appropriate language to extend Geometry. The idea, then, is to translate geometrical notions to algebraic notions. Once the suitable algebraic interpretation has been founded between all possible algebraic interpretations related to a geometrical notion, there occurs frequently that the noncommutative case presents situations and results that did not appear in the classical geometrical situation but gives new ideas on it.NEWLINENEWLINEWith all this in mind, the author commences presenting some geometrical notions and its algebraic interpretation. The classical coordinate ring \(A({\mathbb S}^2)\) on the sphere \({\mathbb S}^2\) is used as a geometrical construction that leads us to the algebraic notion of Hopf algebras. Some examples of noncommutative geometry are presented; for instance, quantum groups obtained from the deformation of algebras of functions of classical groups are described as some examples of Hopf algebras belonging to this context.NEWLINENEWLINEThe translation of differential calculus on Lie groups to the noncommutative case is also treated in accordance with the ideas introduced by Woronowicz. Some notions of noncommutative principal bundles also appear.NEWLINENEWLINEQuantum groups of Drinfeld-Jimbo are introduced as another way to obtain deformations of algebras. In this case, the starting point is a simple Lie algebra. This type of algebra is not easy to deform, so its enveloping algebra is chosen to be deformed by using a deformation parameter. The paper ends making an approach to quantum groups from representation theory of finite algebras.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00009].