The introduction to the theory of algebraic and topological quantum groups, as presented by P. Podleś and E. Müller, provides a foundational overview of the subject. The authors recall essential concepts from Hopf algebra theory, compact (matrix) quantum groups, and their actions on compact quantum spaces. They also discuss key examples, including the classification of quantum $SL(2)$-groups, their real forms, and quantum spheres. Additionally, they explore quantum $SL_q(N)$-groups and quantum Lorentz groups. The physical motivations for quantum groups are highlighted, particularly in the context of integrable models, conformal field theory, and physical models based on quantized space-time. The authors explain how quantum groups can help address issues in Quantum Field Theory, such as the problem of black hole formation when considering very small volumes. The chapter on polynomials on classical groups of matrices delves into the algebraic structures associated with these groups, including the Hopf algebra structure and the properties of corepresentations. It covers the definitions and properties of Hopf algebras, Hopf *-algebras, and irreducible corepresentations. The chapter also discusses the representation theory of Hopf algebras, including the decomposition of representations and the concept of unitary corepresentations. Finally, the authors provide examples of quantum groups, focusing on quantum $SL(2)$-groups. They detail the construction of these groups using Hecke algebras and the symmetrization operator, and prove that there are two standard deformations of quantum $SL(2)$-groups, distinguished by the parameter $q$. The chapter concludes with a proof that these deformations are unique up to isomorphism.The introduction to the theory of algebraic and topological quantum groups, as presented by P. Podleś and E. Müller, provides a foundational overview of the subject. The authors recall essential concepts from Hopf algebra theory, compact (matrix) quantum groups, and their actions on compact quantum spaces. They also discuss key examples, including the classification of quantum $SL(2)$-groups, their real forms, and quantum spheres. Additionally, they explore quantum $SL_q(N)$-groups and quantum Lorentz groups. The physical motivations for quantum groups are highlighted, particularly in the context of integrable models, conformal field theory, and physical models based on quantized space-time. The authors explain how quantum groups can help address issues in Quantum Field Theory, such as the problem of black hole formation when considering very small volumes. The chapter on polynomials on classical groups of matrices delves into the algebraic structures associated with these groups, including the Hopf algebra structure and the properties of corepresentations. It covers the definitions and properties of Hopf algebras, Hopf *-algebras, and irreducible corepresentations. The chapter also discusses the representation theory of Hopf algebras, including the decomposition of representations and the concept of unitary corepresentations. Finally, the authors provide examples of quantum groups, focusing on quantum $SL(2)$-groups. They detail the construction of these groups using Hecke algebras and the symmetrization operator, and prove that there are two standard deformations of quantum $SL(2)$-groups, distinguished by the parameter $q$. The chapter concludes with a proof that these deformations are unique up to isomorphism.