Topological tensor representations of $\mathfrak{gl}(V)$ for a space $V$ of countable dimension

To Yuri Ivanovich Manin on the occasion of his 85th birthdayThe Lie algebra gl(V ) is the Lie algebra of all endomorphisms of a countable-dimensional complex vector space V .We define a tensor category T gl(V ) of topological representations of the Lie algebra gl(V ), so that V , V := Hom C (V, C), and the adjoint representation gl(V ) are objects of T gl(V ) .This makes T gl(V ) an analogue of the category of finite-dimensional modules over the finite-dimensional Lie algebra gl(n).Our main result is that the category T gl(V ) is antiequivalent as a symmetric monoidal category to the category T gl() of tensor representations of the Lie algebra gl() of finitary infinite matrices.The category T gl() has been studied extensively and is known to have a universality property as a tensor category.