"Quaternion algebras and quadratic forms"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | * let | + | * let <math>F</math> be a field |
| − | * consider a quaternion algebra defined by | + | * consider a quaternion algebra defined by <math>F[i,j]/(i^2=a,j^2=b,ij=-ji)</math> |
* we denote it as | * we denote it as | ||
| − | + | :<math>\left(\frac{a,b}{F}\right)</math> | |
| − | * 4 dimensional algebra over | + | * 4 dimensional algebra over <math>F</math> with basis <math>1,i,j,k</math> and multiplication rules <math>i^2=a</math>, <math>j^2=b</math>, <math>ij=-ji=k</math>. |
* it is an example of a central simple algebra (see [[Brauer group]]) | * it is an example of a central simple algebra (see [[Brauer group]]) | ||
| − | * it is either a division algebra or isomorphic to the matrix algebra of | + | * it is either a division algebra or isomorphic to the matrix algebra of <math>2\times 2</math> matrices over <math>F</math>: the latter case is termed split |
==quaternion algebra as a quadratic space== | ==quaternion algebra as a quadratic space== | ||
| − | * let us consider the algebra | + | * let us consider the algebra <math>A=\left(\frac{a,b}{F}\right)</math> |
| − | * we regard it as a quadratic space associated with the quadratic form | + | * we regard it as a quadratic space associated with the quadratic form <math>(1,-a,-b,ab)</math> |
==Hilbert symbol== | ==Hilbert symbol== | ||
| − | * In this case the algebra represents an element of order 2 in the [[Brauer group]] of | + | * In this case the algebra represents an element of order 2 in the [[Brauer group]] of <math>F</math>, 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. |
2020년 11월 13일 (금) 17:05 판
introduction
- let \(F\) be a field
- consider a quaternion algebra defined by \(F[i,j]/(i^2=a,j^2=b,ij=-ji)\)
- we denote it as
\[\left(\frac{a,b}{F}\right)\]
- 4 dimensional algebra over \(F\) with basis \(1,i,j,k\) and multiplication rules \(i^2=a\), \(j^2=b\), \(ij=-ji=k\).
- it is an example of a central simple algebra (see Brauer group)
- it is either a division algebra or isomorphic to the matrix algebra of \(2\times 2\) matrices over \(F\): the latter case is termed split
quaternion algebra as a quadratic space
- let us consider the algebra \(A=\left(\frac{a,b}{F}\right)\)
- we regard it as a quadratic space associated with the quadratic form \((1,-a,-b,ab)\)
Hilbert symbol
- In this case the algebra represents an element of order 2 in the Brauer group of \(F\), 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
- http://www.math.virginia.edu/~ww9c/kranec.pdf