On Herstein's Lie isomorphism conjectures
by
Wallace Martindale
University of Massachussetts at Amherst
Coauthors: Blau, Beidar, Bresar, Chebotar

Let R and S be prime rings with involution, with respective skew elements K and L, with respective extended centroids C and D, and let

\alpha: [K, K]/[K, K] \cap C --> [L, L]/[L, L] \cap D

be a Lie isomorphism. The main result is (roughly speaking) that \alpha is determined by a related associative isomorphism, with the exception of several low- dimensional counterexamples. This result in essence settles the most difficult of the conjectures on Lie isomorphisms made by Herstein in his 1961 AMS Hour Talk (his framework was the more restricted one of simple rings with involution.)

Date received: January 27, 1999