Quaternion algebras and quadratic forms
imported>Pythagoras0님의 2013년 11월 29일 (금) 08:45 판
introduction
- consider an algebra defined by $F[i,j]/(i^2=a,j^2=b,ij=-ji)$
- it is an example of a central simple algebra (see Brauer group)
- classification of quaternion algebras over fields
- division algebra
- matrix algebra
Hilbert symbol
- The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules \(i^2=a\), \(j^2=b\), \(ij=-ji=k\). In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
expositions
- Lewis, David W. 2006. “Quaternion Algebras and the Algebraic Legacy of Hamilton’s Quaternions.” Irish Mathematical Society Bulletin (57): 41–64. http://www.maths.tcd.ie/pub/ims/bull57/S5701.pdf
- Quaternion algebras and quadratic forms, Master's thesis, Zi Yang Sham, University of Waterloo
- www.math.virginia.edu/~ww9c/kranec.pdf