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Synopsis of Linear Associative Algebra. a Report on Its Natural

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Synopsis of Linear Associative Algebra. a Report on Its Natural SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA A REPORT ON ITS NATURAL DEVELOPMENT AND RESULTS REACHED UP TO THE PRESENT TIME BY JAMES BYRNIE SHAW Professor of Mathematics in the James Millikin University WASHINGTON, D. C. : Published by the Carnegie Institution of Washington 1907 CARNEGIE INSTITUTION OF WASHINGTON PUBLICATION No. 78 Z&t OT (gafftmort (preeg BALTIMORE, HD., D. A. CONTENTS. INTRODUCTION. ... ERRATA. Page. 11. Line read 13, for \e tj \'' |c,,|'. 15. In the foot-notes change numbering as follows : for 1 read 2, for 2 read 3, for 3 read 4, for 4 rend 1 . 26. Line 21, for A" read A,, . 33. Line e read 15, for Aes k Ae^,, 34. Line 6, for [?, (,,) read [rn((p^. 49. Line 6, for njj'j., read m' 53, 54. In the table for r>6 in every instance change r2 to r 3, and r 3 to r 4. = / case (27), however, read e, (211) (12 r 3). 57. Line read t . 8, for t, t 59. Line 33, remove the period after A. 67. Line 12, insert it comma (,) after "integer". to *. 68. Lines 9 and 10, change / 71. Line 17, in type III for e n read e l( . l l 72. Last line, for aqa~ read aqa~ . In 2* 6 73. Line 3 from bottom, for jk read jk . 94. Line 7, for Si'j' read Si*j'\ l '.' 1. L.ist in the second column of the determinant and third line S. line, for j~ /' <f> <f> l read S . .j~ y* <j> 100. Line 12, for = read < = <. 106. Some of these cases are equivalent to others previously given. 3 = = 107. Line from bottom, e., read for ~(221) e, (211). * 2 ' a- 116. Line 25 should read p = '. 'J 124. Note 3, add: cf. BEEZ S. 128. Line 1 1, for = 1 .... k, read i=\....Ji,. SYNOPSIS OF LINEAR ASSOCIATIVE ALGEBRA INTRODUCTION. 1 This memoir is genetic in its intent, in that it aims to set forth the present state of the mathematical discipline indicated by its title: not in a comparative study of different known algebras, nor in the exhaustive study of any particular algebra, but in tracing the general laws of the whole subject. Developments of individual known algebras may be found in the original memoirs. A partial bibliography of this entire field may be found in the Bibliography of the 2 Quaternion Society, which is fairly complete on the subject. Comparative 8 studies, more or less complete, may be found in HANKEL'S lectures, and in 4 CAYLEY'S paper on Multiple Algebra. These studies, as well as those men- tioned below, are historical and critical, as well as comparative. The phyletic 6 development is given partially in STUDY'S Encyklopadie article, his Chicago 7 Congress* paper, and in CARTAN'S Encyclopedic article. These papers furnish numerous expositions of systems, and references to original sources. Further historical references are also indicated below. 8 In view of this careful work therefore, it does not seem desirable to review the field again historically. There is a necessity, however, for a presentation of the subject which sets forth the results already at hand, in a genetic order. From such presentation may possibly come suggestions for the future. Attention will be given to chronology, and it is hoped the references given will indicate priority claims to a certain extent. These are not always easy to settle, as they are sometimes buried in papers never widely circulated, nor is it always possible to say whether a notion existed in a paper explicitly or only implicitly, consequently this memoir does not presume to offer any authori- tative statements as to priority. The memoir is divided into three parts : General Theory, Particular Sys- tems, Applications. Under the General Theory is given the development of the subject from fundamental principles, no use being made of other mathematical disciplines, such as bilinear forms, matrices, continuous groups, and the like. 'Presented, in a slightly different form, as an abstract of this paper, to the Congress of Arts and Sciences at the Universal Exposition, St. Louis, Sept. 22, 1904. ' Bibliography of Quaternions and allied systems of mathematics, Alexander Macfarlane, 1904, Dublin. 'HAMKBL 1. References to the bibliography at the end of the memoir are given by author and number of paper. CAYLET 9. 'ST0DT8. STUDY 7. 'CABTAN 3. BEXAN 2, GIBBS 2, R. GRAVES 1, HAOEN 1, MACFABLANE 4. 1 5 ASSOCIATIVE ALGEBRA 6 SYNOPSIS OF LINEAR 1 of sets. The We find the first Buch general treatment in HAMILTON'S theory of in this was made by first extensive attempt at development algebras w:iy S was It lias been critic- BENJAMIN PEIRCE . His memoir really epoch-making. who has undertaken to extend Peirce's method, ally examined by HAWKKS", 4 treatment of a similar character was by showing its full power . The next to several theorems of CARTAN", who used the characteristic equation develop the semi-simple, or Dedekind, much generality. In this development appear The important theorem and the pseudo-nul, or nilpotent, sub-algebras. very the use of double that the structure of every algebra may be represented by is the ultimate the first factor being quadrate, the second non-quadrate, units, who reaches. The latest direct treatment is by TABER", proposition he Peirce had reexamines the results of PEIRCE, establishing them fully (which them to domain for the coordinates. not done in every case) and extending any the are not only in the field of [His units however linearly independent or field coordinates, but for any domain ] have been followed Two lines of development of linear associative algebra continuous It was besides this direct line. The first is by use of the group. 7 The method was followed PoiNCARE who first announced this isomorphism. 8 and non ScHEFFERS who classified algebras as quaternionic quaternionic. by , can be so that In the latter class he found "regular" units which arranged in terms of those which the product of any two is expressible linearly to order five follow both. He worked out complete lists of all algebras 9 who added the theorems that quater- inclusive. His successor was MoLiEN , and that algebras nionic algebras contain independent quadrates, quaternionic He did can be classified according to non-quaternionic types. not, however, CARTAN. reach the duplex character of the units found by C.S. PEIRCE" The other line of development is by using the matrix theory. ' in it sooner. The first noticed this isomorphism, although embryo appeared u 13 . The former shows that the line was followed by SHAW and FROBENICS determines its units, and certain of the direct equation of an algebra quadrate the other units form a which with the quadrates units ; that nilpotent system The is thus made a sub- may be reduced to certain canonical forms. algebra of the associative units used in these canonical forms. algebra under the algebra has a DEDEKIND whose FROBENIUS proves that every algebra sub-algebra, factors in the of the This is the semi- equation contains all equation algebra, showed that the units form a simple algebra of CARTAN. He also remaining units be nilpotent algebra whose may regularized. of these It is interesting to note the substantial identity developments, will be in the order aside from the vehicle of expression. The results given the method of derivation. The of development of the paper with no regard to references will cover the different proofs. 2. HAWKES 1, 3, 4. > HAMILTON 1. B. PBIRCE 1, 3. HAWKES ' SCHEFFEBS 8. CARTAN 2. TABER4. POINCAKE 1. 1, 2, IOC. 8. PEIBCEl, 4. "SHAW 4. "FHOBENIUS 14. INTRODUCTION 7 The last chapter of the general theory gives a sketch of .the theory of general algebra, placing linear associative algebra in its genetic relations to linear general algebra. Some scant work has been done in this development, the line of 1 particularly along symbolic logic. On the philosophical side, which this treatment general leads up to, there have always been two views of complex algebra. The one regards a number in such an algebra as in a reality duplex, triplex, or multiplex of arithmetical numbers or expressions. so-called The units become mere umbrae serving to distinguish the different 2 coordinates. This seems to have been CAYLEy's view. It is in essence the view of most writers on the subject. The other regards the number in a linear algebra as a single entity, and multiplex only in that an equality between two such numbers implies n equalities between certain coordinates or functions S of the numbers. This was HAMILTON'S view, and to a certain extent GRASS- MANN'S.* The first view seeks to derive all properties from a multiplication table. The second seeks to derive these properties from definitions applying to all numbers of an algebra. The attempt to base all mathematics on arith- metic leads to the first view. The attempt to base all mathematics on algebra, or the theory of entities defined by relational identities, leads to the second view. It would seem that the latter would be the more profitable from the standpoint of utility. This has been the case notably in all developments along this line, for example, quaternions and space-analysis in general. HAMILTON, and those who have caught his idea since, have endeavored to form expressions for other algebras which will serve the purpose which the scalar, vector, conjugate, etc., do in quaternions, in relieving the system of reference to any unit-system. Such definition of algebra, or of an algebra, is a develop- ment in terms of what may be called the fundamental invariant forms of the algebra.
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