Quantum representation theory and Manin matrices I: The finite-dimensional case

We construct a theory that describes a quantum (non-commutative) analogue of representations within the framework of “non-commutative linear geometry” set out in the work of Manin [Quantum groups and noncommutative geometry, Univ. Montréal, Centre de Recherches Mathématiques, Montréal, QC, 1988]. For this purpose, the concept of an internal $\mathrm{hom}$-functor is generalized to the case of parameterized adjunctions, and we construct a general approach to representations of monoids for a symmetric monoidal category with a parameter subcategory. A quantum theory of representations is then obtained by applying this approach to the monoidal category of a certain class of graded algebras with the Manin product, where the parameterizing subcategory consists of connected finitely generated quadratic algebras. We formulate this theory in the language of Manin matrices. We also obtain quantum analogues of the direct sum and tensor product of representations. Finally, we give some examples of quantum representations.