Jordan Algebras and Mutation Algebras. Homotopy and Von
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ANNALES SCIENTIFIQUES DE L’UNIVERSITÉ DE CLERMONT-FERRAND 2 Série Mathématiques SANTOS GONZALES JIMENEZ Jordan algebras and mutation algebras. homotopy and von Neumann regularity Annales scientifiques de l’Université de Clermont-Ferrand 2, tome 97, série Mathéma- tiques, no 27 (1991), p. 177-191 <http://www.numdam.org/item?id=ASCFM_1991__97_27_177_0> © Université de Clermont-Ferrand 2, 1991, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’Université de Clermont- Ferrand 2 » implique l’accord avec les conditions générales d’utilisation (http://www. numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 177- JORDAN ALGEBRAS AND MUTATION ALGEBRAS. HOMOTOPY AND VON NEUMANN REGULARITY Santos GONZALES JIMENEZ (*) The aim of the present paper is to present Jordan algebras in the context of Santili’s Mechanics. We also study the homotope algebra of a non-associative algebra (not necessarily associative algebra) with special attention to the homotope algebra of a Boolean algebra. Later the relation between homotopy and Von Neumann regularity is considered, mainly for Jordan algebras. Finally the idempotent (Jordan) algebras are studied. DEFINITIONS AND NOTATIONS If A is any non associative algebra there are two associated algebras A- and A+ having the same underlying vector space as A but with products [x,y] = xy-yx and x.y = 1 (x Y + Y x) respectively where we denote by juxtaposition the initial product on A. The algebra A is said Lie-admissible if A- is a Lie-algebra, that is [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 holds for every x, y, z in A, and Jordan-admissible if A+ is a Jordan algebra, that is x.( Y .x2) _ (x. Y ).x2 for every x, y. The algebra A is called flexible if the identity (x,y,x) = 0 holds for all x, y in A where (a,b,c) = (ab)c - a(bc) is the associator of a, b, c. It is known ([21], p.141 ) that if A is flexible and Jordan-admissible then A is a non-commutative (not necessarily commutative) Jordan algebra and conversely. Also, A is said to be power-associative if the subalgebra generated by any arbitrary element x in A is associative. In any algebra A, the distributive and scalar laws imply that the mappings Rx : y H yx, L : y H xy are linear transformation on the vector space of A. Obviously 1) A is commutative iff Rx = Lx for every x. 2) A is associative iff LXRY = RyLx for every x, y in A. 3) A is flexible iff LXRX = RXLX for every x. 4) A is Lie iff Rxy = RxRy- R yRx and Rx = _Lx* (*) Supported by a grant of Institut de Mathdmatiques (University of Montpellier) and by C.A. Investigacion Cientifica y Tdcnica, n° 0778-84. 178 5) A is noncommutative Jordan iff La , Ra , and commute. L a 2 R a 2 6) A is Lie-admissible iff R[x,y]-L[x,y] = [Rx - Lx , If e is an idempotent of a flexible power-associative algebra A, then A = Al + All2 -+- Ao (Peirce decomposition) ([3] p. 562) with Ai = {xE Alex = xe = ix}, i = 0,1 and Alf2 = {x~A ) I ex+xe = x } . Then A I and Ao are zero or orthogonal subalgebras of A. Moreover Ai Al/2 + AI-i ’ i = 0,1. Finally by we denote a subalgebra ; by + the direct sum of vector spaces and by ? the direct sum of subalgebras. 1. JORDAN ALGEBRAS IN HADRONIC MECHANICS In recent years an increasing number of mathematicians and experimentatl physicists tried to achieve a generalization of Atomic Mechanics, specifically conceived for the structure of strongly interacting particles (hadrons). This new mechanics is called Hadronic Mechanics and these studies include works mainly by theoretical physicists (Santilli, Mignani, Trostel, Okubo, Fronteau,...), experimental physicists (Ranchs, Solbodrian, Louzett,...) and mathematiciens (Myung, Osborn, Benkart, Tomber, Oehmke, ...). The Hadronic Journal, Nonantum, Massachusetts (USA), under the editor-ship of R.M. Santilli has played an important role in the development of Hadronic Mechanics and recently the mathematical journal : "Publications in Algebras, Groups and Geometries" under the editor-ship of H.C. Myung. In this physical context the Santilli’s generalization of Heisenberg’s equation of motion dx/dt = for time-development of any observable x (see G. Loupias, this Colloque), where 9 is Planck’s constant divied by 2n and i = V-1 is the imaginary unit, is dx/dt = i)Í (xpH-Hqx), where p and q are arbitrary fixed nonsingular operaotrs. This equation leads to a new product x*y =xpy - yqx defined on the same linear structure of the associative algebra A and where by juxtaposition we denote the associative product in A. This new algebra is called the (p,q)-mutation of A and is denoted by A(p,q). The algebraic structure and physical applications of A(p,q) have been investigated in some details by Santilli, Myung, Ktorides, osbom, Tomber, Kalnay... It seems to be an interesting problem to investigate the existence of a unit element in the mutation algebra. So Kalnay in ([9], p. 15) says : "Let A be an associative algebra over the field of the complex numbers C with unit element (in particular A could be a C*- algebra). The fundamental realization A(p,q) of the Lie-admissible algebras is due to Santilli. We shall choose a subalgebra A* c A(p,q), so A* is also Lie-admissible. The motivation for working with A* will be the quantum algebra : Nambu quantum algebras 179 needs a unit element and sometimes it is easier to find it in a proper subspace of A(p,q) than in A(p,q) itself, since in the subalgebra A* the multiplication table is smaller than in A(p,q), ..." For this reason in [5], we take up the problem of the existence of a unit element in any mutation algebra. Our aim is to get necesary and sufficient conditions on A, p and q so as to guarantee the existence of a unit element in A(p,q). The main result is that A(p,q) contains a unit element if and only if A has a unit element, p-q is invertible in A and pxq = qxp for every element x in A. If A(p,q) contains a unit element it is proved [5] that A(p,q) is flexible and Jordan-admissible (that is, A(p,q)+ is a Jordan algebra), so A(p,q) is a non-commutative Jordan algebra. Also is proved the following Theorem [5]. If A contains a unit element 1 and p-q is invertible in A then the following properties are equivalent : 1) A(p,q) has a unit. 2) A(p,q) is isomorphic to A(s+l,s) whith s in the center of A (we consider s = (p-q and a : A(s+l,s) e A(p,q) given by a(x) = 3) A(p,q) is generalized quasi-associative. 4) A(p,q) is flexible. 5) A(p,q) is power associative. 6) A(p,q) satisfies the third power identity ((x,x,x)* = 0). An interesting consequence of the above theorem is the following. It known that the expoential function exp x = I xl/i! , x = 1 is definable in a nonassociative but i=0 power-associative algebra A, and that this function plays an important role in the structure theory of nonassociative algebras. H.C. Myung [17] studies the exponential function in the (p-q)-mutation algebra of an associative algebra A with unit over a field F of characteristic 0. He assumes that A(p,q) is a power-associative algebra with a unit element 1. Obviously by our theorem above the interesting Myung’s results are always true for any mutation algebra with identity. Now, we have the following two natural questions : 1) When is the (p,q)- mutation A(p,q) a Lie algebra ? and 2) What happens when the mutation process is reiterated ? With respect to the first question, since A(p,q) is a Lie-admissible algebra, it will be a Lie-algebra if and only if it is an anticommutative algebra, that is x*y = xpy - yqx = -y*x = -ypx + xqx. If r = p-q, it is clear that A(p,q) is an anticommutive algebra if and only if the homotope algebra A(r) (the algebra with the same underlying vector space A and the new multiplication defined by x.y = xry) is anticommutative. So the question is : when does an associative algebra A possesses an element r # 0 such that A(r) is 180 anticommutative ? In this sense we obtain in [6] that the mutation algebras A(p,q) which are Lie algebras are the algebras A(p,q) or "are very near" to them. Finally, with respect to the second question. I studied in [8] the mutation algebra of a nonassociative algebra. Obviously, to obtain the flexibility and Lie-admissibility of the new algebra, it would be necessary to impose conditions over the elements p and q. Define the commutative center K(A) = {xE A I [x,y] = 0 V y e A }, the left nucleus N j (A) = (xe A I (x,y,z) = 0, V x,ze A }, the right nucleus Nr(A) _ {ze A I (x,y,z) = 0, V x,yE A), the middle nucleus Nm(A) = (ye A I (x,y,z) = 0, V x,ze A) , the nucleus N(A) = N j(A)nNm(A) and the center Z(A) = K(A)nN(A).