Structure and representations of small quantum groups

This thesis is devoted to the study of algebraic structures arising from representation theory of a small quantum group at a root of unity in the relation to the structure of its center.;We describe the block decomposition and multiplicative structure of a subalgebra in the center Z of a small quantum group. It is a sum of a previously known subalgebra Z' and a new one Z&d15; , obtained by Hopf-algebraic methods from the Grothendieck ring of the category of finite dimensional modules. We identify the intersection Z&d15; with the annihilator of the radical of Z&d15; and we prove that it is isomorphic to the algebra of characters of projective modules over the small quantum group. We show that the properties of Z&d15; are in many respects similar to those of the restriction of the Verlinde algebra over the small quantum group. We apply the theory of quantized tilting modules to show that the subalgebra Z&d15;+Z' of the center surjects to the algebras of endomorphisms of certain projective modules. Similar methods allow to conclude that the center Z coincides with the constructed subalgebra Z&d15;+Z' in case of rank = 1 and is larger than that in general.