  In this paper, we study the generalized (co)homology Hopf algebras of the
loop spaces on the infinite classical groups, generalizing the work due to
Kono-Kozima and Clarke. We shall give a description of these Hopf algebras in
terms of symmetric functions. Based on topological considerations in the first
half of this paper, we then introduce a universal analogue of the factorial
Schur $P$- and $Q$-functions due to Ivanov and Ikeda-Naruse. We investigate
various properties of these functions such as the cancellation property, which
we call the $\mathbb{L}$-supersymmetric property, the factorization property,
and the vanishing property. We prove that the universal analogue of the Schur
$P$-functions form a formal basis for the ring of functions with the
$\mathbb{L}$-supersymmetric property. By using the universal analogue of the
Cauchy identity, we then define the dual universal Schur $P$- and
$Q$-functions. We describe the duality of these functions in terms of Hopf
algebras.
