  We construct Quantum Representation Theory which describes quantum analogue
of representations in frame of "non-commutative linear geometry" developed by
Manin. To do it we generalise the internal hom-functor to the case of
adjunction with a parameter and construct a general approach to representations
of a monoid in a symmetric monoidal category with a parameter subcategory.
Quantum Representation Theory is obtained by application of this approach to a
monoidal category of some class of graded algebras with Manin product, where
the parameter subcategory consists of connected finitely generated quadratic
algebras. We formulate this theory in the language of Manin matrices and obtain
quantum analogues of direct sum and tensor product of representations. Finally,
we give some examples of quantum representations.
