Loy, Richard J.

English

Loy Richard J.

University of Szeged

A note on the preceding paper by J. B. Miller By RICHARD J. LOY in Clayton (Victoria, Australia)*) 1. Introduction Let 21 be an associative algebra over a field K of zero characteristic, 2lft[iJ, the algebra, over 21, of polynomials of degree =5 k in a commutative indeterminate t with the usual multiplication modulo the principal ideal (tk+v). We consider (algebra) homomorphisms of 21 into 2tfc[i]. Much of the definitions and notation, of [1] will be used, without further explanation. Suppose H is a homomorphism of 21 into 2It[i] so that, if a£21, H(a) = a0 + tai + t2a2'+ ... +tkak. Writing a0 = (p(a), ai = Fl(a) (/ = 1, 2, ..., k) it is clear that the maps <p, F, are linear transformations'over 21. Furthermore, since H is a ' homomorphism it follows,, if a, b £ 21, that (i) (p{ab) = (p{a)(p{b), (ii) . F^ab) = 2 Fj( a)Fi-j(b), where F0 = cp. The problem is to obtain a representation for H in terms of transformations on 21 of some given type. This has been done in [1] under the supposition that (p-is the identity endomorphism, I, on 21. For completely general endomorphisms the problem appears intractible but under suitable restrictions a solution can be-obtained. To be more specific, let {a)) (n = 1, 2, ..., k). . Thus the e-homomorphisms of [1] are /-homomorphisms in this nomenclature.. *) The author is a General Motors—Holden's Limited Research Fellow. •234 R. J. Loy An operator Z> €23(91) will be called a (/»-derivation if it commutes with cp and satisfies D(ab) = D(a)cp(b) + cp(a)D(b) (a, bf^l). Thus F t of (ii) is a (^-derivation if H is a ¡p-homomorphism. Finally, the endomorphism cp will be called averaging if cp{a<p(b))=cp(a)cp{b) = cp(cp(a)b) (<7, ¿>€9X). •Clearly any idempotent endomorphism is averaging and, conversely, if 21 has an identity, or if the range of <p contains an element which is not a left (or right) divisor •of-zero, then cp is idempotent. The results of [1] are extended to (p-homomorphisms for (p an idempotent •endomorphism and for cp an averaging endomorphism. The author wishes to express his thanks to Professor J . B . MILLER for suggesting the possibility of extending the results of [1] and for his constant encouragement .and guidance in preparation of this paper. 2. Representations of <p-homomorphisms Lemma. Let ip €23(91) be an averaging endomorphism and Z> €23(21) a <p-deriv-ation. Then for a, ¿»€21, «=1,2,..., cp(D"(ab)) = -'(/->)} • Proof . The result is clear for n= 1. For n — 2 we have, if a, ¿>€2f, .<p{D\ab)) = cp(D(D(a)cp(b) + cp(a) D(b)j) = = <p (D\a) cp2(b) + cp (D(a)) cp (D (b)) + cp (D (a)) <p(D (b)) + cp2(a) D\b)) = = cp(D\a)b + 2D{a)D(b) + aD2(b)) : since <P (<P (*) = <P {xcp iy)) = cp (A-) ,p(y) = cp (xy) if x, y€2l . An inductive argument gives the general case. C o r o l l a r y . With cp, D as in the lemma define <p exp D as the (formal) sum ~ 1 y. — cpD". If cp exp D defines an operator in 23(2t), in particular if D is nilpotent, ,n-o n\ jhen this operator is an endomorphism. Proof . Follows from the lemma in the obvious manner. T h e o r e m . Let cp € 23(91) be a given averaging endomorphism and Dx, D2, . .., ...., Dt€23(21) given cp-derivations. Let <p be the operator in Zk +,(23) with cp along .the leading diagonal and zero elsewhere. Then fp exp (Dk+,), as defined above, is the matrix in J t + 1 (93) of a cp-homomorphism of 2t into 2lt[/] with F, = cpD{. Note on the preceding paper 235 Conversely, given a cp-homomorphism H of 21 into with cp idempotent, its matrix Fk+l satisfies <pFk+1=tp e x P (At + i) where DX,D2, ..., Dk€23(21) are (p-derivations and D{ can be taken as F , . , Further-more the (pDt are uniquely determined. P r o o f . The proof of the first part is exactly as in [1]. For the partial converse, note that if H is a (p-homomorphism of 21 into 2lk[t], with (p idempotent, then Hep is a 7-homomorphism of cp(21) into <p(2t)fc[/]. Thus by [1] the matrix Fk+lq> of Hq> satisfies (iii) Fk+l<p = exp(\k+i) where Fk+i is the matrix of H, and the operators Ai,A2,...,Ak of At + i are derivations on <p(21)- For i=\,2,...,k define Bi by n f \ _ { 0 i f a £ker 0?)' U i W U ( a ) if aOm{<p) and extend D t linearly to the whole of 2t. The resulting operator is well defined since 21 = ker ((p)ffiim (cp) (see §3 below). But then if a j = X j + y j , x y€ker (cp), ^•e im (cp) for . /=1 ,2 , D-iaya2) = Di(xxx2 + yix2 + x^^ + y^y^ = AO'iJ'2) since ker {cp) is a two-sided ideal. Since im (cp) is a subalgebra A O I « 2 ) = = = ^i(yi)y2 + yi^i(y2) = JDiia^yiaJ + viaJDiiyJ; moreover, Di=Dicp = (pDi and so Dt is a (^-derivation on 21, / = 1, 2, ..., k. Also, if a £2t, I>;(a) — A¡((p(a)), so Ai(p = Di. By (iii) then, since y>2=y> and A ¡(p = (pA ¡cp for each i, Fk+Xq> = exp (\k + x)q> =) = y> exp (Dk+,) and the result follows. R e m a r k s . 1. The above result remains valid if the base field D has nonzero characteristicp, provided k<p. This restriction ensures that <pexpDk+1 is defined. 2. If it is supposed that q>Fn = Fn(p = F„ (« = 1, 2, ..., k) then 'the process /»' of [1] can be applied to give results analogous to those in [l]'with rp-derivations in place of derivations. 3. The result for k = «> generalizes in the same manner. 236 R. J. Loy: Note on the preceding paper 3. Existence of (/»-derivations Lemma. An algebra 2t admits an idempotent endomorphism if and only if it admits a (vector space) direct sum representation 91 = 3 © © where 3 is a two-sided ideal and 93 is a subalgebra. P r o o f . If ), 23 = im(<p). Then 3 is a two sided ideal and 93 is a subalgebra. If a£9I , a = (a — (p(a)) + <p(a) so 91 = 3 + 93. Since <p is zero on 3 and is the identity on 23 it follows that the sum is direct. Conversely, if 21 = 3 © © for some two-sided ideal 3 and subalgebra 23, define a transformation (a)=y. Then cp is clearly idempotent and linear. Also, if a, = x{ , a2=x2+y2 <P(a 1^2) = <P(x\X2 + yix2 + xiy2 + yly2) = = vOiVz) = y\yi = <p(fl\)(p(a2). Thus (p is an idempotent endomorphism. T h e o r e m . Let 21 be an associative algebra which has a direct sum representation 9( = 3 ©93 as in the lemma, with 23 non-commutative. Then 21 admits a non-zero (p-derivation D such that (pD = Dip =D, <p being defined as in the lemma. Proof . Let 2( be any element such that cp(a) is not in the centre of 23. Straightforward calculation shows that the operator D defined by D(x) — (p(ax — xa), x 6 9(, has the desired properties. C o r o l l a r y . Let A-,A2,... be inner derivations of 21, D , , Z)2, ... the cor-responding (p-derivations as defined in the preceding theorem, that is, Di = (pA-t, i=\,2, ... . Then (p exp (D) —(pF where F is an inner endomorphism in X„(2l) determined by the A,,A2, ... as in [1 ] . P r o o f . By § 4 of [1], exp (A)/i = exp (C) A exp ( - C), A £ 2„(2l), whence ( exp (D) = q> exp (yA) = q> exp A, and the result follows. Bibliography [1] J. B. MILLER, Homomorphisms, higher derivations, and derivations of associative algebras, Acta Sci. Math. 28 (1967), 221—232. (Received September 21, 1966)

A note on the preceding paper by J. B. Miller

http://acta.bibl.u-szeged.hu/14155/1/math_028_fasc_001_002_233-236.pdf

A note on the preceding paper by J. B. Miller

Abstract

Similar works

Full text

Available Versions

University of Szeged