CHAPTER 1
A divisio n algebr a i s a centra l simpl e algebr a
In thes e lecture s a divisio n algebr a i s alway s a finite dimensiona l associativ e
algebra ove r som e field where ever y no n zer o element ha s a multiplicativ e inverse .
That is , a divisio n algebr a D i s a vecto r spac e ove r a field F wit h a produc t
D(gpD D which is associative, has a unit I G D , an d for which every 0 ^ d G D,
there i s a d~ l G Ö suc h that dd~ l = d~ 1d 1. Sinc e F i s isomorphic t o F 1 C D
we will alway s assum e F C D.
It seem s that th e first nontrivia l divisio n algebr a (nontrivia l meanin g noncom -
mutative) wa s defined b y Hamilton . I t i s instructive t o recal l Hamilton' s example .
Let H b e th e rea l algebr a linearl y spanne d b y l,i,j,k wher e i 2 = j 2 = k 2 = 1
and ij k = —ji. I t i s no t har d t o se e (bu t a bi t tedious ) tha t thi s define s
an associativ e algebr a structur e o n H L A s usual , w e vie w th e rea l field R C H I
by identifyin g r wit h r 1. Furthermore , ther e i s a rea l linea r automorphis m
er : H + H define d b y j(ri + r^i + r$j + r^k) r\ T2i r$j r^k whic h i n
addition satisfie s er (aß) = cr(ß)a(a). Furthermore , defin e n(a) = aa(a) an d not e
that n(a) = r\ - f r\ + r\ + r\ G M wher e a = ri + r2 i + j + r^k. I t i s no w
easy t o se e tha t H I is a divisio n algebra , becaus e n(a) = 0 implie s a = 0 , an d s o
a~l a(a)/(n(a)).
It may be that th e reader would like a more careful proo f that th e multiplicatio n
of quaternion s i s associative . On e wa y o f doin g thi s i s t o embe d H I into a know n
associative algebra . I n fact , on e ca n d o just this . Conside r th e algebr a M2(C ) o f
two b y two matrice s ove r th e comple x field. Le t i , j b e th e matrices :
/v^l 0 W O 1 \
\ 0 -y/=i)\-l DJ
Then th e rea l algebr a generate d b y i , j i s easil y see n t o b e isomorphi c t o H L I n
addition, th e nor m define d abov e i s just th e restrictio n o f the determinant .
Hamilton's exampl e illustrate s som e theme s tha t wil l recu r throug h ou t th e
lectures. I t i s fairl y eas y t o defin e a n algebra . W e coul d hav e define d Hamilton' s
algebra ove r an y field. Th e resul t would , i n general , onl y b e a s o calle d centra l
simple algebra . T o sho w H I was a divisio n algebr a w e neede d th e mor e specia l
property tha t a certai n quadrati c for m ha d a n o nontrivia l zeroes . I n general ,
constructing centra l simpl e algebra s wil l b e eas y an d showin g the y ar e divisio n
algebras wil l b e deeper . Th e analog y on e ca n kee p i n min d i s tha t i t i s eas y t o
write dow n a polynomial , eve n on e wit h distinc t roots . I t i s deepe r t o sho w i t i s
irreducible.
Just lik e the quaternions , al l divisio n algebra s wil l be show n t o b e subring s o f
matrices. Als o like the quaternions, al l division algebra s will inherit a structure lik e
the determinan t fro m th e overlyin g matrix ring . I n the languag e o f section four , al l
5
http://dx.doi.org/10.1090/cbms/094/02
Previous Page Next Page