Duals of Simple Two-sided Vector Spaces

Let K be a perfect field and let k ⊂ K be a subfield. In [4, Theorem 3.2], 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.