Reischer, Corina

Simovici, Dan A.

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On Some Generalizations of Lie Algebras

Institute of Mathematics AS CR, v. v. i.

Matematický časopisCorina Reischer; Dan A. SimoviciOn Some Generalizations of Lie AlgebrasMatematický časopis, Vol. 22 (1972), No. 1, 19--28Persistent URL: http://dml.cz/dmlcz/126795Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1972Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMatematický časopis 22 (1972), No. 1 OJa S€ME CrENEKALIZATIONS OF LIE ALGEBRAS COMWA REISCHER, DAN A. SIMOVICI, Iassi (Romania) § 1. Wegik lite psendo-algebra. Let K ke a commutative field and L a vector space over K. Definition. 1 is a weak Lie pseudo-algebra (w. L. p.-aj if, to every pair of vectors x, g/ in L, tfoere corresponds a vector ze L, catted the product of x and y, z — - [x, y\ satwjfying the following axioms: i°. I>> r 4- *] == fc> 2/] -H [*> *1 - lx> °] 2°. [* -t-y, ^ - [as, z] + [y, ^] - [o, 2] 3°. [a:, ocy] = <L[X, y] - (a — l)[a?, o] -f <s, a>y, <«, a> e Ar 4 ° . [a;, ass] = c* 5°. The^e «dfr?/ fiyv E K such that for every x9 y, ze L, [fogfL»] +• [[&/, 2], a;] +- [[*> «], y] — MH^ V\> o] + + 0 , 4 0] + [fe *]> 0]} - *([*, *] + [*, 2/] + [0, *]) - o. CONSEQUENCES FROM THE DEFINITION 1. From theaxioios 1, 2, 4 i t is possible to infer the anticommutativity of the multip-licatieon: (1) [x, y] = - [y, x], if the charsacteiiistic of the field K is different from two. 2. The consequence 1 and the third axiom involve: (2) [cue, y] = a[x9 y] - (a - l)[o, y] — (y, <x>. 3. If we take in the 5-th axiom x = y = 0, we obtain (1 - V ) [ 0 , 3 ] = 0. Since in genoul [c*, z] -^ 0, i t follows tha t v = 1. In the 5-th axiom there rests only the (^onstaait \i called „the constant of type" . 4. J xzt>\ ys = \ y> [a?., Ž/Í] — (n — 1) \ [o, y/J í=l j= 1 i = l í = l J = l 19 II. > Ь-í- o] — (m — 1) *=i § 2. Special classes of w. L. p,-a and examples If [x, o] = [o, x] = o, V x sL, we obtain Lie's pseudo-algebras intro-duced by H e r z [2] and when (x>, a ) — o, V x e L, V a e K, the weak Lie's algebra (w. L. a). Finally, if [x, o] = [o, x] = o and (x, a ) = o, V a: e L, V a e /\r, we obtain the usual Lie algebras. We shall give some examples of w. L. a and w. L. p.-a. 1. Let si be a weak algebra [1], tha t means, a linear algebra in which the distributivity laws are replaced by : x(y + z) = xy + x-z — x - o, (x + y)z = xz + yz — o • z. If si is endowed with the product: (3) [x, y] = xy — yx, wre obtain a w. L. a with the constant of the, type /i = o. This is the weak Lie algebra associated to a weak algebra. 2. Let 0 be the set of functions: f:[GF{2)]*-+[GF(2)]. By using the inner operation: (4) [/> g] = f(xl, • • • > ffn-l, fl^l , • • • , xn)) ® giXl, • • • , #n_i, / fa l , . . . , Xn)) , where the sign ® means the addition modulo 2, we obtain a w. L. a. Indeed, using the common notations from the logical algebra, every / belong-ing to 0 may be written as: f=xnfl ® (1 0 xn)fo = xn(fl®fo) © /o, where / i and /o are respectively / (# i , . . ., xn_i, o) and / (# i , . . ., xn_\, 1). Then: [X y] = (xn(gi © flfo) ® go)(fi © /o) ® /o ® ® (xn(h © /o) ® /o)(flri ® go) © ,9oi = 0o/i © /oflri ® /o © go. 20 We obtain immediately the relation^: [f, 9 0 K\ = [/, gr] © [/, h] 0 [/, e], [ / © gr, A] = [/, A] e for, A] 0 [o, A], [f,f] = o, and f ' [[/, g], M © [[g, h],f] © [[h,f], g] © [/, o] © [gr, o] © [h, o] = o, (Here a = — a (mod 2) ai*d [/, o] = /o) . Using the properties of this algebra we obtained some applications in the switching theory [5, 6]. 3. Let i b e a distributive algebra in the usual sense. If we introduce the product [x, y] = xy — yx + x — y, we obtain a w. L. a. We have: [x, y + z] = [x, y] + [x, z] — [x, o], [x + y,z] = [x, z] + [y, z] - [o, z]. [x, x] = o; [x, o] = x, and [[*, v\ *] + [[y> z]> *] + [[*, %l y] + {[ix> yl °] + [[*> zl °] + + [[«, x], o]} — {[x, o] + [y, o] + [z, o]) = o. For this w. L. a the constant of type is /x = 1. 4. Let ^(r) be the field of meromorphic functions of the variable T in a simply connected region and Rn an n-dimensional vector space of ?^-tuples, defined over ^ ( T ) . Let {ut}, i = 1, n be a basis in Rn. We suppose that Rn is differentiably closed, namely the vectors ut verify the equations du( . . - . - , . (5) — = CLtjUj{T). dr If u e Rn , then ^ -= t^a*, a? G ^ ( T ) . 21 We introduce the mapping T : Rn-> Rn, by dui d — - a* + Ui -dr dT a. (6) Tu = a. + ut — I t is clear tha t : T(u + v) = Tu + Tv. Then dX T(Xu) = XTu + — u . dx dui Because formula (6) contains the term oLi the transformation T depends dT on the basis chosen in Rn. I t is clear tha t there are many transformations T. Let us consider the matrix M\ U of the basis, U = I u% 1 ; because m are vectors the U-type is n X n. \un) If we take a new basis, related to the basis U by the relation U = AV, where A = (A(j) is an invertible n X n matrix, from (5) it follows t h a t : dAij dvj — Vj + Aij — = CLihAhkVk , dT dT or: dvj ( dAik \ Aij ——- = — —— + KihAhk \Vk, dT y dT J and if we denote by B = (Bjf) the inverse of the matrix A, we obtain the new coefficients of the equation (5): / dApj \ &ij — Bip ( — ~ ; + u-phAhj). \ d T / Denote by 3~ the set of all transformations T. Let us consider the operation [.,.]: #~ X &^F, 22 defined by (7) [T, U] = To U- UoT + T- U, where (To U)u = T(Uu). To make £T a w. L. p.-a. we interpret addition and scalar multiplication as the ordinary addition of two transformations and multiplication of a trans-formation by a scalar and use the product defined by (7). Thus [T + U, V]u = ((T +U)o V)u - (V o (T + U))u + + (T + U)u -Vu=(To V)u + (U o V)u - (V o T)u -- (V o U)u + Tu+Uu-Vu = = (T o V - V oT + T - V)u + (U o V - V o U + U - V)u + + Vu = ([T, V] + [U, T])u + Vu. Since we obtain [0, V] = - V, [T + U, V] = [T, V] + [U, V] - [0, V]. Now, by using (7) and (6), we have: [aT, U]u = (OLT o U)u - (U o OLT)U + a.Tu - Uu = = ((OLT O U)U - (aU o T)U + da/dTTtt) + OLTU — Uu = = (OLT O U)U — (OLU O T)U - da/dT Tu + OLTU — Uu = = (aT o U)u - (at/ o T)u + OLTU — OLUU + OLUU — — da/dT Tu— Uu = = a[T, U]u - (a - 1)[0, U]u - <U, a> Tu. For V T e Z7~ by comparing with the relation (2), we have" <T, a> = da/dT. Finally, for the axiom 5, let us calculate: [[T, U], V] + [[U, V], T] + [[V, T], U] = i) = [TU - UT + T - U, V] + [UV -VU+U-V,T] + + [VT - TV + V - T, U] = = (TU - UT + T - U)V - V(TU - UT + T - U) + TU-UT + T-U-V + + (UV - VU + U - V)T - T(UV - VU + U - V) + + UV -VU +U - V -T + + (VT - TV + V - T)U - U(VT - TV + V - T) + X) Here for simplicity we write TU for T o U. 23 + VT -TV +V -T - U = = VU - TU + TV - VT + UT - UV - U - T - - V = = TV-VT-T—V+VU-UV+V—U+UT — TU + + U-T—U — T—V= '' l" = [T, V] + [V, U] + [U, T] + [0, T] + [0, U] + [0, V]. Thus we obtain [[T, U], V] + [[U, V], T] + [[V, T], U] + [T, 0] + [U, 0] + [V, 0] + + [[T, U], 0] + [[U, V], 0] + [[V, T], 0J = 0. § 3. The transformation <. , . > in a w. L. p.-a For the transformation < . , . > : L x K-> K, the w. L. p.-a acts like a do-main of operators over the field K. Now, we shall deal with some properties of these transformations. Proposition 1. If the characteristic of the field K is not two, then <., .> : TJ X X K -> K is a bilinear application. We note next tha t according to the axiom 2, [x + y, ctz] = [xfctz] + [y, ca] — [o, <xz], on the other hand according to the axioms 3 and 2, [x + y, CLZ] = ct[x +.y, z] — (a — l)[x + y, o] + <x + y, CL)Z = = K[X, Z] + «.[y, z] — a[o, z] — (a — l)[x, o] — v — (a — I)[y, o] + <x + y, a>* = = CL[X, Z] — (a — l)[x, o] + (x, <z)z + a[y, z] — — (a — I)[y, o] + <y, K)Z — a[o, z] — <p, a)z. Hence, <.r + y, CL)Z = (x, a>z + (y, *)z — (o, CL)Z. This relation holds for V z e L, therefore: (x + y, a> = <a;, a> + (y, a> — <o, a>. If we set y = — x, taking into account proposition 2, when the characte-ristic of the field is different from two, we obtain <0, a> = 0 and so <# + y, a> = (x, a> + <y, a>. Hence [x, (a + /%] = [a:, a?/ + 0y] = [x, ay] + [x, fy] - [a;, o] = = a[x% 2/] - (a - l)[x, o] + (x, x)y + /?[^, y] - (t? - l ) [r , o] + 24 + <s, P>y — [x, o] = i - (a + P)[x9 y] - (a + /? ~ l)[x9 o] + <x9 *)y + (x, 0>y. On the other hand, [x, (a + P)y] = (a + 0)[x9 y] - (a + (3 - 1)[*, o] + <s, a + /3>y> therefore <a, a + 0> = <a, a> + <#, /?>. Proposition 2. 7/ ^e characteristic of the field K is not two, then <(3y,a> = (5<y,a>. Let. us calculate the expression . [ax, fly] = §[ax, y] — (/? — l)[o.r, o] + <ox, /?># = = (l(a[x, y] — (a — l)[o, y] — <y, a>x) — — (/? — l)a[#, o] + <a:r, /?> y, therofore [ax, fy] = a|8[a:, y] — /?(a — l)[o, y] — <y, a>/fo — — a(/? — l)[z, o] + <aa;, /J>y = a/?[x, y] — a(/? — l)[tf, o] — /?(a — l)[o, y] — <y, a>fa + <aar, /?>y =-= */?([*. 2!] — [». °] - [°. 2/]) + «[*. °1 + P[o, y] -— <y. «>/?* + <ox, /3>y. Then whence [/%, aæ] = — [aж, /?j/] = aß([y, x] - [y, o] - [o, x]) + + ß[У, o] + a[o, x] — <x, ß>ay + <ßy, a>x; or <y, a>flx — <ax, $>y = </Jy, a>a: — <x, fi>ay, (P<y, a> — <py, a»x + («.<x, p> - <ax, fi»y = o. As this relation holds V x, y e L, we get the necessary equality.. Proposition 3. We have (8) <x, a/3> = <x, a>/3 + <x, /3>a, /jerece the elements of a w. L. p.-a may be interpreted like differentiations oven the field K. 25-Indeed [x, a/Sy] = [x, a(py)] = a[x, 0y] - (a - l)[.r, o] + <x, a)fy = a(/3[.r, y] - (0 _ i)[„, 0 ] _|_ (x> ^y) _ - (a - l)[.r, o] + (x, a)py = = after, y] - a(/? - l)[.r, o] + <.r, ,S>ay - (a - l)[.r, o] + + <.r, a>/?y = after, y] - (a/S - l)[.r, o] + <x, p)ay + <.r, a>/Sy. On the other hand [x, (a/%] = after, y] - (aj8 - l)[.r, o] + <x, a/S>y. Comparing these two expressions we obtain the relation (8). Proposition 4. <x, ±n) = o,x e L, where n = 1 + . . . + l, (n times), 1 being the unity in K. If in axiom 3 we put a = 1, it is easy to see tha t <x, 1> y _ o, V y e L, hence <.r, 1> = o. Thus <.r, o> = [.r, o] — [x, o] = o. Thus <x, - 1) = <.r, — 1> + <.r, 1> = <.r, 1 - l> = <x, o) = o. Taking into account the proposition 1, we obtain P. 4. Corollary [x, -y] = [y, x] + 2[.r, o] [ - x, y] = [y, x] + 2 [o, y ] , Proposition 5. <.r, — a> = — <x, a). Indeed: <x, - a) = <x, ( - l ) a > = <x, - 1>« + <.r, « > ( - l ) = - <x, a), using the propositions 3 and 4. Proposition 6. For every aeK and x, y, eL, the following holds: (9) fi«x, a) - <y, a))[z, o] + «[.r, y], a) - <x, <y, « » + <y, <x, a)))z = o. Let us consider now the axiom 5: [[x, y], az] + [[y, az], x] + [[az, x], y] _ M{[[x, y], o] + [[y, az], o] + [[az, x],o]} - ([o, .r] + [o, y] + [o, az]) = o. Hence, since: [[x, y], az] = a[[x, y], z] - (a - l)[[x, y], o] + <[x, y], a)z, [[y, az], x] = a[[y, z]r x] _ <y, a >[ 0 , cr] + <cr, «>[y, o] -- <*, «>[-y, 2] + <2/, a>[z, .r] + (1 - a)[[y, 0 ] , .r] — <.r, <y, a » z , [[az, x], y] = a[[z, x], y] + <x, a)[o, y] + <y, a>[ 0 , .r] + 2 6 + (1 — a)[[o, x], y] - <y, a>[z, a;] — <.*, a>[z, y] + <y, <ar, a » z , and [[y, az], o] = a[[y, z], o] — (a — l)[[y, o], o] -f- <y, a>[z, o], [[az, a], o] = a[[z, a], o] — (a — l)[[o, a], o] — <ar, a>[z, o], it follows that: «([[*» 2!]> «1 + Ely. «1, * ] + [[z. «]» 2/1) + (<[.*, y], a> — (x, <y, a}} + + <y, <*, a»)z + (1 — a)([[o, a], y] + [[a:, y], o] + [[y, o], a]) — — /*([[*, y]» o] + «[[y, z], o] + a[[z, as], o] — (a — l)[[y, o], o] — — (a — -)[[o» * ] , o] + <y, a>[z, o] — <«, a> [z, o]) — [o, a?] — — [o,y] — <x[o, z] = o, and using axiom 5, we obtain relation (9). Corollary. If p, = o or [z, o] — o we obtain Herz's relation <[*, y], a> = <.r, <y, a » — <y, <x, a » . § 4. The adjoint transformation in a w. L. p.-a We introduce in a w. L. p.-a the adjoint transformation by: ad(a;)y == [x, y] — [x, o] — [o, y]. Proposition 7. In a w. L. p. a. ad is a pseudo-linear transformation. Indeed, we have: ad (x)ay = [(x, ay] — [x, o] — [o, <xy] = = a[», y] — (a — l)[x, o] — [x, o] + (x, a>y — — a[o, y] = a|>, y] — a[.r, o] — a[o, y] + <x, ct}y = = a ad (a?)y + <x, a>y, аnd hence аd {x)(y + z) = [ж, y + z] — [x, o] — [o, y + z] = = [x, y] + [x, z] — [x, o] — [x, o] — [o, y] — [o, z] = = аd (x)y + аd(ж)z, аd (x)(ay + ßz) = a аd (x)y + /З аd (ж)z + <.r, a>y + <ж, jS>z. If <#, a> -= o, V a; G X, then ad is a linear transformation (the case of weak Lie algebras). 27 Proposition 8. There hold the following equalities: ' ! ad (oLx)y = a ad (x)y — <j/, OL)X, and • : • • ; , . • . ad (x -f i/)2 = ad (.x)z -f ad (y)z. The proof of these relations is an immediate one. If in the set of the adjoint transformations of a weak Lie algebra we introduce the ordinary addition and multiplication by a scalar, and use the product : ! s ([ad (x), ad (y)])z = (ad [x, y])z, then this set becomes a weak Lie algebra. REFERENCES [1] CLIMESCU, Al.: Anneaux faibles. Bull. Inst. Polit. din Iasi, 7 (f. 1-2), 1961, 1^-5. [2] HERZ, J . C : Pseudo algebres de Lie. C r. Acad, sci., 256, No 14, I, 1935-1937; I I , 2289-2291. [3] JACOBSON, N.: Lie Algebras. New York, 1962. [4] JACOBSON, N.: Pseudo-linear transformations. Annals of Math., 38, 1937, 485-507 . [5] REISCHER, C - S I M O V I C I , D.: Associative algebraic structures in the set of the Boolean functions and some applications in automata theory. I E E E Trans, on Compu-ters, c-20,3 1971, 9—14, [61 SIMOVICI, D.: Sur la commutativite de la superposition des fonctions booleonnes et certaines proprietes ^'iteration. Analele st. Univ. Iasi, 15, 1969, 449 — 455. Received December 16, 1969 Departmt of Iassi Romania ent of Mathematics University 28 

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On Some Generalizations of Lie Algebras

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