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NONASSOCIATIVE QUATERNION ALGEBRAS OVER S. PUMPLUN, V. ASTIER
 

Summary: NONASSOCIATIVE QUATERNION ALGEBRAS OVER
RINGS
S. PUMPL¨UN, V. ASTIER
Abstract. Non-split nonassociative quaternion algebras over fields were
first discovered over the real numbers independently by Dickson and Al-
bert. They were later classified over arbitrary fields by Waterhouse.
These algebras naturally appeared as the most interesting case in the
classification of the four-dimensional nonassociative algebras which con-
tain a separable field extension of the base field in their nucleus. We
investigate algebras of constant rank 4 over an arbitrary ring R which
contain a quadratic ´etale subalgebra S over R in their nucleus. A gen-
eralized Cayley-Dickson doubling process is introduced to construct a
special class of these algebras.
Introduction
Let k be a field. A non-split nonassociative quaternion algebra over k
is a four-dimensional unital k-algebra A whose nucleus is a separable qua-
dratic field extension of k. Non-split nonassociative quaternion algebras
were early examples of nonassociative division algebras which are neither
power-associative nor quadratic and were first considered by Dickson [D] in
1935, and by Albert [A] in 1942, both times over the reals. In 1987, Water-

  

Source: Astier, Vincent - Department of Mathematics, University College Dublin

 

Collections: Mathematics