Let C be a symmetric monoidal K-linear category, where K is an algebraically closed field of characteristic zero. Deligne proved that if C satisfies a certain finiteness assumption, then C is necessarily equivalent to the category Rep_K-G, of the finite dimensional rational representations of G. In this talk I will explain how one can use this results to study finite dimensional structures over K. By finite dimensional structures we mean here structures such as algebras, graded algebras, and also embeddings of projective varieities into projective spaces. For such a structure W I will explain a construction of a symmetric monoidal category C_W which is a complete invariant of W. This category will be defined over a subfield K_0 of K, and it will be a form of the category Rep_K-Aut(W). I will then explain how can one use C_W to study different forms of W and invariants (also in the geometric invariant theory sense) of W.
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