weyl groups of fine gradings on matrix algebras octonions and the albert algebra given a grading a g gag on a nonassociative algebra a by an abelian group g we have two subgroups of aut a the automorphisms that stabilize each component ag as a subspace and the automorphisms that permute the components by the weyl group of we mean the quotient of the latter subgroup by the former in the case of a cartan decomposition of a semisimple complex lie algebra this is the automorphism group of the root system i e the so called extended weyl group a grading is called fine if it cannot be refined we compute the weyl groups of all fine gradings on matrix algebras octonions and the albert algebra over an algebraically closed field of characteristic different from 2 in the case of the albert algebra

composition algebras with large derivation algebras abstract the finite dimensional flexible composition algebras include the hurwitz algebras composition algebras with unit element but also other interesting classes of algebras the para hurwitz and the okubo algebras the above mentioned algebras present many symmetries and this is reflected in their large derivation algebras in the present paper we study the opposite question what can be said about the composition algebras if we have some information about their derivation algebras our main result is the classification of all the composition algebras with such large derivation algebras

links between functions and subdifferentials abstract a function in a class f x is said to be subdifferentially determined in f x if it is equal up to an additive constant to any function in f x with the same subdifferential a function is said to be subdifferentially representable if it can be recovered from a subdifferential we identify large classes of lower semicontinuous functions that possess these properties

group gradings on associative algebras abstract let r g g rg be a g graded ring we describe all types of gradings on r if g is torsion free and r is artinian semisimple if r is a matrix algebra over an algebraically closed field f then we give a description of all g gradings on r provided that g is an abelian group in the case of an abelian group g we also classify all finite dimensional graded simple algebras and finite dimensional graded division algebras over an algebraically closed field of characteristic zero

gradings on semisimple algebras abstract the classification of gradings by abelian groups on finite direct sums of simple finite dimensional nonassociative algebras over an algebraically closed field is reduced by means of the use of loop algebras to the corresponding problem for simple algebras this requires a good definition of free products of group gradings

new simple lie superalgebras in characteristic 3 abstract symplectic respectively orthogonal triple systems provide constructions of lie algebras respectively superalgebras however in characteristic 3 it is shown that this role can be interchanged and that lie superalgebras respectively algebras can be built out of symplectic triple systems respectively orthogonal triple systems with a different construction as a consequence new simple finite dimensional lie superalgebras as well as new models of some nonclassical simple lie algebras over fields of characteristic 3 will be obtained

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

on the cartan jacobson theorem abstract the well known cartan jacobson theorem claims that the lie algebra of derivations of a cayley algebra is central simple if the characteristic is not 2 or 3 in this paper we have studied these two cases with the following results if the characteristic is 2 the theorem is also true but if the characteristic is 3 the derivation algebra is not simple we have also proved that in this last case there is a unique nonzero proper seven dimensional ideal which is a central simple lie algebra of type a2 and the quotient of the derivation algebra modulo this ideal turns out to be isomorphic as a lie algebra to the ideal itself the original motivation of this work was a series of computer aided calculations which proved the simplicity of derivation algebras of cayley algebras in the case of characteristic not 3 these computations also proved the existence of a unique nonzero proper ideal which turns out to be seven dimensional in the algebra of derivations of split cayley algebras in characteristic 3

r commutative geometry and quantization of poisson algebras abstract an r commutative algebra is an algebra a equipped with a yang baxter operator r a a a a satisfying m mr where m a a a is the multiplication map together with the compatibility conditions r a 1 1 a r 1 a a 1 r id m m id r 2 r 1 and r m id id m r 1 r 2 the basic notions of differential geometry extend from commutative or supercommutative algebras to r commutative algebras examples of r commutative algebras obtained by quantization of poisson algebras include the weyl algebra noncommutative tori quantum groups and certain quantum vector spaces in many of these cases the r commutative de rham cohomology is stable under quantization

nondifferentiable multiplier rules for optimization and bilevel optimization problems in this paper we study optimization problems with equality and inequality constraints on a banach space where the objective function and the binding constraints are either differentiable at the optimal solution or lipschitz near the optimal solution necessary and sufficient optimality conditions and constraint qualifications in terms of the michel penot subdifferential are given and the results are applied to bilevel optimization problems

fine gradings on simple classical lie algebras abstract the fine abelian group gradings on the simple classical lie algebras including d 4 over algebraically closed fields of characteristic 0 are determined up to equivalence this is achieved by assigning certain invariant to such gradings that involve central graded division algebras and suitable sesquilinear forms on free modules over them

boundedness of certain sets of lagrange multipliers in vector optimization in this paper we establish lagrange multiplier rules in terms of michel penot subdifferential for nonsmooth vector optimization problem a constraint qualification or regularity condition in terms of michel penot subdifferential is given and under this regularity condition the boundedness of certain sets of lagrange multipliers are discussed

the simple non lie malcev algebra as a lie yamaguti algebra the simple 7 dimensional malcev algebra m is isomorphic to the irreducible sl 2 c module v 6 with binary product x y x y defined by the sl 2 c module morphism 2v 6 v 6 combining this with the ternary product x y z x y z defined by the sl 2 c module morphism 2v 6 v 2 sl 2 c gives m the structure of a generalized lie triple system or lie yamaguti algebra we use computer algebra to determine the polynomial identities of low degree satisfied by this binary ternary structure

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

on maximal subalgebras of central simple malcev algebras let f be a field of characteristic different from two and a a central sim ple non lie malcev algebra over f our purpose is to determine the maximal subalgebras of a with respect to inclusion maximal subalgebras of simple lie algebras over an algebraically closed field have been studied by dynkin 3 41 and maximal subalgebras of central simple associative alternative and special jordan algebras have been studied by racine 121 racine s results on octonion algebras will be relevant in our investigation 1

quasidifferentials and maximal normal operators quasidifferentials are studied with the theory of maximal normal operators the quasidifferential of a normally quasidifferentiable function is a pair of upper and lower semicontinuous operators which are special maximal normal operators the function sum of the upper and lower semicontinuous operators is the clarke subdifferential of this function basic calculus and minimal forms of quasidifferentials are also discussed

semismoothness and decomposition of maximal normal operators abstract a function is semismooth if and only if its subdifferential is semicontinuous all the semicontinuous operators form a big linear subspace of the maximal normal operator space by big we mean that this subspace includes all the continuous operators and all the maximal monotone operators all the decomposable maximal normal operators form another big subspace

on some algebras related to simple lie triple systems abstract the set hom der t t f t t is determined for any simple finite dimensional lie triple system t over a field f of characteristic zero it turns out that it contains nontrivial elements if and only if t is related to a simple jordan algebra in particular this provides a new proof of the determination by laquer of the invariant affine connections in the simply connected compact irreducible riemannian symmetric spaces

gradings on tensor products of composition algebras and on the smirnov algebra abstract we give classifications of group gradings up to equivalence and up to isomorphism on the tensor product of a cayley algebra c and a hurwitz algebra over a field of characteristic different from 2 we also prove that the automorphism group schemes of c n and c n are isomorphic on the other hand we prove that the automorphism group schemes of a smirnov algebra t c a 35 dimensional simple exceptional structurable algebra constructed from a cayley algebra c and c are isomorphic this is used to obtain classifications up to equivalence and up to isomorphism of the group gradings on smirnov algebras

fine gradings on simple exceptional jordan pairs and triple systems abstract we give a classification up to equivalence of the fine group gradings by abelian groups on the jordan pairs and triple systems of types bi cayley and albert under the assumption that the base field is algebraically closed of characteristic different from 2 the associated weyl groups are computed we also determine for each fine grading on the bi cayley and albert pairs the induced grading on the exceptional simple lie algebra given by the tits kantor koecher construction

models of the octonions and g2 different models of the cayley algebras and of their lie algebras of derivations are given based on some distinguished subalgebras of the later ones

classification of group gradings on simple lie algebras of types a b c and d abstract for a given abelian group g we classify the isomorphism classes of g gradings on the simple lie algebras of types a n n 1 b n n 2 c n n 3 and d n n 4 in terms of numerical and group theoretical invariants the ground field is assumed to be algebraically closed of characteristic different from 2

convex subcones of the contingent cone in nonsmooth calculus and optimization the tangential approximants most useful in nonsmooth analysis and optimization are those which lie between the clarke tangent cone and the bouligand contigent cone a study of this class of tangent cones is undertaken here it is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone there are such convex subcones which are more accurate approximants than the clarke tangent cone and possess an associated subdifferential calculus that is equally strong in addition a large class of convex subcones of the contingent cone can replace the clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program however the clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven overall the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones

gradings on symmetric composition algebras abstract the group gradings on the symmetric composition algebras over arbitrary fields are classified applications of this result to gradings on the exceptional simple lie algebras are considered too

division composition algebras through their derivation algebras abstract composition algebras are algebras with a nondegenerate quadratic multiplicative form in this paper we construct five families of division composition algebras and prove that any division composition algebra with nonabelian derivation algebra belongs to one of these families as a by product we obtain new examples of real division algebras

the differentiability of real functions on normed linear space using generalized subgradients abstract the modification of the clarke generalized subdifferential due to michel and penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space the g teaux differentiability of any real function can be deduced from the g teaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative for any distance function on a space with uniformly g teaux differentiable norm the clarke and michel penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for g teaux differentiability however on a banach space with rotund dual the frechet differentiability of a distance function implies that it is a convex function a mean value theorem for the modified generalized subdifferential has implications for g teaux differentiability

a return mapping algorithm for elastoplastic and ductile damage constitutive equations using the subloading surface method summary in this paper is presented a return mapping algorithm for an elastoplastic damage model that couples the constitutive equations of an unconventional plasticity model with the phenomenological description of ductile damage in the continuum damage mechanics framework this approach combines the advantages of describing the mechanical property degradation by using the lemaitre model with the realistic accumulation of plastic deformation in cyclic mobility problems from the subloading surface model most of the damage models for ductile failure focus on monotonic loading conditions whereas the present model aims to extend the investigation to cyclic loading and low cycles fatigue problems a cutting plane algorithm is adopted to describe the material behavior that consider both isotropic and kinematic hardening laws two simple numerical studies show that the elastoplastic and damage model is implemented correctly displaying a quadratic rate of convergence for local equilibria and precise solutions even for large prescribed strain increments

composite nonsmooth programming with gâteaux differentiability this paper examines constrained nonsmooth optimization problems where the objective function and the constraints are compositions of locally lipschitz functions and g teaux differentiable functions but are not necessarily frechet differentiable or strict differentiable lagrangian necessary and sufficient optimality conditions are presented for various classes of composite programs these are obtained by constructing appropriate convex approximations for composite functions the lagrange multipliers are also characterized in terms of subgradients of the value function under appropriate conditions

integration of multivalued operators and cyclic submonotonicity we introduce a notion of cyclic submonotonicity for multivalued operators from a banach space x to its dual we show that if the clarke subdifferential of a locally lipschitz function is strictly submonotone on an open subset u of x then it is also maximal cyclically submonotone on u and conversely that every maximal cyclically submonotone operator on u is the clarke subdifferential of a locally lipschitz function which is unique up to a constant if u is connected in finite dimensions these functions are exactly the lower c 1 functions considered by spingarn and rockafellar

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

numerical computation of algorithmic consistent tangent moduli in large strain computational inelasticity abstract an algorithm for the numerical computation of so called algorithmic or consistent tangent moduli in finite inelasticity is presented these moduli determine the sensitivity of algorithmic expressions for stresses with respect to the change in total deformation they serve as iteration operators by application of newton type solvers for the iterative solution of non linear initial boundary value problems in finite inelasticity the underlying concept of the numerical computation is a perturbation technique based on a forward difference approximation which reduces the computation of the tangent moduli to a multiple stress computation the algorithmic procedure is material independent and surprisingly simple it is outlined for a lagrangian as well as an eulerian geometric setting and applied to model problems of finite elasticity and finite elastoplasticity

a series of algebras generalizing the octonions and hurwitz radon identity we study non associative twisted group algebras over mathbb z 2 n with cubic twisting functions we construct a series of algebras that extend the classical algebra of octonions in the same way as the clifford algebras extend the algebra of quaternions we study their properties give several equivalent definitions and prove their uniqueness within some natural assumptions we then prove a simplicity criterion

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

simple graded commutative algebras we study the notion of graded commutative algebra for an arbitrary abelian group the main examples are the clifford algebras already treated in h albuquerque and s majid 2002 2 we prove that the clifford algebras are the only simple finite dimensional associative graded commutative algebras over r or c our approach also leads to non associative graded commutative algebras extending the clifford algebras

computational approach to the simplicity of f 4 o s in the characteristic two case we present in this paper a computational approach to the study of the simplicity of the derivation lie algebra of the quadratic jordan algebra h3 os denoted by f4 os when the characteristic of the base field is two we will show not only a collection of routines designed to find identities and construct principal ideals but also a philosophy of how to proceed studying the simplicity of a lie algebra we have first implemented the quadratic jordan structure of h3 os into the computer system mathematica computing the derivation lie algebra of the quadratic jordon algebra h3 os at any characteristic preprint 2001 and then determined the generic expression of an element of the lie algebra f4 os der h3 os see 41 once the structure of f4 os is completely described it is time to analyze the simplicity by using the strategy mentioned if the characteristic of the base field is not two the lie algebra is simple but if the characteristic is two the lie algebra is not simple and there exists only one proper nonzero ideal i which is 26 dimensional and simple as a lie algebra in order to prove this last affirmation we have used again the set of routines to show the simplicity of the ideal and that it is isomorphic to f4 i which is also a simple lie algebra this isomorphism is constructed from a computed cartan decomposition of both lie algebras

gradings on the real forms of the albert algebra of g2 and of f4 we find all the fine group gradings on the real forms of the albert algebra and of the exceptional lie algebras g2 and f4

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

a quantum octonion algebra using the natural irreducible 8 dimnensional representation and the two spin representations of the quantum group uq d4 of d4 we construct a quantum analogue of the split octonions and study its properties we prove that the quantum octonion algebra satisfies the q principle of local triality and has a nondegenerate bilinear form which satisfies a q version of the composition property by its construction the quantum octonion algebra is a nonassociative algebra with a yang baxter operator action coming from the rmatrix of uq d4 the product in the quantum octonions is a uq d4 module homomorphism using that we prove idenitities for the quantum octonions and as a consequaence obtain at q 1 new representation theory proofs for very well known identities satisfied by the octonions in the process of constructing the quantum octonions we introduce an algebra which is a q analogue of the 8 dimensional para hurwitz algebra

the maximal normal operator space and integration of subdifferentials on nonconvex functions on definit des operateurs normaux maximaux et des fonctions primales sur des ensembles normaux on trouve que la normalite maximale est conservee dans plusieurs operations et que les operateurs normaux maximaux ont de bonnes proprietes de fermeture et de limitation

gradings on octonions abstract all the different ways of grading any cayley dickson or octonion algebra as well as the associated central simple malcev algebras will be obtained

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

semiregularity and generalized subdifferentials with applications to optimization the michel penot subdifferential of a locally lipschitzian function is the principal part of the clarke subdifferential it coincides with the g derivative at differentiable points a locally lipschitzian function can be determined by its michel penot subdifferential uniquely up to an additive constant though this cannot be done by its clarke subdifferential if the set of abnormal points is not negligible a set valued operator is the michel penot subdifferential of a locally lipschitzian function if and only if it is a seminormal operator satisfying a cyclical condition various calculus rules hold for the michel penot subdifferential equalities hold for these rules at a point under semiregularity which is weaker than regularity for a locally lipschitzian function in a separable banach space semiregularity holds everywhere except for a haar zero set applications in optimization are discussed

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

models of the lie algebra f4 abstract we describe the models of the exceptional lie algebra f 4 which are based on its semisimple subalgebras of rank 4 the underlying fact is that any reductive subalgebra of maximal rank of a simple lie algebra induces a grading on this algebra by means of an abelian group in such a way that the nontrivial components of the grading are irreducible modules

a zassenhaus conjecture and cpa structures on simple modular lie algebras abstract commutative post lie algebra structures on lie algebras in short cpa structures have been studied over fields of characteristic zero in particular for real and complex numbers motivated by geometry a perfect lie algebra in characteristic zero only admits the trivial cpa structure in this article we study these structures over fields of characteristic p 0 we show that every perfect modular lie algebra in characteristic p 2 having a solvable outer derivation algebra admits only the trivial cpa structure this involves a conjecture by hans zassenhaus saying that the outer derivation algebra out g of a simple modular lie algebra g is solvable we try to summarize the known results on the zassenhaus conjecture and prove some new results using the classification of simple modular lie algebras by premet and strade for algebraically closed fields of characteristic p 3 as a corollary we obtain that every central simple modular lie algebra of characteristic p 3 admits only the trivial cpa structure

computing block angular karmarkar projections with applications to stochastic programming we present a variant of karmarkar s algorithm for block angular structured linear programs such as stochastic linear programs by computing the projection efficiently we give a worst case bound on the order of the running time that can be an order of magnitude better than that of karmarkar s standard algorithm further implications for approximations and very large scale problems are given

pauli gradings on lie superalgebras and graded codimension growth abstract we introduce grading on certain finite dimensional simple lie superalgebras of type p t by elementary abelian 2 group this grading gives rise to pauli matrices and is a far generalization of z 2 z 2 grading on lie algebra of 2 2 traceless matrices we use this grading for studying numerical invariants of polynomial identities of lie superalgebras in particular we compute graded pi exponent corresponding to pauli grading

color algebras and affine connections on s6 the classification of algebras having a given lie group as their automorphism group has recently surged as an important problem in the study of general structure theory of algebras particle physics differential geometry and invariant system theory 2 4 11 14 17 191 since the automorphism group aut a of a real or complex algebra a is a lie group whose lie algebra is the derivation algebra der a of a zl 1 this is equivalent to the problem of classifying algebras having a given lie algebra as their derivation algebra symmetries that frequently appear in physics are represented by automorphisms of certain nonassociative algebras a where the largeness of aut a or der a reflects the symmetries such algebras with the symmetry group x 3 c su 3 or the compact exceptional lie group g2 have extensively been investigated in particle physics geometry of affrne connections on a reductive homogeneous space and for the study of nite dimensional real division algebras z 4 14 191 the color algebra of domokos and kiivesi domokos 4 o onion and para octonion algebras and the complex or real okubo algebra s are examples of such algebras a well known theorem of f adams asserts that if the sphere s admits

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

local triality and some related algebras abstract eight dimensional symmetric composition algebras give very nice formulae for triality and local triality these latter formulae will be extended to any algebra in a suitable variety the simple algebras in this variety will be shown to be precisely the symmetric composition algebras over fields while the prime algebras are central orders in the simple ones

gradings on the simple real lie algebras of types g2 and d4 abstract we classify group gradings on the simple lie algebras of types g 2 and d 4 over the field of real numbers or any real closed field fine gradings up to equivalence and g gradings with a fixed group g up to isomorphism