algebras of quotients of jordan algebras abstract we define a jordan analogue of lambek and utumi s associative algebra of quotients and we construct the maximal algebra of quotients for nondegenerate jordan algebras we apply those results to other classes of algebras of quotients appearing in the literature

kac moody monopoles and periodic instantons it is shown that calorons or periodic instantons are the same as monopoles with the loop group as their structure group their twistor correspondences and spectral data are defined the spectral data is shown to determine the general caloron

strong primeness of hermitian jordan systems this paper establishes the strong primeness of all jordan systemsjof hermitian type trapped between ample hermitian elements of a prime associative systemrand its martindale system of symmetric quotientsq r h0 r j h q r this completes the converse of zelmanov s classification of strongly prime jordan systems providing if as well as only if classifications of strongly prime and primitive jordan systems

coordinate algebras of extended affine lie algebras of type a1 abstract the cores of extended affine lie algebras of reduced types were classified except for type a1 in this paper we determine the coordinate algebra of extended affine lie algebras of type a1 it turns out that such an algebra is a unital z n graded jordan algebra of a certain type called a jordan torus we classify jordan tori and get five types of jordan tori

minimal ideals of jordan systems we show that minimal ideals of jordan systems algebras triple systems and pairs are either simple or trivial so answering the question posed by nam and mccrimmon in 1983

coordinatization of triangulated jordan systems we obtain partial coordinatizations of any quadratic jordan triple system containing a triangle e u e i e a split jo ke ku 0 ke z hz k such that e e ez is invertible and u is faithful in the sense that l x e u 0 x 0 this is automatic if j is nondegenerate or t e k we identify a hermitian matrix subsystem jh j e du j2 e2 e h d d 7t and a clifford subsystem j k no k e j q s c we show that a simple j must be all hermitian or all clifford coincides with j or j yielding a shorter proof osborn s capacity 2 theorem for jordan algebras we also can coordinatize most unital bimodules for hermitian matrix systems the impetus for this study came from two sources in presenting a selfcontained treatment of jordan structure theory n jacobson 4 sought a shorter proof of the osborn capacity 2 theorem the proof is not short for linear jordan algebras 2 and is quite involved for quadratic jordan algebras 3 in coordinatizing jordan triple systems the second author needed a version of the capacity 2 theorem making no semisimplicity restriction on the ambient system and without assuming that the tripotents e are division tripotents lo by avoiding semisimplicity assumptions such a formulation also coordinatizes unital bimodules in the triangulated case one cannot expect as precise a coordinatization as for rank n 3 in the latter case if j is a jordan algebra then jz

monopoles and rational maps a note on a theorem of donaldson in a recent paper donaldson gave a description of the moduli space of su 2 monopoles in terms of rational maps this was done indirectly via the associated solution of nahm s equations we give here an interpretation of these rational maps in terms of the monopole s spectral curve and then as scattering data for the monopole itself

trivial minimal ideals of jordan systems abstract we show that in jordan systems algebras triple systems and pairs monomials containing two elements of a trivial minimal ideal vanish so improving the answer given by anquela and cortes in j a anquela t cortes minimal ideals of jordan systems invent math 168 2007 83 90 to the problem of nam and mccrimmon in n s nam k mccrimmon minimal ideals in quadratic jordan algebras proc amer math soc 88 4 1983 579 583 inspired by the problem posed in 1971 by zhevlakov in the dniester notebook unsolved problems in the theory of rings and modules second ed 1976 see filippov et al 1993 5

metric in the moduli of su 2 monopoles from spectral curves and gauss manin connection in disguise we show here that from the metric of the manifold m20 i e the reduced moduli of su 2 2 monopoles in yang mills higgs theory one can recover the respective moduli of spectral curves using the method gauss manin connection in disguise this work is a step toward creating an inverse process of finding the metric of any mk0 from spectral curves this is a thirty year old problem that we hope to shed some light on

local and subquotient inheritance of simplicity in jordan systems abstract in this paper we prove that the local algebras of a simple jordan pair are simple jordan pairs all of which local algebras are simple are also studied showing that they have a nonzero simple heart which is described in terms of powers of the original pair similar results are given for jordan triple systems and algebras finally we characterize the inner ideals of a simple pair which determine simple subquotients answering the question posed by o loos and e neher 1994 j algebra166 255 295

the symmetric rouche s form for robustness the objective of this work is to bring to control literature the strengthened and symmetric version of the classical rouche s form the author discusses the need for rouche s form as a tool for the analysis of linear time invariant control systems a robustness criterion is given and its corresponding geometrical interpretation is obtained a simplified implementation of the symmetric form is obtained and its relation to passivity is highlighted an application to time delay systems is given

algebra norms on tensor products of algebras and the norm extension problem abstract we show that if a is a finite dimensional simple associative algebra with involution over the field k of real or complex numbers whose hermitian part h a is of degree 3 over its center if b is a unital algebra with involution over k and if is an algebra norm on h a b then there exists an algebra norm on a b whose restriction to h a b is equivalent to applying zel manovian techniques we prove that the same is true if the finite dimensionality of a is relaxed to the mere existence of a unit for a but the unital algebra b is assumed to be associative we also obtain results of a similar nature showing that for suitable choices of algebras a and b over k the continuity of the natural product of the algebra a b for a given norm can be derived from the continuity of the symmetrized product

gauss manin connection in disguise calabi yau threefolds we describe a lie algebra on the moduli space of non rigid compact calabi yau threefolds enhanced with differential forms and its relation to the bershadsky cecotti ooguri vafa holomorphic anomaly equation in particular we describe algebraic topological string partition functions bf f g rm alg g geq 1 which encode the polynomial structure of holomorphic and non holomorphic topological string partition functions our approach is based on grothendieck s algebraic de rham cohomology and on the algebraic gauss manin connection in this way we recover a result of yamaguchi yau and alim lange in an algebraic context our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of calabi yau threefolds correspond to coordinates on such a moduli space we discuss the mirror quintic as an example

local orders in jordan algebras abstract making use of results on general algebras of quotients of jordan algebras we study a notion of local order based on the version for linear jordan algebras of the ideas of fountain and gould 14 as adapted to the jordan context by fernandez lopez and garcia rus in 7 in particular we characterize the set of lesieur croisot elements of a nondegenerate jordan algebra as those elements of the jordan algebra lying in the socle of its maximal algebra of quotients and apply this relationship to extend to quadratic jordan algebras the results of fernandez lopez and garcia rus on local orders in nondegenerate jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors

algebras whose multiplication algebra is pi or gpi abstract we obtain nonassociative extensions of some basic results in the associative pi and gpi theories such as regev s tensor product pi theorem kaplansky s primitive pi theorem posner s prime pi theorem amitsur s primitive gpi theorem and martindale s prime gpi theorem

jordan centers and martindale like covers abstract in this paper we show that the scalar center of a nondegenerate quadratic jordan algebra is contained in the scalar center of any of its martindale like covers

a criterion for estimating robust time delays for closed loop stability a formula for estimating robust time delays for closed loop stability is presented continuous single input single output linear time invariant systems represented by their rational transfer functions which are not necessarily stable are considered the proposed formula is derived using the symmetric rouche form an illustrative example is included and some closing remarks concerning the merits of the formula are made

quadratic jordan systems of hermitian type abstract in this paper we extend zel manov s classification of linear jordan systems triple systems and pairs of hermitian type to quadratic jordan systems over an arbitrary ring of scalars using associative triple systems of the first kind for what we believe to be a more natural tool to describe special jordan triples we show that an i special prime quadratic jordan triple system the ideals of which remain semiprime with a nonzero hermitian part lies between an ample subspace of hermitian elements in a prime associative triple system and those in its martindale system of symmetric quotients modulo some extra definitions the structure of strongly prime jordan pairs of hermitian type follows almost immediately from the jordan triple result

monopoles and their spectral data a new definition of spectral data of a monopole is given for any compact lie or kac moody group it is shown that the spectral data determines the irreducible monopole in the case of maximal symmetry breaking the spectral data is shown to reduce to an earlier definition in terms of algebraic curves indexed by the nodes of the dynkin diagram of the group the structure of solutions to nahm s equations corresponding to the monopole is discussed

minitwistors and 3d yang mills higgs theory we construct a minitwistor action for yang mills higgs ymh theory in three dimensions the feynman diagrams of this action will construct perturbation theory around solutions of the bogomolny equations in much the same way that mhv maximally helicity violating diagrams describe perturbation theory around the self dual yang mills equations in four dimensions we also provide a new formula for all tree amplitudes in ymh theory and its maximally supersymmetric extension in terms of degree d maps to minitwistor space we demonstrate its relationship to the roiban spradlin volovich witten rsvw formula in four dimensions and show that it generates the correct mhv amplitudes at d 1 and factorizes correctly in all channels for all degrees we construct a minitwistor action for yang mills higgs ymh theory in three dimensions the feynman diagrams of this action will construct perturbation theory around solutions of the bogomolny equations in much the same way that mhv maximally helicity violating diagrams describe perturbation theory around the self dual yang mills equations in four dimensions we also provide a new formula for all tree amplitudes in ymh theory and its maximally supersymmetric extension in terms of degree d maps to minitwistor space we demonstrate its relationship to the roiban spradlin volovich witten rsvw formula in four dimensions and show that it generates the correct mhv amplitudes at d 1 and factorizes correctly in all channels for all degrees

derivations and automorphisms of jordan algebras in characteristic two a jordan algebra j over a field k of characteristic 2 becomes a 2 lie algebra l j with lie product x y x y and squaring x 2 x 2 we determine the precise ideal structure of l j in case j is simple finite dimensional and k is algebraically closed we also decide which of these algebras have smooth automorphism groups finally we study the derivation algebra of a reduced albert algebra j h3 o k and show that der j has a unique proper nonzero ideal vj isomorphic to l j k 1j with quotient der j v j independent of o on the group level this gives rise to a special isogeny between the automorphism group of j and that of the split albert algebra whose kernel is the infinitesimal group determined by vj

zelmanov s prime theorem for quadratic jordan algebras e i zelmanov has recently shown that any prime linear jordan algebra without nil ideals but no finiteness conditions imposed is either a homomorphic image of a special algebra or is a form of the 27 dimensional exceptional algebra in particular all division algebras and all simple unital algebras are either derived from associative algebras or are 27 dimensional over their centers this effectively ends the search for infinite dimensional exceptional algebras in the present paper zelmanov s result is extended to quadratic jordan algebras jordan algebras were invented in the 1930s in the search for an exceptional algebraic setting for quantum mechnics an algebraic system which behaved like the usual algebra of quantum mechanical observables hermitian operators on hilbert space but was exceptional in the sense that its structure was not determined behind the scenes by some unobservable associative algebra in their pioneering paper of 1933 91 jordan von neumann and wigner classified the finite dimensional formally real linear jordan algebras and a a albert showed 11 that the only simple algebra in the list which was exceptional was a certain 27 dimensional algebra of 3 x 3 hermitian matrices with entries from an gdimensional cayley algebra all subsequent investigations of exceptional jordan algebras led back to this same albert algebra the next breakthrough in the structure of jordan algebras was n jacobson s 1965 introduction 6 j of inner ideals and the resultant quadratilication of the theory in terms of the quadratic product xyx instead of the linear product xy vx the final breakthrough was the astounding 1979 result of a young russian mathematician e i zelmanov that the only simple exceptional linear jordan are albert algebras thus the search for an infinite dimensional setting for exceptional quantum mechanics is doomed to failure the only simple exceptional structure allotted to mortals is the 27 dimensional albert algebra this came as a complete surprise to western researchers who hoped the free jordan algebra might lead to an 291 002 i 8693 82 060297 30 02 00 0

commuting u operators in jordan algebras for elements x y in a non degenerate non unital jordan algebra over a commutative ring the relation x y 0 is shown to imply that the u operators of x and y commute uxuy uyux the proof rests on the ze 0 lmanov mccrimmon classication 15 of strongly prime quadratic jordan algebras

imbedding jordan systems in primitive systems we show that special jordan systems j which are free of torsion over a scalar ring imbed in primitive hence strongly prime jordan systems j j heart j with simple primitive heart we show examples of exceptional jordan systems which cannot be imbedded in a nondegenerate system

tits kantor koecher algebras of strongly prime hermitian jordan pairs abstract in this paper we describe the tits kantor koecher algebras of strongly prime hermitian jordan pairs in terms of prime associative pairs and their martindale pairs of symmetric quotients

moufang sets from groups of mixed type abstract we consider the existence of moufang sets related to certain groups of mixed type this way we obtain new examples of moufang sets and new constructions of known classes the most interesting class of new examples is related to the moufang quadrangles of type f 4 and to the ree tits octagon over a nonperfect field and the root groups of each member have nilpotency class three

local pi theory of jordan systems ii abstract in this paper we study jordan systems having nonzero local algebras that satisfy a polynomial identity we prove that in nondegenerate jordan systems the set of elements at which the local algebra is pi is an ideal and that if it is nonzero it coincides with the socle when the system is primitive

estimating the robust dead time for closed loop stability a method based on the small gain theorem is provided for estimating the robust dead time for closed loop stability the method lends itself to generalization for multivariable systems and also to graphical implementation some examples are provided to demonstrate the use of the method and the results are compared to the exact critical values of dead time

the kurosh problem for jordan nil systems over arbitrary rings of scalars abstract we show that quadratic jordan nil algebras of bounded degree and more generally quadratic jordan nil algebras satisfying a polynomial identity are locally nilpotent we also extend these results to jordan pairs and triple systems

outer fractions in quadratic jordan algebras abstract using new techniques of zelmanov c martinez improved on work of jacobson mccrimmon and parvathi to give a necessary and sufficient ore type condition for an arbitrary linear jordan algebra with no 2 or 3 torsion to have an algebra of fractions in this paper we extend to quadratic algebras the concept of algebras of outer fractions with respect to an ore monad and describe necessary and sufficient ore type conditions for the embedding in such an algebra of fractions the details of the actual embedding will appear in a subsequent paper

the ring of fractions of a jordan algebra abstract we derive a necessary and sufficient ore type condition for a jordan algebra to have a ring of fractions

jordan quadruple systems abstract we define jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the jordan tetrad a b c d a b c d d c b a as a quadrilinear operation on associative algebras we find further identities in degree 10 which are not consequences of the defining identities we introduce four infinite families of finite dimensional jordan quadruple systems and construct the universal associative envelope for a small system in each family we obtain analogous results for the anti tetrad a b c d a b c d d c b a our methods rely on computer algebra especially linear algebra on large matrices the lll algorithm for lattice basis reduction representation theory of the symmetric group noncommutative grobner bases and wedderburn decompositions of associative algebras

contact metric 5 manifolds cr twistor spaces and integrability the cr twistor space is defined over a contact metric 5 manifold m like the 4 dim twistor theory the integrability of the almost cr twistor structure is discussed in terms of the weyl conformal curvature and also the scalar curvature of m

the jordan algebras of a lie algebra abstract we attach a jordan algebra l x to any ad nilpotent element x of index of nilpotence at most 3 in a lie algebra l this jordan algebra has a behavior similar to that of the local algebra of a jordan system at an element thus l x inherits nice properties from l and keeps relevant information about the element x

the maximal symmetric ring of quotients abstract the maximal symmetric ring of quotients q r as defined by utumi is a symmetric version of the maximal ring of quotients of r for the most part we study this ring when r k g is a group algebra for example we show that if g is a free product of groups and if r k g is a domain then q r is usually equal to r on the other hand there are certainly groups for which q r is properly larger than r and we construct a number of such examples

on the construction of monopoles we show that any self dual su 2 monopole may be constructed either by ward s twistor method or nahm s use of the adhm construction the common factor in both approaches is an algebraic curve whose jacobian is used to linearize the non linear ordinary differential equations which arise in nahm s method we derive the non singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions

local lesieur croisot theory of jordan algebras in this paper we study jordan algebras having nonzero local algebras that satisfy the property of being lesieur croisot i e being orders in nondegenerate jordan algebras of finite capacity we will prove that the set of the elements of a nondegenerate jordan algebra at which the local algebra is lesieur croisot is an ideal

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

herstein s theorems and simplicity of hermitian jordan systems abstract in this paper we extend herstein s first construction relating associative and jordan ideals to pairs and triple systems as a consequence we show that an associative pair or triple system is simple if and only if its jordan symmetrization is simple we also generalize herstein s second construction to ample subsystems of associative algebras pairs and triple systems which provides information on their simplicity when the associative structure is simple

primitivity in finitely generated special quadratic jordan algebras abstract primitive jordan algebras were introduced by zelmanov in the linear case 13 and hogben and mccrimmon in the quadratic case 4 a classification theorem for quadratic primitive jordan algebras in the spirit of the general classification of prime nondegenerate jordan algebras 9 is proven in 2 in that result besides albert and clifford type algebras there appear special algebras containing an ideal of the form h0 a for a primitive associative algebra a with involution to prove this result a key fact is that for a prime a primitivity can be transferred from h0 a to a and back the first part of this result stems from a more general theorem asserting that any tight associative envelope of a primitive jordan algebra is primitive 2 this can be extended to envelopes making use of the concept of primitivity any tight associative envelope of a primitive jordan algebra is primitive the aim of this paper is to examine the reciprocal of that theorem for finitely generated jordan algebras we prove in section 4 that a finitely generated special jordan algebra having a primitive envelope is primitive in the case of h a the proof makes uses of the herstein second construction 7 for a semiprime a with involution any nonzero ideal of h a contains some h0 i with i a nonzero ideal of a our proof will follow a similar pattern we prove in section 2 that the herstein mccrimmon construction can be extended to prime finitely generated jordan algebras with nonzero hermitian part for that we make use of some basic properties of hermitian ideals as defined in 9 in section 3 we treat algebras an anti hermitian type called here simply anti hermitian algebras separately and we collect the previous results in section 4 to prove our main theorem

primitive jordan pairs and triple systems abstract in this paper we give a characterization of primitivity of jordan pairs and triple systems in terms of their local algebras as a consequence of that local characterization we extend to jordan pairs and triple systems most of the known results about primitive jordan algebras in particular we describe primitive jordan pairs and triple systems over an arbitrary ring of scalars in the sense of the structure of primitive quadratic jordan algebras by j a anquela t cortes and f montaner 1995 j algebra172 530 553 5 1

the structure of primitive quadratic jordan algebras in this paper we give a complete description of primitive quadratic jordan algebras following the classification of strongly prime quadratic jordan algebras k mccrimmon and e i zel manov adv math 69 no 2 1988 133 222 the proof is based in two results of independent interest we show that every p i primitive jordan algebra is simple and unital with nonzero socle and prove that associative tight tight envelopes of special primitive jordan algebras are also primitive primitive as a consequence of the latter fact we see that an associative algebra a is one sided primitive if and only if a is primitive and an associative algebra a with involution is primitive if and only if an ample subspace h 0 a is primitive

zel manov polynomials in quadratic jordan triple systems abstract zel manov polynomials are elements of a free special jordan system which are both hermitian their values look like hermitian elements and clifford they do not vanish on systems t containing h 3 h m 3 t these polynomials decide the classification of strongly prime special jordan systems t is either clifford or hermitian according as some zel manov polynomial does or does not vanish on t the existence of such polynomials has been established by mccrimmon and zel manov for quadratic jordan algebras and by zel manov for linear jordan triple systems and pairs in this paper we carry out the construction for quadratic jordan triple systems and pairs

polynomial identities and nonidentities of split jordan pairs abstract we show that split jordan pairs over rings without 2 torsion can be distinguished by polynomial identities with integer coefficients in particular this holds for simple finite dimensional jordan pairs over algebraically closed fields of characteristic not 2 we also generalize results of drensky and racine and of rached and racine on polynomial identities of respectively jordan algebras and jordan triple systems

algebras of quotients of nonsingular jordan algebras abstract we define a jordan analogue of johnson s associative algebra of quotients and study it under a suitable condition of nonsingularity of the jordan algebra which we call strong nonsingularity in particular we prove the existence and describe the maximal algebras of quotients of prime strongly nonsingular jordan algebras

associative systems of left quotients abstract in this paper we construct the maximal left quotient system of every pair of modules in some generalized matrix rings these rings can be seen as 3 graded algebras or as superalgebras we show the relation among the three different notions of left quotients

goldie theory for jordan algebras it is shown that zelmanov s version of goldie s conditions still characterizes quadratic jordan algebras having an artinian algebra of quotients which is nondegenerate at the same time jordan versions of the main notions of the associative theory such as those of the uniform ideal uniform element singular ideal and uniform dimension are studied moreover it is proved that the nondegenerate unital jordan algebras of finite capacity are precisely the algebras of quotients of nondegenerate jordan algebras having the property that an inner ideal is essential if and only if it contains an injective element 2002 elsevier science usa

the etale tits process of jordan algebras revisited the etale tits process which was called the toral tits process by petersson racine 12 may be viewed as a jordan theoretical method to construct associative algebras with involution more specifically starting from a cubic etale algebra e and a quadratic etale algebra l both over an arbitrary base field as well as from invertible elements u e b l having the same norms the etale tits process produces an absolutely simple jordan algebra j j e l u b of degree 3 and dimension 9 which by structure theory must be the symmetric elements of a central simple associative algebra of degree 3 with involution of the second kind in addition e identifies canonically with a subalgebra of j

platonic hyperbolic monopoles we construct a number of explicit examples of hyperbolic monopoles with various charges and often with some platonic symmetry the fields are obtained from instanton data in mathbb r 4 that are invariant under a circle action and in most cases the monopole charge is equal to the instanton charge a key ingredient is the identification of a new set of constraints on adhm instanton data that are sufficient to ensure the circle invariance unlike for euclidean monopoles the formulae for the squared higgs field magnitude in the examples we construct are rational functions of the coordinates using these formulae we compute and illustrate the energy density of the monopoles we also prove for particular monopoles that the number of zeros of the higgs field is greater than the monopole charge confirming numerical results established earlier for euclidean monopoles we also present some one parameter families of monopoles analogous to known scattering events for euclidean monopoles within the geodesic approximation

primitivity in jordan systems is ubiquitous abstract the main result of the paper asserts that if a jordan pair v v v is primitive at some 0 b v then it is primitive at any 0 b v also if a jordan triple system t is primitive at some 0 b t then it is primitive at any 0 b t as a tool similar results concerning one sided primitivity and primitivity of associative pairs and triple systems are established equivalences between our definitions of primitivity and some other appearing in the literature are also obtained as a consequence

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

maximal algebras of martindale like quotients of strongly prime linear jordan algebras abstract in this paper we prove the existence and give precise descriptions of maximal algebras of martindale like quotients for arbitrary strongly prime linear jordan algebras as a consequence we show that zelmanov s classification of strongly prime jordan algebras can be viewed exactly as the description of their maximal algebras of martindale like quotients as a side result we show that the martindale associative algebra of symmetric quotients can be expressed in terms of the symmetrized product i e in purely jordan terms

idempotents and peirce gradings of jordan algebras abstract in this paper we find a necessary and sufficient condition for a peirce grading of a jordan algebra j to come from a peirce decomposition with respect to an idempotent of a jordan algebra j containing j as a subalgebra we also show that the above condition holds automatically when j is nondegenerate

diagonalization in jordan pairs as every student of linear algebra knows a rectangular matrix over a division ring can be diagonalized by elementary row and column operations similarly there are normal forms for alternating and hermitian matrices which can be achieved by simultaneous row and column operations since matrices of this kind form the main examples of jordan pairs it is natural to ask whether similar results hold in the jordan setting we show that this is in fact the case study the obstruction to diagonalizability the defect and also prove that nondegenerate jordan pairs admit a rank function sharing many properties with classical matrix rank let v v v be a jordan pair and s e e a set of orthogonal idempotents the s diagonal elements are those in xi 1 v e now suppose that v is nondegenerate and satisfies dcc on principal inner ideals an element x e v a i is called diagonalizabfe if x is s diagonal for some s consisting of division idempotents to see that for matrices this is the same as the usual notion of diagonalizability suppose that v has in addition act on principal inner ideals then v contains a frame that is a finite set f of orthogonal division idempotents such that v f 0 for rectangular or hermitian matrices over division rings f can be taken to consist of the diagonal matrix units e eii whereas for alternating matrices over a field e eli1 2ie zi the jordan analogues of elementary row and column operations are the inner automorphisms b v e v e and v e v e where e e f which generate the group of f elementary automorphisms of v 6 any set of orthogonal division idempotents can be transformed up to association into f by an f elementary automorphism 6 th 21 it follows that x is diagonalizable if and only if cp x is f diagonal for some elementary automorphism cp of v this shows the equivalence with the usual definition

the zelmanov approach to jordan homomorphisms of associative algebras abstract zelmanov s work on prime jordan algebras leads to an idempotent free version of martindale s theorem on the extension of jordan homomorphisms and derivations from the hermitian elements h r of an associative algebra of degree 2 with involution to associative homomorphisms and derivations on r the condition that j h r be of degree 2 is replaced by the intrinsic condition that j z j h a consist entirely of values of zelmanov polynomials

nondegenerate jordan algebras satisfying local goldie conditions abstract a structure theorem is given for nondegenerate jordan algebras j satisfying the ascending chain condition on annihilators of a single element and such that j contains no infinite direct sum of inner deals inside the inner ideal generated by each element x j as a consequence of this theorem and of the main results of a previous paper by the authors j algebra 174 1995 1024 1048 it is obtained that such jordan algebras j are precisely the local orders in nondegenerate jordan algebras satisfying dcc on principal inner ideals and without non artinian quadratic ideals which extends to local orders the zel manov theorem for goldie jordan algebras

monopoles and baker functions the work in this paper pertains to the solutions of nahm s equations which arise in the atiyah drinfield hitchin manin nahm construction of solutions to the bogomol nyi equations for static monopoles this paper provides an explicit construction of the solution of nahm s equations which satisfy regularity and reality conditions the lax form of nahm s equations is reduced to a standard eigenvalue problem by a special gauge transformation these equations may then be solved by the method of baker krichever this leads to a compact representation of the solutions of nahm s equations the regularity condition is shown to be related to the monodromy of the gauge reduced linear operator hitchin showed that the solutions of nahm s equations can be characterized by an algebraic curve and some data on that curve here this characterization reduces to a transcendental equation involving certain loop integrals of a meromorphic differential donaldson coordinatized the moduli space ofk monopoles by a class of rational maps from the riemann sphere to itself the data of a baker function is equivalent to this map this method gives an apriori construction of the known two monopole solutions we also give a generalization of the two monopole solution to a class of elliptic solutions of arbitrary charge these solutions correspond to reducible curves with elliptic components and the associated donaldson rational function has a simple partial fraction expansion

nahm s equations and the classification of monopoles solutions of nahm s system of ordinary differential equations are produced by variational methods this leads to an explicit parametrisation of the solutions to the bogomolny equation over 3