COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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Identities in Jordan algebras C. M. GLENNIE THE main part of this paper is the calculation of the dimensions of certain subspaces of Jordan algebras. From a knowledge of these dimensions we deduce a theorem on identities in Jordan algebras. This is given in the third and fmal section. In the first section we set up some notation and give some preliminary results. The results are not new but it is convenient to gather them together here. The second section gives the statement and proof of the main theorem. The reader should consult the preceding paper by L. J. Paige in this volume for background material. 1. We shall work throughout over a fixed but arbitrary field of characteristic zero and shall not refer to the ground field again. The restriction on the characteristic can almost certainly be relaxed but this would require further investigation which we have not carried out. We shall be working in certain free Jordan and free associative algebras and shall use a, b, c, . . . to denote the free generators. In particular places we shall writep, 4, I, . . . instead of a, b, c, . . . when the result we are stating remains true if the variables are permuted or if we wish to indicate a typical monomial. The element pqrsfsrqp in an associative algebra will be denoted by pqrs, and called a tetrad. Similarly pqmt + tsrqp is pqrst and so on. Tetrads such as abed, dcba, acef, fzin which the letters appear in alphabetical or reversed alphabetical order will be called ordered tetrads. As associative products occur only under bars we shall also use juxtaposition to denote the Jordan product ip? Products in the Jordan algebras will be left normed, i.e. xyz means (x~)z and so on. We use the following notation. -W M(n) N(n) SW subspace of the free Jordan algebra on n generators spanned by monomials linear in each generator, subspace of the free special Jordan algebra on II generators spanned by monomials linear in each generator, subspace of the free associative algebra on iz generators spanned by the &n! elements W arising from the n! monomials w linear in each generator, (n Z= 2) subspace of L(n) spanned by monomials pw where w is a monomial linear in each of the generators other than p, 307
308 T(n) U(n) 0) [VI R-4 C. M. Glennie (n 2 3) subspace of L(n) spanned by monomials pqw where ~1 is a monomial linear in each of the generators other than p and q, (n 3 2) subspace of s(n) spanned by monomials pw with p =I= a, (n a 3) subspace of T(n) spanned by monomials pqw with p =l= a andq + a, subspace spanned by the subset W of a vector space, the mapping y + yx, x and y elements in the Jordan algebra under consideration, P(x, Y, 3 RWKv4 + R(YVW) + WWY), Qtx, Y, 4 R(Yz)RW + RWR(Y) + Rtx~)R(z), Sk Y, 4 RWWR~Y) + R~YWWW. With the above notation the linearized form of the Jordan identity xyy2 = xy2y is xPcV, z, t> = ~Q(Y, z, t) (1) or xR(yzt) = xP(y, z, t)-xS(y, z, t). (2) From (1) and (2) we have at once XNYZO = xQtv, z> t>-~S(Y, z, 2). (3) It is clear that M(n) G N(n). We have also LEMMA 1. For n 3 3, U(n)+ V(n) = L(n). Proof. Let w E L(n). Then w is a sum of elements aRwhere R is a monomial in operators R(x) and each x is a monomial in some of the generators b, c, . . . . If x contains more than two generators then by (2) R(x) can be expanded as a sum of words R(y) where each y contains fewer generators than X. Repeating such expansions as often as necessary gives the result. COROLLARY. 5’(n) +T(n) = L(n) and S(n) + V(n) = L(n). LEMMA 2. dim S(n) =Z n dim L(n - I), dim T(n) G &t(n - 1) dimL(n - 2), dim U(n) =Z (n - 1) dim Len - l), dim V(n) == g(n - l)(n - 2) dim L(n - 2). Proof. The proofs of these inequalities follow at once from the definitions of S(n), etc. The following relations, in which p, q, r, . . . denote distinct elements from b, c, d, . . . and x is a monomial in the remaining generators, are either clear from the definitions of the operators or follow easily from (1) (2) (3) and previous relations in the set. xQ(p, q, r) E U(n) (4) xP(P, 4, r> E u(n) (5) xS(P, 4, r> E u(n) (6) Mw-) E u(n) (7) Identities in Jordan algebras 309 xtqrtst)) - xQ(q, r, st) E u(n) (8) x(qr)(st) - xQ(q, r, st) E U(n) (9) xp(9W)--ptp, 9r9 st> E 0) (10) x(qrp)(st) + x(stp)(qr) - xQ(p, qr, st) E U(n). (11) The following lemmas are due to Cohn. Proofs will be found in [l]. LEMMA 3. abed-(sgn n)pqrs E M(4) where p, q, r, s is the permutation z of a, b, c, d. LEMMA 4. M(n) + [W(n)] = N(n) for n = 1, . . . , 7 where W(n) = 4 (the empty set) for n = 1, 2, 3, W(4) = {abed), W(5) = .- {pqrst}, W(6) __ = - - - = {pqrstufpqrsut, -. pqrs(tu), pqrstu-pqrsut}, W(7) = {pqrstuv+pqrsutv, pqrs(tu)v, pqrstuv-pqrsutv}. In the cases n = 5, 6, 7 the set is to include all elements obtained by replacing p, q, r, . . . by any permutation of a, b, c, . . . such that pqrs is an ordered tetrad. Let U be a subspace of the vector space V and W = {wl, . . . , w,} be a subset of V. If ri (i = 1, . . . , m) denotes the relation ~ iiijWj E U j=l amongst the elements of W and R = {rl, . . . , rm} we shall call the mX n matrix A = (;iij) the word-relation matrix for Wand R. We have LEMMA 5. dim (U+[ W]) =Z dim U+(n-rank A). Proof. Let r = rank A. We can find r elements from W each expressible as a linear combination of some element in U and the remaining n-r elements of W. So U+ [ w] is spanned by any basis of U together with n-r elements from W, and the result follows. 2. THEOREM 1. For n = 1, . . . , 7, dim L(n) = dim M(n). The dimensions are respectively 1, 1, 3, 11, 55, 330, 2345. Proof. The mapping a - a, b - b, etc., can be extended to a linear transformation of L(n) onto M(n). So dim M(n) 4 dim L(n). For each n we now find a number d(n) such that dim L(n) G d(n) and dim M(n) 3 d(n). It follows at once that dimL(n) = dim M(n) = d(n). We shall use w(n) to denote the number of elements in W(n). For simplicity we write L for L(m) and so on when dealing with the case n = m. n= 1. Take d = 1. L is spanned by a single monomial. So dim L =S d. By Lemma4,M=N.SodimM=dimN= Iad. n = 2. Take d = 1. L is spanned by the single monomial ab. SO dimL
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COMPUTATIONAL PROBLEMS IN ABSTRACT
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vi Contents E. KRAUSE and K. WESTON
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X Preface I am indebted to the auth
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4 J. Neubiiser 2.3.5. Added in proo
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8 J. Neubiiser Investigations of gr
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12 J. Neubiiser For 1 es 4 r, ws is
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16 J. Neubiiser Investigations of g
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Some examples using coset enumerati
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40 C. M. Campbell Some examples usi
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44 N. S. Mendelsohn for example, in
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48 H. Jiirgensen Calculation with e
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60 V. Fe&h and J. Neubiiser 2. The
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64 L. Gerhards and E. Altmann Autom
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68 L. Gerhards and E. Altmann Autom
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72 L. Gerhards and E. Altmann ni E
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76 W. Lindenberg and L. Gerhards Se
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80 W. Lindenberg and L. Gerhards Se
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84 K. Ferber and H. Jiirgensen The
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7 The construction of the character
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92 John McKay defining multiplicati
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96 John McKay Construction of chara
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100 John McKay In the table, c indi
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104 C. Brott and J. Neubiiser irred
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108 C. Brott and J. Neubiiser eleme
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112 J. S. Frame The characters of t
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116 J. S. Frame 3. The decompositio
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TABLE 5. The Z, character block for
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132 R. Biilow and J. Neubiiser Deri
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A search for simple groups of order
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140 Marshall Hall Jr. Simple groups
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148 Marshall Hall Jr. otherwise no
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152 Marshall Hall Jr. easily found
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160 Marshall Hail Jr. This leads to
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164 Marshall Hall Jr. b= (OO)(Ol, 0
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168 Marshall Hall Jr. 11. R. BRAUER
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172 Charles C. Sims THEOREM 2.3. Th
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176 Charles C. Sims has a set of im
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180 Degrc - No. Order t N (-63 Gene
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An algorithm related to the restric
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A module-theoretic computation rela
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192 A. L. Tritter expressed in the
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196 A. L. Tritter a question is mos
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200 John J. Cannon stack. The Cayle
Page 107 and 108: , I The computation of irreducible
Page 109 and 110: 208 P. G. Ruud and R. Keown product
Page 111 and 112: 212 P. G. Ruud and R. Keown TX wher
Page 113 and 114: 216 P. G. Ruud and R. Keown 4. L. G
Page 115 and 116: 220 N. S. Mendelsohn where it is kn
Page 117 and 118: 224 Robert J. Plemmons such class [
Page 119 and 120: 228 Robert J. Plemmons ! TOTALS Sem
Page 121 and 122: 232 Takayuki Tamura 8” = 0B if an
Page 123 and 124: 236 Takayuki Tamura Case ,Q = {e).
Page 125 and 126: 240 Takayuki Tamura The calculation
Page 127 and 128: 244 Takayuki Tamura We calculate46
Page 129 and 130: 248 Takayuki Tamura Immediately we
Page 131 and 132: 252 Takayuki Tamura Let U be a subs
Page 133 and 134: 256 Takayuki Tamura TABLE 8. AI1 Se
Page 135 and 136: I The author and R. Dickinson have
Page 137 and 138: 264 Donald E. Knuth and Peter B. Be
Page 155 and 156: 300 Lowell J. Paige Non-associative
Page 157: 304 Lowell J. Paige We may lineariz
Page 161 and 162: 312 C. M. Glennie where in each cas
Page 163 and 164: 316 A. D. Keedwell of the elements
Page 165 and 166: A projective coqfiguration J. W. P.
Page 167 and 168: The uses of computers in Galois the
Page 169 and 170: 328 W. D. Maurer mod p; for a facto
Page 171 and 172: 332 J. H. Conway Enumeration of kno
Page 173 and 174: J. H. Conway Enumeration of knots a
Page 175 and 176: 340 J. H. Conway -thus our symmetry
Page 177 and 178: 344 J. H. Conway 2 string links to
Page 179 and 180: 348 J. H. Conway 3 and 4 string lin
Page 181 and 182: 352 Non-alternating J. H. Conway L
Page 183 and 184: 356 J. H. Conway Alternating 11 cro
Page 185 and 186: H. F. Trotter Computations in knot
Page 187 and 188: 364 H. F. Trotter to a. Schubert [1
Page 189 and 190: 368 Shen Lin sequence of squares, t
Page 191 and 192: 372 Harvey Cohn We also consider GZ
Page 193 and 194: 376 Harvey Cohn In the case of Fig.
Page 195 and 196: 380 Harvey Cohn always dictated by
Page 197 and 198: 384 Hans Zassenhaus A real root cal
Page 199 and 200: 388 Hans Zassenhaus the elements A,
Page 201 and 202: 392 Hans Zassenhaus It is clear fro
Page 203 and 204: 396 R. E. Kalman Invariant factors
Page 205 and 206: 400 List of participants DR. J. J.
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