  In two dimensional fluid, there are only two classes of swimming ways of
micro-organisms, {\it i.e.}, ciliated and flagellated motions. Towards
understanding of this fact, we analyze the swimming problem by using
$w_{1+\infty}$ and/or $W_{1+\infty}$ algebras. In the study of the relationship
between these two algebras, there appear the wave functions expressing the
shape of micro-organisms. In order to construct the well-defined quantum
mechanics based on $W_{1+\infty}$ algebra and the wave functions, essentially
only two different kinds of the definitions are allowed on the hermitian
conjugate and the inner products of the wave functions. These two definitions
are related with the shapes of ciliates and flagellates. The formulation
proposed in this paper using $W_{1+\infty}$ algebra and the wave functions is
the quantum mechanics of the fluid dynamics where the stream function plays the
role of the Hamiltonian. We also consider the area-preserving algebras which
arise in the swimming problem of micro-organisms in the two dimensional fluid.
These algebras are larger than the usual $w_{1+\infty}$ and $W_{1+\infty}$
algebras. We give a free field representation of this extended $W_{1+\infty}$
algebra.
