An Introduction to Nonassociative Algebras

Richard D. (Richard Donald) Schafer

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• Publié par: aftoi
• Publié le: 08 décembre 2010
• Nombre de lectures: 34
• Langue: English

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The Project Gutenberg EBook of An Introduction to Nonassociative Algebras, by R. D. Schafer

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Title: An Introduction to Nonassociative Algebras

Author: R. D. Schafer

Release Date: April 24, 2008 [EBook #25156]

Language: English

Character set encoding: ASCII

*** STARTOFTHISPROJECTGUTENBERGEBOOKNONASSOCIATIVEALGEBRAS ***

AN INTRODUCTION TO NONASSOCIATIVE ALGEBRAS

R. D. Schafer

Massachusetts Institute of Technology

An Advanced Subject-Matter Institute in Algebra Sponsored by The National Science Foundation

Stillwater, Oklahoma, 1961

Produced by David Starner, David Wilson, Suzanne Lybarger and the Online Distributed Proofreading Team at http://www.pgdp.net

Transcriber’s notes This e-text was created from scans of the multilithed book published by the Department of Mathematics at Oklahoma State University in 1961. The book was prepared for multilithing by Ann Caskey. The original was typed rather than typeset, which somewhat limited the symbols available; to assist the reader we have here adopted the convention of denoting algebras etc by fraktur symbols, as followed by the author in his substantially expanded version of the work published under the same title by Academic Press in 1966. Minor corrections to punctuation and spelling and minor modifications to layout are documented in the L A TEX source.

iii

These are notes for my lectures in July, 1961, at the Advanced Subject Matter Institute in Algebra which was held at Oklahoma State University in the summer of 1961. Students at the Institute were provided with reprints of my paper, Structure and representation of nonassociative algebras(Bulletin of the American Mathematical Society, vol. 61 (1955), pp. 469–484), together with copies of a selective bibliography of more recent papers on non- associative algebras. These notes supplement§§3–5 of the 1955 Bulletin article, bringing the statements there up to date and providing detailed proofs of a selected group of theorems. The proofs illustrate a number of important techniques used in the study of nonassociative algebras.

Stillwater, Oklahoma July 26, 1961

R. D. Schafer

I. Introduction

By common consent a ringRis understood to be an additive abelian group in which a multiplication is defined, satisfying (1) (xy)z=x(yz) for allx, y, zinR and (2) (x+y)z=xz+yz, z(x+y) =zx+zy for allx, y, zinR, while an algebraAover a fieldFis a ring which is a vector space over Fwith (3)α(xy) = (αx)y=x(αy all) forαinF,x, yinA, so that the multiplication inAis bilinear. Throughout these notes, however, the associative law (1) will fail to hold in many of the algebraic systems encountered. For this reason we shall use the terms “ring” and “algebra” for more general systems than customary. We define aringRto be an additive abelian group with a second law of composition, multiplication, which satisfies the distributive laws (2). We define analgebraAover a fieldFto be a vector space over Fwith a bilinear multiplication (that is, a multiplication satisfying (2) and (3)). We shall use the nameassociative ring(orassociative algebraring (or algebra) in which the associative law (1) holds.) for a In the general literature an algebra (in our sense) is commonly referred to as anonassociative algebrain order to emphasize that (1) is not being assumed. Use of this term does not carry the connotation that (1) fails to hold, but only that (1) is not assumed to hold. If (1) is actually not satisfied in an algebra (or ring), we say that the algebra (or ring) isnot associative, rather than nonassociative. As we shall see in II, a number of basic concepts which are familiar from the study of associative algebras do not involve associativity in any way, and so may fruitfully be employed in the study of nonassociative algebras. For example, we say that two algebrasAandA 0 overFare isomorphicin case there is a vector space isomorphismx↔x 0 between them with (4) (xy) 0 =x 0 y 0 for allx, yinA. 1

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INTRODUCTION

Although we shall prove some theorems concerning rings and infinite-dimensional algebras, we shall for the most part be concerned with finite-dimensional algebras. IfAis an algebra of dimensionnover F, letu 1 , . . . , u n be a basis forAoverF. Then the bilinear multiplica- tion inAis completely determined by then 3 multiplication constants γ ijk which appear in the products n (5)u i u j = X γ ijk u k , γ ijk inF. k=1 We shall call then 2 equations (5) amultiplication table, and shall some- times have occasion to arrange them in the familiar form of such a table: u 1 . . . u j . . . u n u 1 . . . u i . . . P γ ijk u k . . . .. u n . The multiplication table for a one-dimensional algebraAoverFis given byu 2 1 =γu 1 (γ=γ 111 ). There are two cases:γ= 0 (from which it follows that every productxyinAis 0, so thatAis called azero algebra), andγ6 In the latter case the element= 0.e=γ −1 u 1 serves as a basis forAoverF, and in the new multiplication table we havee 2 =e. Thenα↔αeis an isomorphism betweenFand this one-dimensional algebraA. We have seen incidentally that any one-dimensional algebra is associative. There is considerably more variety, however, among the algebras which can be encountered even for such a low dimension as two. Other than associative algebras the best-known examples of alge- bras are the Lie algebras which arise in the study of Lie groups. ALie algebraLoverFis an algebra overFin which the multiplication is anticommutative, that is, (6)x 2 = 0 (implyingxy=−yx), and theJacobi identity (7) (xy)z+ (yz)x+ (zx)y= 0 all forx, y, zinL

INTRODUCTION

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is satisfied. IfAis any associative algebra overF, then thecommutator (8) [x, y] =xy−yx satisfies (6 0 ) [x, x] = 0 and (7 0 ) h [x, y], z i + h [y, z], x i + h [z, x], y i = 0. Thus the algebraA − obtained by defining a new multiplication (8) in the same vector space asAis a Lie algebra overF. Also any subspace ofAwhich is closed under commutation (8) gives a subalgebra ofA − , hence a Lie algebra overF. For example, ifAis the associative algebra of alln×nmatrices, then the setLof all skew-symmetric matrices inAis a Lie algebra of dimension 2 1 n(n−1). The Birkhoff-Witt theo- rem states that any Lie algebraLis isomorphic to a subalgebra of an (infinite-dimensional) algebraA − whereA In the generalis associative. literature the notation [x, y] (without regard to (8)) is frequently used, instead ofxythe product in an arbitrary Lie algebra., to denote In these notes we shall not make any systematic study of Lie al- gebras. A number of such accounts exist (principally for characteristic 0, where most of the known results lie). Instead we shall be concerned upon occasion with relationships between Lie algebras and other non- associative algebras which arise through such mechanisms as thederiva- tion algebra. LetAbe any algebra overF. By aderivationofAis meant a linear operatorDonAsatisfying (9) (xy)D= (xD)y+x(yD all) forx, yinA. The setD(A) of all derivations ofAis a subspace of the associative algebraEof all linear operators onA the commutator [. SinceD, D 0 ] of two derivationsD,D 0 is a derivation ofA,D(A) is a subalgebra of E − ; that is,D(A) is a Lie algebra, called thederivation algebraofA. Just as one can introduce the commutator (8) as a new product to obtain a Lie algebraA − from an associative algebraA, so one can introduce a symmetrized product (10)x∗y=xy+yx in an associative algebraAto obtain a new algebra overFwhere the vector space operations coincide with those inAbut where multipli- cation is defined by the commutative productx∗yin (10). If one is

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INTRODUCTION

content to restrict attention to fieldsFof characteristic not two (as we shall be in many places in these notes) there is a certain advantage in writing (10 0 )x∙y= 1 2 (xy+yx) to obtain an algebraA + from an associative algebraAby defining products by (10 0 ) in the same vector space asA. ForA + is isomorphic under the mappinga→ 2 1 ato the algebra in which products are defined by (10). At the same time powers of any elementxinA + coincide with those inA: clearlyx∙x=x 2 , whence it is easy to see by induction on nthatx∙x∙ ∙ ∙ ∙ ∙x(nfactors) = (x∙ ∙ ∙ ∙ ∙x)∙(x∙ ∙ ∙ ∙ ∙x) =x i ∙x n−i = 2 1 (x i x n−i +x n−i x i ) =x n . IfAis associative, then the multiplication inA + is not only com- mutative but also satisfies the identity (11) (x∙y)∙(x∙x) =x∙[y∙(x∙x)] for allx, yinA + . A (commutative)Jordan algebraJis an algebra over a fieldFin which products arecommutative:

(12)xy=yx

and satisfy theJordan identity

for allx, yinJ,

(11 0 ) (xy)x 2 =x(yx 2  all) forx, yinJ. Thus, ifAis associative, thenA +  is any sub-is a Jordan algebra. So algebra ofA + , that is, any subspace ofAwhich is closed under the symmetrized product (10 0 ) and in which (10 0 ) is used as a new multi- plication (for example, the set of alln×nsymmetric matrices). An algebraJoverFis called aspecial Jordan algebrain caseJis isomor- phic to a subalgebra ofA + for some associativeA shall see that. We not all Jordan algebras are special. Jordan algebras were introduced in the early 1930’s by a physi- cist, P. Jordan, in an attempt to generalize the formalism of quantum mechanics. Little appears to have resulted in this direction, but unan- ticipated relationships between these algebras and Lie groups and the foundations of geometry have been discovered.

INTRODUCTION

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The study of Jordan algebras which are not special depends upon knowledge of a class of algebras which are more general, but in a certain sense only slightly more general, than associative algebras. These are thealternativealgebrasAdefined by the identities

(13)x 2 y=x(xy) and (14)yx 2 = (yx)x

for allx, yinA

for allx, yinA,

known respectively as theleftandright alternative laws. Clearly any associative algebra is alternative. The class of 8-dimensionalCayley algebras(orCayley-Dickson algebras, the prototype having been dis- covered in 1845 by Cayley and later generalized by Dickson) is, as we shall see, an important class of alternative algebras which are not as- sociative. To date these are the algebras (Lie, Jordan and alternative) about which most is known. Numerous generalizations have recently been made, usually by studying classes of algebras defined by weaker iden- tities. We shall see in II some things which can be proved about com- pletely arbitrary algebras.