A TYPE OF PRIMITIVE ALGEBRA*

Transcription

1 A TYPE OF PRIMITIVE ALGEBRA* BT J. H. M. WEDDERBURN I a recet paper,t L. E. Dickso has discussed the liear associative algebra, A, defied by the relatios xy = yo(x), y = g, where 8 ( x ) is a polyomial i x which is ratioal i the field F i which A is defied. I this paper Dickso shows for = 2 ad = 3 that, whe 6 ad g are properly chose, A is primitive, i. e., every elemet i it with the exceptio of zero has a iverse: A is i fact i the case =2a direct geeralizatio of quaterios. So far as I am aware o algebra other tha these two ad fields has bee proved to be primitive; hece it is of cosiderable iterest to fid that for ay value of, 6 ad g ca be so chose as to make Dickso's algebra primitive. I discussig primitive algebras we may without ay real loss of geerality assume that ay elemet commutative with every other elemet of the algebra is a scalar, i. e., a elemet of the field F. For, all such elemets geerate a commutative subalgebra B ad, as i the theory of groups, we ca fid a complex C for which A = BC= CB. We ca therefore regard A as a algebra whose coefficiets lie i the field F exteded by the elemets of the commutative primitive algebra B ad, i this algebra, scalars are the oly elemets commutative with every other elemet. A primitive algebra will be called ormal whe reduced i this maer. It follows from the geeral theory of liear associative algebras^ that a ormal primitive algebra is of order ti2 ad that, whe the field is sufficietly exteded, it is equivalet to the simple matric algebra epq(p,q=l,2, ",), so that its idetical equatio is of degree. We shall ow ivestigate the cosequeces that result from the assumptio that the ormal primitive * Preseted to the Society, December 31, tthese Trasactios, vol. 15 (1914), pp j See, for istace, my paper i the Proceedigs of the Lodo Mathematical Society, ser. 2, vol. 6 (1907), pp , where refereces are give. 162

2 1914] A TYPE OF PRIMITIVE ALGEBRA 163 algebra A cotais a elemet for which the group of the correspodig idetical equatio is the cyclic group of order. 1. Let A he. a ormal primitive algebra of order 2 ad let x be a elemet of it for which the group of the idetical equatio / ( x ) = 0 is cyclic. If i > 21,»> are the roots of x, the correspodig primitive idempotet elemets are #r err (»-&) (»- r-l)(x-$r+1)... (g- j.) (ír- íl) ' (fr- Ér-OUr" fc+l) ( ~ f ) (r-1,2,,) ad* whe F is exteded by the roots of x, A cotais a matrict algebra ( evq ) which does ot reduce to ei, e2,,e, sice the x would be commutative with every elemet of A. We may therefore suppose ei2 + 0, so that there is a elemet yi, ratioal i F, for which ei yi e2 is ot zero. The cojugates of eiyie2 i F are er yi er+i (r = 1, 2,, ), oe of which ca therefore be zero. If we set the elemet er yi er+i = Vr, r+i er, r+i, V = 12 Vr, r+l er, r+l r=l is ratioal sice it is the sum of the cojugates of ex yi e2; its th power is evidetly the scalar tji2 23 i = g, say, which is therefore ratioal. Sice where X = UkrBr, r=\ Xy = Z) r TJr, r+l Cr, r+l = yxi, I Xl = r-l «r The elemet xi is commutative with x ad is therefore a ratioal polyomial i x, say 6 ( x ), sice the idetical equatio, / ( x ) = 0, has o repeated roots. Evidetly x ad y geerate A whe expressed i its matric form ad therefore also i its ratioal form. We have therefore the followig Theorem. A ormal primitive algebra which cotais a elemet x whose group is cyclic is geerated by x ad a elemet y which satisfies the relatios * See Proceedigs of the Lodo Mathematical Society, I. c, p. 97. t A matric algebra is oe for which a basis ( epg ) ca be chose for which erq e = 0, q + r, ei>«ev = epr i 2err = 1. Some of the elemets e, may be zero, but if so the algebra is reducible.

3 164 J. H. M. WEDDERSBURN [April xy = y8(x), y = g, where 8 (x) ad g are ratioal* i the field of the coefficiets ad y is the first power of y which is commutative with x. Coversely if x ad y satisfy these coditios, the group of x is the cyclic group of order. This may be show as follows. Sice xy = y8(x), therefore xyr = f8r(x), where 8r (x) deotes the fuctio 8 iterated r times. If 6r (x) = x, the yt is commutative with x, which is cotrary to the give coditios uless r =. The matric form of y ca be cosiderably simplified by choosig the uits epq i such a way that r\t, r+i et, T+i is replaced by er, r+i for r= 1, 2,, 1, e i beig chose so that ei2 e23 6-i, ei = 11 This chage i the basis of (epq) leaves e (e = 1, 2, ) ualtered that x has the same form as before while y becomes so y = ei2 + e e-i, + ge\ It is easily show from the forms give above for x ad y that ay elemet, z, of A ca be expressed uiquely i the form y"-1 h0 + y~2hi+-h yh-2 + A_i, where hr (r = 0, 1,, 1) are ratioal polyomials i x. Now ad yr = ei, r+i + e2, T e-r, + g (e-r+\, i+ +er) y XT = r + 1 e,, r + i + Çr+2 Cl, r+2 + * ' *, so that i the matric form of ay elemet of A all the coefficiets to the left of the pricipal diagoal are multiplied by g, ad if ay particular coefficiet is zero all the coefficiets i the same diagoal are zero. 2. We shall ow determie the coditios which g must satisfy i order that a algebra of Dickso's type shall be primitive. Let us cosider a elemet of the form _ z = yr + 2/^1Äi+-r-yÄ^-i+A, (fc + 0) * We may evidetly modify y so that g is a algebraic iteger of the field.

4 1914] A TYPE OF PRIMITIVE ALGEBRA 165 ad let ki, k2,, k-r be polyomials i x to be determied later, the iy-r + y-r-1 a + + k-r)z = 9 + y»"1 hi + y"~2 h2 + + y»"' hr + y"-1 T h y' + y"-2 jr+1 kiy*-1 hi + +y~2 y~' k2 yr -\-1-. If we put ki = yt Ai y~r, which is a kow ratioal polyomial i x, the coefficiet of y~~l is zero. Similarly the coefficiet of y~2 vaishes if Id- yh y^1 hi y~r - yt h2 y~t, which is also a kow polyomial i x, ad so o till the coefficiet of yr is reached, after which the process termiates. We ca therefore i geeral determie the k's so that the product commeces with a term y*~' h (x) (s < r) ad, after multiplyig o the right by 1 / A (x), which exists if h +- 0, we have a umber which begis with the term y7-*. If this umber has a iverse, z will also have oe, sice, by a well-kow theorem o matrices, if a product has a iverse, the same is true of each factor. We have therefore oly two types of elemets to cosider: first, those which may be reduced to the form y + <p (x) by repeated applicatios of the process give above; ad secod elemets for which at some stage the product is idepedet of y. If y + <p ( x ) is expressed i its matric form, its determiat is easily see to be V ( 1 ) <P ( & ) * 0 ( i ) 9, a so that, if g is ot the orm of ay ratioal fuctio of, ay elemet of this type has a iverse. Suppose ow that z is a elemet of the secod of the two types metioed above, so that there exists a elemet zi = y"-r + y"-^1 ki +-h k-r, such that zi z is idepedet of y. The determiatio of the coefficiets ki, ', k-^r is wholly idepedet oî g, as the trasform of ay polyomial i x by a power of y depeds solely o 6 ( x ). We may therefore set ziz = g + k, where k is idepedet of g, which may be regarded as a variable scalar. If k is ot itself a scalar, g + k certaily has a iverse ad therefore z has also. We may assume therefore that A; is a scalar, so that the matrix correspodig to Zi differs from the adjoit of the matrix correspodig to z merely by a scalar factor which must be a itegral fuctio of g sice Zi, z ad g + k are all itegral i g. Sice the determiat of the adjoit of a matrix is a power* of the determiat of that matrix, the determiat of z,

5 166 wedderbur: a type of primitive algebra say z, must be a power of g + k, ad as ] z is obviously of the rth degree i g, begiig with the term ( 1 )-r gr, we have s - (-iy-*(g + ky. But, whe g = 0, z \ becomes the orm of fa so that (- iy-rkr= N(fa). Uless the scalar g + k is zero, z has a iverse. I the cotrary case, gr N ( fa). Hece, if o power of g less tha the th is the orm of a ratioal polyomial i x, every elemet of the algebra, except zero, has a iverse.* "The existece of such ratioal umbers g follows from Satz 33 of Hubert's Bericht, Jahresbericht der Deutsche Mathematiker-Vereiigug, vol. 4 (1897), p. 198.