  Let $K$ be a perfect field and let $k \subset K$ be a subfield. In previous
work of the second author and C. Pappacena, left finite dimensional simple
two-sided $k$-central vector spaces over $K$ were classified by arithmetic data
associated to the extension $K/k$. In this paper, we continue to study the
relationship between simple two-sided vector spaces and their associated
arithmetic data. In particular, we determine which arithmetic data corresponds
to simple two-sided vector spaces with the same left and right dimension, and
we determine the arithmetic data associated to the left and right dual of a
simple two-sided vector space. As an immediate application, we prove the
existence of the non-commutative symmetric algebra of any $k$-central two-sided
vector space over $K$ which has the same left and right dimension.
