A New Approach to the Representation Theory of the Symmetric Groups, IV. $\mathbb Z_2$-Graded Groups and Algebras; Projective Representations of the Group $S_n$

We start with definitions of the general notions of the theory of $\mathbb Z_2$-graded algebras. Then we consider theory of inductive families of $\mathbb Z_2$-graded semisimple finite-dimensional algebras and its representations in the spirit of approach of the papers [14], [21] to representation theory of symmetric groups. The main example is the theory of the projective representations of symmetric groups.