A construction of finite-dimensional faithful representation of Lie algebra

We give a natural proof of the Ado theorem.Comment: 2 page

Structures of boson and fermion Fock spaces in the space of symmetric functions

We realize the Weil representation of infinite dimensional symplectic group and spinor representation of infinite-dimensional group $GL$ by linear operators in the space of symmetric functions in infinite number of variables.Comment: 32 page

Hua type integrals over unitary groups and over projective limits of unitary groups

We discuss some natural maps from a unitary group U(n) to a smaller group U(n-m) (these maps are versions of the Livshic characteristic function). We calculate explicitly the direct images of the Haar measure under some maps. We evaluate some matrix integrals over classical groups and some symmetric spaces (values of the integrals are products of Gamma-functions). These integrals generalize Hua Loo Keng integrals. We construct inverse limits of unitary groups equipped with analogues of the Haar measure and evaluate some integrals over these inverse limits.Comment: 20 pages; the final version; evaluation of some normalizing constants is adde