"Quaternion algebras and quadratic forms"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
* let $F$ be a field | * let $F$ be a field | ||
| − | * consider a quaternion algebra | + | * 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 <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 $2\times 2$ 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== | ==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. |
| 18번째 줄: | 25번째 줄: | ||
* 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 | * 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 | ||
* [http://uwspace.uwaterloo.ca/bitstream/10012/3656/1/second2.pdf Quaternion algebras and quadratic forms], Master's thesis, Zi Yang Sham, University of Waterloo | * [http://uwspace.uwaterloo.ca/bitstream/10012/3656/1/second2.pdf Quaternion algebras and quadratic forms], Master's thesis, Zi Yang Sham, University of Waterloo | ||
| − | * www.math.virginia.edu/~ww9c/kranec.pdf | + | * http://www.math.virginia.edu/~ww9c/kranec.pdf |
[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
2014년 1월 18일 (토) 04:34 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