Z.5I M3 UC-NRLF $C lb5 IDD Zbc "Clnipersits of Cbtcago INVARIANTIVE CHARACTERIZATIONS OF LINEAR ALGEBRAS WITH THE ASSOCIATIVE LAW NOT ASSUMED A DISSERTATION Submitted to the Faculty op the Ogden Graduate School OF Science I^f Candidacy for the Degree of Doctor of Philosophy department of mathematics j BY CYRUS COLTON MACDUFFEE Private Edition Distributed By THE UNIVERSITY OF CHICAGO LIBRARIES CHICAGO, ILLINOIS Reprinted from The Transactions of the American Mathematical Society, Vol. 23, No. 2. March 1923 Zbc Tantversiti? of Cbicago INVARIANTIVE CHARACTERIZATIONS OF LINEAR ALGEBRAS WITH THE ASSOCIATIVE LAW NOT ASSUMED A DISSERTATION Submitted to the Faculty of the Ogden Graduate School OF Science in Candidacy for the Degree OF Doctor of Philosophy department op mathematics BY CYRUS COLTON MACDUFFEE Private Edition Distributed By THE UNIVERSITY OF CHICAGO LIBRARIES CHICAGO, ILLINOIS Reprinted from The Transactions of the American Mathematical Society, Vol. 23, No. 2. March 1923 EXCK.---.fHI > »\ » » INVARIANTIVE CHARACTERIZATIONS OF LINEAR ALGEBRAS WITH THE ASSOCIATIVE LAW NOT ASSUMED* BY CYRUS COLTON MACDUFFEE I. Introduction We consider linear algebras in \vhich neither the commutative nor the asso- ciative law of multiplication is assumed, and whose coordinates and constants of multiplication are numbers of a general field F. A rational integral invariant, or covariant, is a rational integral function of the constants of multiplication, or of the constants of multiplication and the coordinates of the general number, which has the invariantive property under the total group of linear homogeneous transformations on the units. If an invariantive function also actually involves the units, it has been called a vector covariant by Professor O. C. Hazlett.f who shows that every rational integral vector covariant can be obtained as a covariant of the general number of the algebra and a fundamental set of ordinary covariants. In Section II of this article, it is shown how vector covariants may be formed directly from the constants of multiplication without assuming the knowledge of any ordinary covariants or invariants. To do this, the notion is introduced of a hypercomplex determinant whose elements obey neither the associative nor the commutative law of multiplication, and a few simple properties of such hyper- complex determinants are derived. From the vector covariants and the charac- teristic determinants of the algebra, ordinary relative invariants may easily be found. In Section III the linear algebra in three units, one of which is a principle unit, is considered. Invariants and covariants of the algebra are calculated by the method of Section II, and a set of ten of these functions is shown to form a com- plete system of invariants and covariants from the standpoint of Lie. In Section IV it is shown that for the example of Section III the arithmetic invariant denoting the rank can be replaced by a rational integral covariant. The generic case is defined as the case for which certain three covariants are * Presented to the Society, March 25, 1921. t These Transactions, vol. 19 (1918), p. 408. (135) 136 C. C. MACDUFFEE [March different from zero, and this case is reduced by rational transformations in the general field to a canonical form. The parameters of the canonical form are characterized by invariants. II. Invariants and vector covariants 1. Notations. Consider the general linear algebra in n units whose multi- plication table is given by (1) eiCj = YlkZ" yuk^k {i, / = 1, . . ., m), and whose general number is x = ^'il"xiei, where the yjj^ and Xj are num- bers of a general field F. Consider the transformation of units (2) T: e/ = J^fJl a^^ej Z? ^ | a,;,- 1 ?^ . (i = 1, . . . , «), also with coefficients a,;,- in F. This transformation T induces upon the con- stants 7y-4 of multiplication and the coordinates Xi a transformation 5 which carries 7,;,^ into yijk and Xi into x, in such a way that (3) eWj = ^IZ" y'ijke'k, and (4) E:"% = ^ = ^'=Z':i^'^^ According to Professor Hazlett, * a vector covariant is defined to be a function of the units and the constants of multiplication and the coordinates of the form 'vilijk; x/yBs) (i, j,k,r,s = \, ..., n), such that, under a transformation T of the units and its induced transformation S, there exists a relation of the form (5) v' = v{y'ijk; xl; e's) = D" v{y,jk; x/, e^ {i, j, k, r, s = 1, . . ., n). The integer n is called the weight of the vector covariant v. As Miss Hazlett noted, f it follows readily from (1) and (3) that every vector covariant is expressible linearly in the units, i. e., (6) ^' = T^:^ie:. v = Y:rjiViei, where v'i is the same function of the y'ij^ and x'^ that f, is of the 7,;,^ and the x,. * These Transactions, vol. 19 (1918), p. 408. t Ibid., p. €16. 1922] UNBAR ALGEBRAS 137 2. Theorem 1. The coefficients Vj of every vector covariant of weight n expressed in the form v = SjC" ^,-^,- are transformed cogrediently apart from the factor D~'' mith the coordinates Xi of the general number x = 2)~"xj^,- under a linear trans- formation of determinant D. From (5) and (6) we have Making use of (2), we find that On account of the linear independence of the units Cj, which occur only where shown explicitly, (7) v;= D-" 2];:>.' «.>• • {j = !,:.., ft). Similarly from (4) and (2) we have (8) Xj = YPlxla^j (/=!,...,«). Comparing (7) with (8), we see that the theorem is proved. 3. Hypercomplex determinants.* On account of the fact that multiplication is usually neither commutative nor associative, a determinant whose elements are hypercomplex numbers must be defined more precisely than a determinant whose elements are ordinary numbers. We define the general hypercomplex determinant of the nth order (9) • D = an ... ai„ ani a„„ [oiia22 • . . a„„] to be the sum of n ! terms of the type (-1)* [ai,-, 02,, ••• a„ij in which ii, 4 i„is an arrangement of 1, 2, . . .,n derived from the natural order by k interchanges. The first subscripts must occur in their natural order in every term. The brackets indicate that the method of grouping the factors is arbitrary, but the same method is to be used in each term. A particular * Determinants whose elements are quaternions and hence associative but not commuta- tive were considered by Cayley, On certain results relating to quaternions, The Philosoph- icalMagazine, vol. 26 (1845) pp. 141-145; and also by C. J. Joly, second edition of Hamilton's Elements of Quaternions, vol. 2, Appendix 1, p. 361. 138 C. C. MACDtnPFEE [March hypercomplex determinant of the nth order is obtained by replacing the brackets inclosing the leading term by a particular grouping of that term, in which case it is understood that every term of the expansion is to be grouped in that way. Thus there are as many particular hypercomplex determinants of the nth order as there are ways of grouping n ordered factors. 4. Lemma 1. Every hypercomplex determinant merely changes sign upon the interchange of any two columns. Consider the determinant (9). Its terms may be arranged in gC^!) pairs of the type (10) (-1)* [axi^ ... Or, (-1)*+' [au. . . Qri. a,, where the two terms in each pair differ only in having their i, and i^ subscripts interchanged. Since each is obtainable from the other by one interchange of subscripts, they are opposite in sign. Let us denote D with its rth and 5th columns interchanged by D'. Evidently this interchange leaves the first sub- scripts in their natural order but interchanges the second subscripts v and i, wherever they occur, and so interchanges the absolute values of the terms in each pair (10). Thus every term in D with its sign changed is equal to a term of D' and vice versa. Then D' = —D. This argument depends in no way upon the manner of grouping, so the lemma holds for all hypercomplex deter- minants. No analogous theorem exists concerning the interchange of two rows. Lemma 2. Any hypercomplex determinant two of whose columns are identical is zero. For by the interchange of the two identical columns the determinant is un- altered, and yet changes in sign. 5. Lemma 3. A hypercomplex determinant the elements of whose jth column are binomials a^j -\- bij is equal to the sum of two determinants identical with the first except that the jth column of the one is composed of the a,j- while the jth column of the other is composed of the b^j. The corresponding theorem holds for rows. Let us set D = an • • • ihr + Clr) flln a»l • • • {bnr + O • • • flnx A = ail i>lr • (^In Obi • • • b„r . . a„„ A = On ... Clr ... Oin flnl • • • C„r . . . a„ 1922] LINBAR ALGSBRAS 139 where it is understood that the manner of grouping is arbitrary, but the same manner must be used consistently in every term of each determinant. It follows from the distributive law that (11) a„- (Psi^ + Csi) . . . a„i^ = au^ . . . bsi^ + au ... Cs, for all values of the subscripts and for all methods of grouping. It holds in particular for i^ = r and 5 = 1, . . ., n where r is fixed. Let k denote the num- ber of interchanges necessary to obtain the order ij, . . . , i„ from the natural order 1, . . ., n. Then, multiplying (11) by ( — 1) and summing for ii, . . ., i„, we have D = Di + Di, which was to be proved. We prove the corresponding theorem for rows by noting that (11) holds when 5 is fixed and i^ ranges over the values 1, . . .,n, and summing as before. 6. Theorem 2. The determinant V = ei ... e„ ei is a vector covariant of weight 1 of the linear algebra in n units ei, manner of grouping. We set V' = Under transformation (2) this becomes . , e„ for every V ^:;:>i,., ...^;::>„,,.., Z';:r«iu^^....Z'::r««^n^.. From Lemma 3 we see that V, = ^ = bo + biei + 62^2, ^1^2 = Co + Ciei + €202, eiCi = c?o + die\ -f- d2e2, where the coefficients are numbers of F. By applying the linear transformation of units • /, c-v ^1 = ao + ociBi -\- a2^2, ^ ' €2 = Po + /3iei + ft^2, where the coefficients are numbers of F such that D ^ 01182 — «2/3i is different from zero, the principal unit is left invariant and the general number and multi- plication table become (13) and (14) respectively with each letter primed. That is, the transformation (15) of units induces upon the coefficients the following transformation : x'a = Xo + D-\a2&o - oio^2)xi + D-^(aopi — a;i/3o)^2, x[ = D-i ^a-Ti - £>-» fiiX2, X2 = — D~^ UiXi + D~^ aiXi, a'o= A+ D--^{a2po - "oft)^ -f- D-^aoft - ai/3o)C, a[ = Z?-i P2B - D-^fiiC, a'2 = -D-i ajS -f- Z)-i aiC, (16) ^'o = -E + D-^(a20o - ao02)F + r>-i(aoft - a,0o)G, b[ = Z?-i ftF - D-i ^iG, b!, = - Z?-i a2F + Z?-» aiG, c'„ = H + D-^{a20o - aoffi)! + D-'{ao0i - a,0o)J, c[ = D-^ 02l - -D-i 0ij, C2 = - D-^ a2l + D-' aj, d'o = K + D-\a2^o - ao02)L + Z?-i(aoft - ai(3o)M, d[ = D-» ^L - D-i /3iM, d'2 = - D-^a2L + D-'aiM, 142 C. C. MACDUFFEE [March where A = ao^ + ai^'ao + 02^60 + cL\aiPa + aias^o, B ^ a-^a\ + a-^h\ + 2aoai + a\aif:\ + ctiai^x, C ^ 01.^0.2 + oi^hi + 2aoa2 + aia2C2 + «i«2rf2> £ = ^0' + /Si^ao + fii'ba + /Si^s^o + &iM<>, F = |3i=ai + ^2^61 + 2/3o|3i + /Siftci + /SiMi, 6^ = /3i%2 + ^^h + 2/3o;32 + PACi + fiyl3^2, H = ao/3o + ai/SiCo + 02/3260 + aiftco + a2/3ido, 7 = ai/3o + ao/3i + ai/SiOi + 02/3261 + ai/32Ci + a20idi, J = a2/3o + ao/32 + ai/3ia2 + 02/3262 + ai/32C2 + a^ffich, K = Qo/3o + oi/Jiao + 02/8260 + a2/3iCo + ai^^o, L = ao/3i + oi/3o + oijSiOi + 02/3261 + 02/3iCi + oi/32(ii, M = 00P2 + 02180 + ai/3ia2 + 02/8262 + 02/3iC2 + ai^zdi. 10. The Lie group. The generators* of the infinitesimal transformation corresponding to the finite transformation (16) are found to consist of six partial differential equations which it is not necessary to give here in detail. The first term of each equation involves only the Xi, and is the only term in which the Xi occur. Since these equations are generators of an infinitesimal group corresponding to the finite group of transformations (16), they form a fortiori a complete system of partial differential equations. The six equations in fifteen variables have nine functionally independent solutions, and these solutions form a complete set of absolute invariants and covariants of the linear algebra (14). Moreover, if the terms involving Xo, Xi, X2 in these equations be deleted, there results another complete system of partial differential equations, the generators of the group corresponding to the group of transformations in (16) on the con- stants of multiplication only. This system has six functionally independent solutions, the six absolute invariants of the linear algebra. They are six of the nine solutions of the first complete system. We see then that there are exactly six absolute invariants and three absolute covariants in a complete system of invariants and covariants for the algebra. There are then no more than seven functionally independent relative invariants. A complete system will be exhibited of three absolute covariants and seven relative invariants whose jacobian does not vanish identically. 11. Invariants of the characteristic determinants. The right- and left-hand characteristic determinants (12) become 5(co) = - o)' -h Aoj^ - Tco -f- A, 5'(a)) = - 0)5 -f- A'co" - r'w -f- A', • Lie-Scheffers, Vorlesungen iiber continuierliche Gruppen, Leipzig, 1893, p. 716, et seq. 1922] LINEAR ALGEBRAS 143 where A = Sjco + (ai + ct)xi + {di + b^X2, r = Zxi? + {ayCi — a2Ci — ao)xi^ + (Wi — hdi — 60)^2^ + (2ai + 2c^Xiix-i. + (262 + 2di)xtiXi + (0162 — 02^1 + c-4i — Cid2 —do — Co)xiXt, (17) A = a;o' + (02^0 — aoC2)xi' + (6i(io — Mi)«2' + (ciiCi — OjCi — Co) xoxr^ + (ai + C2)xo^Xi + (62 + c/i)x:o^X2 + (Wi — hd^ — 60) xoXi^ + (coC?2 — Cido + 02^0 — 00^2 + OoCi — aiCo)xi''x2 + (60^2 — Wo + Qo^i — aibo + CiC?o — Codi)xix^'^ + {c^di — cA + aib2 — 0261 — co — do)^0^1^2> and A', V, A' are obtained from A, T, A respectively by interchanging c, and d,-, for i = 0, 1, 2. Each of these six expressions satisfies the equations of §10 and is an absolute covariant of the algebra. The invariants of the six ternary forms (17) are invariants of the algebra. A and A' have no invariants. F and T' have one invariant each, their hessians. The ternary cubic forms A and A' each have two relative invariants, the 5 and T of Aronhold.* But only six of the seven invariants have been accounted for by this means, even if all six should prove to be independent. Thus it is evident that the invariants obtainable as the invariants of the coefficients of the char- acteristic determinants are not sufficient to form a complete system. 12. Additional invariants by the method of Section II. By Theorem 2 of §6, the hjrpercomplex determinant 1 ei €2 1 e\ 62 1 ei e2 V ^ 1 e\ €2 = ^1^2 ~ B2P1 1 ei €2 is a vector covariant of weight 1. By the multiplication table (14) this can be expressed as (18) F = (co - rfo) + (ci - d,)ei + (C2 - ^2)^2- Now by Theorem 1, §2, the coefficients of the units in (18) are transformed co- grediently apart from the factor D~^ with the coefficients Xo, Xi, X2 of the general number, and hence when substituted for these variables in (17) give relative invariants of the algebra. f Thus we have Ai = 3(co — do) + (ai + C2)(ci — di) + (di + 62) (^2 — £^2), A[ = 3(co - do) + (fli + d2)(,ci — di) + (ci + 62) (C2 - £^2), r2 = 3(co — do)^ + (aiC2 — a2Pi — ao)(ci — diY + • • , Tj = Z{co -^ doY + (aA — a^i — ao){c\ — diY + • • , A3 = (co — doY + (02^0 — aoC2)(ci — diY + , A3 = (co — doY + (.Oido — aod2){ci — diY + • • • . (19) * Salmon, Higher Plane Curves, Dublin, 1897, pp. 191-192. t Throughout this article the subscript on the symbol for an invariant indicates its weight. 144 C. C. MACDUFFEE [March Each of these forms is transformed into a function of itself by the differential operators of §10. Evidently there are functional relations between the twelve relative invariants which these methods yield, since it has been shown that there can be but seven functionally independent invariants. This redundance makes unnecessary the use of the complicated 5 and T invariants of A and A'. The 5 invariant of A — A' is quite simple however, and will be used. 13. A complete system of invariants. If will be shown that the following three covariants and seven invariants form a complete system : (20) A, A', e = r - r', H^ = hessian of V, H^' = hessian of T', 54 = 5 of A - A', Ai, Tj, r/, A3. Since we have only ten relations in -fifteen variables, it is sufficient to show that they are independent when five of these variables are put equal to constants. It is found convenient to set ao = ai = C2 = t/s = and a2 = I. In fact this normalization can be made upon T, V, A, A' before H2, H2', and Si are calculated. The ten invariants (20) then reduce to the fairly simple forms: A = 3:1^0 + (^2 + di)xi, A' = Zxa + (62 + Ci)x2, 9 = — (ci — di)xi^ — biici — ^1)^2' — 2(ci — di)xQX2, H2 = 4ci{di^ - Ml + 62^) + Uboci - 3bi^ - 66i(co + d,) -3(co + do)^ Hi' = 4di(ci^ - biCi + 62') + 12bodi - 3bi^ - 66i(co + do) (21) - 3 (Co + d^y, Si = (ci - diY/Sl [3bi{co - doY + 6{ca - d„){c,d^ - c^dO + 3b, (Co - do){ci - di) - {ci - diy (V + 3bo)l Ai = 3(co — do), T2 = 3(co - doY - c-i(ci - dO\ T2' = 3{co - do)' - di(ci - d,Y, A3 = (co — do)^ + co(ci — di)^ — Ci(co — do){ci — di)'^, 14. Independence of the invariants. To prove the independence of these ten invariants it is sufficient to prove that the jacobian (22) d(A, A', e, H2, Hi', Si, Ai, T^, T^', A3) i>{xo, xi, Xi, bo, bi, bi, Co, Ci, do, d\) does not vanish identically. Now only the first three of these polynomials (21) involve xo, Xi, X2, and only the first six involve bo, bi, bi. Hence it follows from considering the Laplace development that the jacobian (22) factors into three factors, viz., 1922] LINEAR ALGEBRAS 145 d(A, A^ e) ^ bjHj, Hi', Si) ^ a(Ai, Fa, ^2^ A,) b{xo, xi, X2) d(6o, 61, 62) d(co, ci, rfo, ^1) It is sufficient to show that no one of these three jacobians vanishes identically in ci, and hence it is sufficient to show that the coefficient of the highest power of Ci in each jacobian is not identically zero. It is then permissible to drop all terms in each element except those involving Ci to the highest power to which it occurs in that element; for the other terms evidently cannot enter into the term of highest degree in Ci in the expansion of the determinant. It is important to be sure, however, that the coefficient of the highest power does not vanish, for this method does not give the coefficients of lower powers correctly. By this method it is easy to show that d(A, A', e) „ , , , , ' = oxict' + lower powers of Ci, d(Xn. X^. X-)) b{Xo, Xu Xi) oiHi, Hi , Si) 16 , , I \ s 1 1 r = Oi(co — da) Ci^ + lower powers of Ci, i>(bo,b„bi) 9 d(Ai, r^, Ti', A3) ^ _ g^^, ^ j^^^^ p^^^^^ ^j ^^_ d(co, ci, do, di) Then the jacobian (22) becomes 96 Xidi{co — do)ci'^* + lower powers of Ci, and the ten polynomials (20) are functionally independent and form a complete system of invariants and covariants of the linear algebra (14) from the stand - point of Lie. IV. Characterization by invariants of a canonical form 15. The rank covariant. Let us consider the algebra in three units 1, d, et whose general number and multiplication table are given by (13) and (14) respectively. The rank* of every such algebra is three or two according as every number does not or does satisfy a quadratic equation. This rank is an arith- metic invariant under every linear transformation of units (15). It will be shown that for this example the arithmetic rank invariant can be replaced by a rational covariant. * There is an equation p(oi) = of lowest degree having X as a right-hand (left-hand) root. The degree of this equation is called the right-hand (left-hand) rank of the algebra. (Cf. Dickson, Linear Algebras, p. 23.) It can be shown that there is at least one number satisfying no equation of lower degree. 146 C. C. MACDUFFEE [March In every algebra of rank 3 with a principal unit, there is some number x which does not satisfy a quadratic equation. Then the powers 1, x, x"^ are linearly independent and may be taken as the units 1, ei, e^. That is, the multi- plication table of every rank 3 algebra in these units has the form (14) with Oo = ai = 0, 02 = 1. Conversely, every algebra of this form is of rank 3. Evidently, then, a necessary and sufficient condition that an algebra (14) be of rank 3 is that it be possible to make the above normalization. From equations (16) it is seen that the conditions on a transformation (15) which shall make ao' = a/ = and 02' = 1 are A + D-\a20o - ao02)B + D-'iao0, - a^C = 0, D-'ftS - D-ij3iC = 0, .-D-^UiB + D-^aiC = 1. These conditions are readily found to be equivalent to (23) /3o = A, ^1 = B, /32 = C, where A, B, C, given in (16), are polynomials in the as and the constants of multiplication. Thus the as can be chosen arbitrarily and the P's calculated from the above relations, provided only that the determinant of the transforma- tion (24) D = Oztti' + (— fli + C2 -f 62)01^02 + (62 — Ci — di)aiai^ — bia^^ be different from zero. Now a\ and a^ can evidently be chosen so that D is not zero unless every coefficient of D vanishes. Then a necessary and sufficient condition that the algebra be of rank 3 is that not every coefficient 02 etc. of (24) vanish. It will now be shown that there exists a covariant $ whose coefficients are the coefficients of (24). From (17) we form the absolute covariant r' - r + I (A2 - A'2) every term of which involves (ci — ") + 4B"C'D' = F, B"2 + SB" = G, 76 = 4 Z^wps - 3 Pr^pi + 6 ^2;^^^ _ 12 nirpq^ + 8 gV' - 27 ^2g2^^ £ = i [81 S4/(C' - P')' - 6(C'' - DO)iC'Do - CD') + \ iC' - b'){H^ - H,') + C'D'iC - D')^], F = (ZJ'Fa - C'H2')/{C' - D') + AC'D'{C' + D') - 3(C'' + £"')^ ^ = (4C^ - 4P0 ^-^^ ~ ^' + 4^'-^'(^' - ^')1' Z = C - Z?», w = Bo{C' - D') - B'iC - Z?"), • f = - i (C - D'), q = - I (C - D'), ;• = _ I B"(,C' - D'), s = I [S'XC - £">) + 2C'D'> - 2C»£)']. Equations (37) determine the invariants 5", J5', B" uniquely. A practical method for determining them is to calculate the two solutions of the first three equations of (37) and retain only the one which satisfies the fourth relation. Thus the invariants of the linear algebra (14) which we have found are suffi- cient to isolate and characterize the generic case. The University of Chicago, Chicago, III. VITA Cyrus Colton MacDuffee was born June 29, 1895, at Oneida, New York. He attended the public grammar school and high school in that city. In 1917 he was graduated from Colgate University with the degree of B.S. The two following years he spent as instructor in mathematics at Colgate. He attended the summer school of Cornell University during the summer of 1917. He en- tered the University of Chicago in June, 1919, and was in residence for nine successive quarters. He attended courses under Professors E. H. Moore, L. E. Dickson, G. A. Bliss, E. J. Wilczynski, H. E. Slaught, J. W. A. Young and K. Laves of the University of Chicago faculty, and during the summer quar- ters he attended courses under Professor M. W. Haskell of the University of California, Professor S. Lefschetz of the University of Kansas, Professor T. Hildebrandt of the University of Michigan, and Professor H. Blumberg of the University of Illinois. In June 1920 he received the S.M. degree, and in Sep- tember 1921 the degree of Ph.D. Since then he has been instructor at Prince- ton University. The writer wishes to express his gratitude to all the members of the mathe- matics department, and especially to Professor L. E. Dickson under whose direction this thesis was written. __J^- the lase date stamped beW. JUN 2 1< SOFeb'SOCD LD 21-i00m-9,>, 47(A5702sl6)47a