  By using Bogoliubov transformations to construct the ground states of
fermionic Bardeen-Cooper-Schrieffer (BCS) superfluids and weakly-interacting
Bose gases supporting Bose Einstein Condensation (BEC), their algebraic
structures and implications can be analyzed in detail. Both ground states are
generalized squeezed coherent states saturating a generalized Heisenberg
uncertainty relation, and they acquire quantized Berry phases when the
corresponding systems are transported along a closed path in their parameter
spaces. While the Berry phase of the BCS ground state depends on the total
particle number, the Berry phase of the BEC ground state depends only on the
particles outside the BEC. The Berry phases are associated with magnetic
monopoles in the parameter spaces and we found that the Dirac quantization
condition is satisfied. Moreover, both ground states are entangled states of
the fermion or boson quanta and we found the entanglement entropy quantifying
the internal correlations. A fixed particle-number approach of fermionic
superfluids does not saturate the generalized uncertainty relation, exhibits
internal entanglement, and gives corresponding Berry phase. In addition, the
algebraic structures of the ground states can be classified by the $q$-deformed
Hopf algebra, $\bigoplus_{\mathbf{k}}h_{q_{\mathbf{k}}}(1)$ for bosons and
$q$-deformed Hopf superalgebra $\bigoplus_{\mathbf{k}}h_{q_{\mathbf{k}}}(1|1)$
for fermions, respectively.
