Today I shall talk about quantum groups. First, I shall tell about my understanding of what quantum groups are. A usual Lie group is simultaneously a smooth manifold and a group. I shall not discuss the group structure at the moment, but I shall talk a little about the structure of a smooth manifold. There exist various definitions of smooth manifolds. One of them is algebraic; it frequently turns out to be most useful from the computational point of view. The general principle of computations in mathematics is that everything must be reduced to algebraic problems, which can be solved algorithmically. How can we replace a construction as geometric as a smooth manifold by a purely algebraic notion? For this purpose, instead of a smooth manifold M, we consider the algebra A(M) of smooth (real-valued) compactly supported functions on M. “Compactly supported” means that each function vanishes outside some compact set. If the manifold is compact, then this requirement is not needed. For compact manifolds, the entire approach looks simpler; the theorems have shorter formulations and simpler proofs. But for the result to be general, I state it for all manifolds.

The algebra A(M) is topological; in this algebra, the notion of limit is defined. Convergence on compact manifolds means the convergence of functions together with all their derivatives. The algebra A(M) completely describes the manifold M. Thereby, all geometry is banished and algebra alone remains.

How can we reconstruct the manifold M? If there is another manifold N and a smooth mapping ø: M → N is given, then we can construct a dual mapping of function algebras ø* : A(N) → A(M).