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

algebras of quotients of graded lie algebras abstract in this paper we explore graded algebras of quotients of lie algebras with special emphasis on the 3 graded case and answer some natural questions concerning its relation to maximal jordan systems of quotients

n ary algebras a review with applications this paper reviews the properties and applications of certain n ary generalizations of lie algebras in a self contained and unified way these generalizations are algebraic structures in which the two entries lie bracket has been replaced by a bracket with n entries each type of n ary bracket satisfies a specific characteristic identity which plays the role of the jacobi identity for lie algebras particular attention will be paid to generalized lie algebras which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized jacobi identity and to filippov or n lie algebras which are defined by fully antisymmetric n brackets that satisfy the filippov identity three lie algebras have surfaced recently in multi brane theory in the context of the bagger lambert gustavsson model because of this filippov algebras will be discussed at length including the cohomology complexes that govern their central extensions and their deformations it turns out that whitehead s lemma extends to all semisimple n lie algebras when the skewsymmetry of the lie or n lie algebra bracket is relaxed one is led to a more general type of n algebras the n leibniz algebras these will be discussed as well since they underlie the cohomological properties of n lie algebras the standard poisson structure may also be extended to the n ary case we shall review here the even generalized poisson structures whose generalized jacobi identity reproduces the pattern of the generalized lie algebras and the nambu poisson structures which satisfy the filippov identity and determine filippov algebras finally the recent work of bagger lambert and gustavsson on superconformal chern simons theory will be briefly discussed emphasis will be made on the appearance of the 3 lie algebra structure and on why the a4 model may be formulated in terms of an ordinary lie algebra and on its nambu bracket generalization

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

lichnerowicz jacobi cohomology in this paper we extend the notion of lichnerowicz poisson cohomology to jacobi manifolds we study the relation of the so called lichnerowicz jacobi cohomology with the basic de rham cohomology and the cohomology of the lie algebra of functions relative to the representation defined by the hamiltonian vector fields a natural pairing with the canonical homology is constructed the relation between the lichnerowicz poisson cohomology of a quantizable poisson manifold and the lichnerowicz jacobi cohomology of the total space of a prequantization bundle is obtained particular cases of cosymplectic contact and locally conformal symplectic manifolds are discussed finally the lichnerowicz jacobi cohomology of a non transitive example is studied

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

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

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

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

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

coxeter elements and root bases abstract let g be a lie algebra of type a d e with fixed cartan subalgebra h root system r and weyl group w we show that a choice of coxeter element c w gives a root basis for g moreover using the results of kirillov and thind 2010 kt we show that this root basis gives a purely combinatorial construction of g where root vectors correspond to vertices of a certain quiver and with respect to this basis the structure constants of the lie bracket are given by paths in this construction is then related to the constructions of ringel and peng and xiao

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

dynamics of generalized poisson and nambu poisson brackets a unified setting for generalized poisson and nambu poisson brackets is discussed it is proved that a nambu poisson bracket of even order is a generalized poisson bracket characterizations of poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and lagrangian submanifolds of product and tangent manifolds respectively

homology and cohomology on generalized poisson manifolds the canonical homology of a generalized poisson manifold is introduced and the two spectral sequences associated with the periodic double complex are studied the generalized poisson cohomology is also discussed and a natural parity with the canonical homology is established the results hold for nambu poisson manifolds

jordan algebras at jordan elements of semiprime rings with involution abstract in this paper we determine the jordan algebras associated to ad nilpotent elements of index at most 3 in lie algebras r and skew r for semiprime rings r without or with involution to do so we first characterize these ad nilpotent elements

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

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

new generalized poisson structures new generalized poisson structures are introduced by using suitable skew symmetric contravariant tensors of even order the corresponding jacobi identities are provided by conditions on these tensors which may be understood as cocycle conditions as an example we provide the linear generalized poisson structures which can be constructed on the dual spaces of simple lie algebras

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

grassmann speciality of jordan supersystems abstract the grassmann envelope s s 0 0 s 1 1 is a useful tool for deciding varietal questions about supersystems s s 0 s 1 it is not useful in deciding simplicity since the envelope is always fraught with trivial ideals coming from trivial superscalars at first glance it would not seem useful in deciding superspeciality either but we will show that it is a more sensitive tool than it seems we say a jordan supersystem js algebra triple or pair is grassmann special if its grassmann envelope is special as an ordinary jordan system over 0 certainly superspeciality js as over implies grassmann speciality js as over 0 but it is not obvious how grassmann speciality influences superspeciality of js nevertheless we show how to transform obstacles to speciality of js over anti superspecial elements into obstacles to speciality of js over 0 anti special elements using grassmann boosters so that non superspeciality js over implies non speciality of js over 0 thus grassmann speciality is the same as superspeciality a supersystem j is a superspecial jordan supersystem over iff js is a special jordan system over 0 it is not clear this is the same as speciality of js over since in general speciality of quadratic jordan systems depends on the scalars we give an example of a jordan algebra which is not special but is special over a scalar subring as a corollary any jordan superalgebra with zero linear extreme radical which is viably evenly 4 interconnected is superspecial thus exceptional superalgebras must be built over even parts of degree 3

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

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

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

involutions on rectangular jordan pairs abstract jordan triple systems are equivalent to jordan pairs with involution in recent work with d amour on triples of clifford type we described involutions on pairs m 1 q generalizing these results in this paper we describe all involutions on nondegenerate pairs of rectangular type a r m f j having a simple artinian coordinate algebra r or more generally a simple unital coordinate algebra such that the form f is unital valued f u v 1 for some u m v m the involutions are of hermitian type determined by an involution anti automorphism with 2 1 on the coordinate ring automorphism type determined by an automorphism on the coordinate ring with 2 inner or of isomorphism type determined by an isomorphism of the necessarily non artinian coordinate ring onto a proper subring with 2 somewhat inner

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

quantum mckay correspondence and global dimensions for fusion and module categories associated with lie groups abstract global dimensions for fusion categories a k g defined by a pair g k where g is a lie group and k a positive integer are expressed in terms of lie quantum superfactorial functions the global dimension is defined as the square sum of quantum dimensions of simple objects for the category of integrable modules over an affine lie algebra at some level the same quantities can also be defined from the theory of quantum groups at roots of unity or from conformal field theory wzw models similar results are also presented for those associated module categories that can be obtained via conformal embeddings they are quantum subgroups of a particular kind as a side result we express the classical or quantum weyl denominator of simple lie groups in terms of classical or quantum factorials calculated for the exponents of the group some calculations use the correspondence existing between periodic quivers for simply laced lie groups and fusion rules for module categories associated with a k su 2

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

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

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

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

associative systems generated by symmetric elements of a system with involution abstract in this paper we study associative systems algebras pairs or triples with involution and we extend to them well known results of herstein see for example 5 in particular we show that any simple associative algebra is either generated by traces or it is a composition algebra for some involution in the case of simple associative pairs or triple systems they are either generated by traces or their sets of symmetric elements satisfy a certain clifford homotope pi moreover we show that the associative subpair generated by any ample subspace inherits regularity conditions such as primeness simplicity and primitivity from an associative pair and that the baer properly nilpotent locally nilpotent and jacobson radicals of the associative subpair generated by any ample subspace are directly inherited from the same radical of the whole pair we end up with a close description of ample subspaces for simple algebras pairs or triple systems generated by traces

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

coxeter elements and periodic auslander reiten quiver in this paper we show that for a simply laced root system a choice of a coxeter element c gives rise to a natural construction of the dynkin diagram in which vertices of the diagram correspond to c orbits in r moreover it gives an identification of r with a certain subset i of i z2h where h is the coxeter number the set i has a natural quiver structure we call it the periodic auslander reiten quiver this gives a combinatorial construction of the root system associated with the dynkin diagram i roots are vertices of i and the root lattice and the inner product admit an explicit description in terms of i finally we relate this construction to the theory of quiver representations

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

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

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

the schouten nijenhuis bracket cohomology and generalized poisson structures newly introduced generalized poisson structures based on suitable skew symmetric contravariant tensors of even order are discussed in terms of the schouten nijenhuis bracket the associated jacobi identities are expressed as conditions on these tensors the cohomological contents of which is given in particular we determine the linear generalized poisson structures which can be constructed on the dual spaces of simple lie algebras

nambu jacobi and generalized jacobi manifolds the geometry of nambu jacobi and generalized jacobi manifolds is studied a large collection of examples is given the characteristic distribution generated by the hamiltonian vector fields on a nambu jacobi manifold is proved to be completely integrable and the induced geometrical structure of the leaves of the corresponding generalized foliation is ellucidated

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

on the zelmanovian classification of prime jb triples complex jb triples are complex banach spaces endowed with a triple product subjected to suitable conditions of algebraic and analytic nature see section 1 for the definition they were introduced by w kaup in order to provide an algebraic setting for the study of bounded symmetric domains in complex banach spaces the open unit ball of every complex jb triple is a bounded symmetric domain ka1 and every bounded symmetric domain in any complex banach space is bi holomorphically equivalent to the open unit ball of a suitable complex jb triple ka3 fundamental examples of complex jb triples are provided by complex c algebras with triple product defined by xyz x 1 2 xy z zy x 1

on the higher order generalizations of poisson structures the characterization of the nambu poisson n tensors as a subfamily of the generalized poisson ones recently introduced and here extended to the odd order case is discussed the homology and cohomology complexes of both structures are compared and some physical considerations are made