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In order to get an irreducible representation of u(M) algebra, we develop a canonical theory of mixed state and discuss the dual property of mutually commutable u(M) algebras which have the Casimir operators in common. With the aid of this dual property, we can obtain an irreducible representation of u(M) algebra which is useful for investigations of nuclear dynamics, where generators of u(M) algebra given by one-body operators of fermion are expressed in terms of bosons. As an example, we derive the Holstein-Primakoff representation of u(M) generators. We also derive the irreducible representation of su(3) algebra. * ) Results on the irreducible representation given here is purely mathematical and independent of the atomic nuclear model. However, the present treatment of u(M) algebra will make clear the physical image or the physical ingredient of the irreducible representation.

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