The left and right dimensions of a skew field over the subfield of invariants

Abstract

© 2017 Elsevier Inc.If H is a Hopf algebra and A an H-module algebra without nontrivial H-stable left or right ideals, then the subalgebra of H-invariant elements AH is a skew field and A may be regarded as a vector space over AH with respect to either left or right multiplications. It is proved in the paper that the left dimension of A over AH is equal to the right dimension under the assumptions that A is semiprimary and dim⁡H<∞. In the case when A is itself a skew field, this answers a question raised by J. Bergen, M. Cohen and D. Fischman.