  Abelian codes and complementary dual codes form important classes of linear
codes that have been extensively studied due to their rich algebraic structures
and wide applications. In this paper, a family of abelian codes with
complementary dual in a group algebra $\mathbb{F}_{p^\nu}[G]$ has been studied
under both the Euclidean and Hermitian inner products, where $p$ is a prime,
$\nu$ is a positive integer, and $G$ is an arbitrary finite abelian group.
Based on the discrete Fourier transform decomposition for semi-simple group
algebras and properties of ideas in local group algebras, the characterization
of such codes have been given. Subsequently, the number of complementary dual
abelian codes in $\mathbb{F}_{p^\nu}[G]$ has been shown to be independent of
the Sylow $p$-subgroup of $G$ and it has been completely determined for every
finite abelian group $G$. In some cases, a simplified formula for the
enumeration has been provided as well. The known results for cyclic
complementary dual codes can be viewed as corollaries.
