Lie groups and quantum groups applied as a tool in finding solutions in some field theories

It is well-known that semi-simple Lie groups and their quantum group counterparts are distinguished by their special mathematical properties. These properties enable one to find exact solutions of this type in many field theories. This paper is devoted to some examples of these theories. Firstly, quantum group solutions of reformulated Sławianowskiʹs Gl(n, R)-invariant theory [3,4] are presented. Secondly, a generalization of Polyakov-like theory is discussed. Next, the role of Lie-group solutions in tetrad theories with an arbitrary Lagrangian dependent only on torsion of teleparallelism connection is described. In conclusion, the role of Lie groups and quantum groups as useful tools for finding exact solutions is stressed.