nonreduced extended affine weyl groups abstract in this paper we study the structure of the weyl groups of nonreduced extended affine root systems we show that similar to the case of reduced types an extended affine weyl group w of type bcl is semidirect product of a finite weyl group w of type bl and a heisenberg like normal subgroup h which is also a characteristic subgroup of w moreover h is of the form h h h 0 where both h and h 0 are normal subgroups of h with h h 0 1 h is naturally isomorphic to the root lattice of a finite root system of type bcl furthermore the semidirect product of w and h is isomorphic to the weyl group of a kac moody affine subroot system of r of type bcl

modules of highest weight type for the algebra of derivations over a rational quantum torus for the lie algebra of derivations over a rational quantum torus we construct a class of irreducible weight modules with all finite dimensional weight spaces which are called the modules of highest weight type generating the notion for the case of commutative torus for the lie algebra of derivations over a rational quantum torus we construct a class of irreducible weight modules with all finite dimensional weight spaces which are called the modules of highest weight type generating the notion for the case of commutative torus

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

twisted quantum affinizations and their vertex representations in this paper we generalize drinfeld s twisted quantum affine algebras to construct twisted quantum algebras for all simply laced generalized cartan matrices and present their vertex representation realizations

derivations and invariant forms of jordan and alternative tori jordan and alternative tori are the coordinate algebras of extended affine lie algebras of types a 1 and a 2 in this paper we show that the derivation algebra of a jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations in the course of proving this result we investigate derivations of the more general class of division graded jordan and alternative algebras we also describe invariant forms of these algebras

finite dimensional irreducible representations of elementary unitary lie algebras over quantum tori abstract we classify all finite dimensional irreducible representations of the elementary unitary lie algebras eu n c q over the quantum torus with anti involution c q for n 5

a relation between the domain topology and the number of minimal nodal solutions for a quasilinear elliptic problem abstract we consider the quasilinear problem div u p 2 u u p 2 u u q 2 u in u 0 on where r n is a bounded smooth domain 1 p n and p q p np n p we show that if is invariant by a non trivial orthogonal involution then for q close to p the equivariant topology of is related with the number of solutions which change sign exactly once the results complement those of castro and clapp nonlinearity 16 2003 579 590 since we consider subcritical nonlinearities and the quasilinear case without any assumption of symmetry we also extend theorem b in benci and cerami arch rational mech anal 114 1991 79 93 for the quasilinear case and prove that the topology of affects the number of positive solutions

a class of integrable modules for the core of eala coordinatized by quantum tori extended affine lie algebras eala are higher dimensional generalizations of affine kac moody lie algebras introduced in 19 they have been further studied in 1 2 5 toroidal lie algebras which are universal central extensions of g c t 1 1 t 1 n g simple finite dimensional lie algebra and are prime examples of cores of ealas which are studied in 4 7 14 16 22 24 25 there are many ealas which allow not only laurent polynomial algebra c t 1 1 t 1 n as co ordinate algebra but also quantum tori jordan tori and the octonians tori as co ordinated algebras depending on type of the lie algebra see 1 3 5 6 26 the structure theory of the eala of type ad 1 is tied up with lie algebra gld c cq where cq is the quantum torus see section 1 quantum tori defined in 21 are non commutative analogue of laurent polynomial algebras in this paper we study integrable modules for the core of the eala corresponding to gld c cq which we call we will also add derivations d1 d2 to and call it here cq is two variable quantum torus defined as the ring of non commutating laurent polynomials c t 1 1 t 1 2 with relation to t1t2 qt2t1 note that if q is generic then is an eala we classify theorem 5 2 irreducible integrable modules see definition 1 9 for where the zero degree center acts non trivially we prove that any such module is isomorphic to a highest weight module or a lowest module up to an automorphism see section 5 let and aff be finite and affine root system of sld c and its affinization consider aff z 2 and consider the subalgebra b corresponding to see section 1 by highest weight modules we mean a module with a weight vector killed by b and the highest weight h c t t 1 given as in theorem 4 1 for any such we define a highest weight module

a family of representations of the lie superalgebra gl1 l 1ˆ cq we construct an action of the lie superalgebras glm n cq on the tensor product of the polynomial algebra and the exterior algebra with infinitely many variables this construction is involving one parameter and we show that it is irreducible if and only if is nonzero we also discuss the module structure and irreducibility over a lie subsuperalgebra graded by the root system a m 1 n 1

a characterization of affine kac moody lie algebras we give a new characterization of the affine kac moody algebras in terms of extended affine lie algebras we also present new realizations of the twisted affine kac moody algebras

on conjugacy of cartan subalgebras in extended affine lie algebras abstract that finite dimensional simple lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as coxeter dynkin diagrams and indecomposable irreducible root systems is arguably one of the most elegant results in mathematics the definition of the root system is done by fixing a cartan subalgebra of the given lie algebra the remarkable fact is that up to isomorphism this construction is independent of the choice of the cartan subalgebra the modern way of establishing this fact is by showing that all cartan subalgebras are conjugate for symmetrizable kac moody lie algebras with the appropriate definition of cartan subalgebra conjugacy has been established by peterson and kac an immediate consequence of this result is that the root systems and generalized cartan matrices are invariants of the kac moody lie algebras the purpose of this paper is to establish conjugacy of cartan subalgebras for extended affine lie algebras a natural class of lie algebras that generalizes the finite dimensional simple lie algebra and affine kac moody lie algebras

irreducible representations for toroidal lie algebras abstract in this work a large number of irreducible representations with finite dimensional weight spaces are constructed for some toroidal lie algebras to accomplish this we develop a general theory of z n graded lie algebras with polynomial multiplication we construct modules by the standard inducing procedure and study their irreducible quotients using the vertex operator technics

covering algebras ii isomorphism of loop algebras this paper studies the loop algebras that arise from pairs consisting of a symmetrizable kac moody lie algebra g and a finite order automorphism of g we obtain necessary and sufficient conditions for two such loop algebras to be isomorphic

hermitian representations of the extended affine lie algebra gl2 cq abstract we use the idea of free fields to obtain highest weight representations for the extended affine lie algebra gl 2 c q coordinatized by the quantum torus c q and go on to construct a contravariant hermitian form we further give a necessary and sufficient condition such that the contravariant hermitian form is positive definite

classification and construction of quasisimple lie algebras abstract we study a class of possibly infinite dimensional lie algebras called the quasisimple lie algebras qsla s and generalizing semisimple and affine kac moody lie algebras they are characterized by the existence of a finite dimensional cartan subalgebra a non degenerate symmetric ad invariant killing form and nilpotent rootspaces attached to non isotropic roots we are then able to derive a classification theorem for the possible irreducible elliptic quasisimple root systems moreover we construct explicit realizations of some of them as untwisted and twisted current algebras generalizing the affine loop algebras

on the number of solutions of nls equations with magnetics fields in expanding domains abstract in this paper we look for multiple weak solutions u c for the complex equation i a x 2 u u f u 2 u in the set r n is a smooth bounded domain 0 is a parameter a is a regular magnetic field and f is a superlinear function with subcritical growth our main result relates for large values of the number of solutions with the topology of the set in the proof we apply minimax methods and ljusternik schnirelmann theory

the alternative torus and the structure of elliptic quasi simple lie algebras of type we present the complete classification of the tame irreducible elliptic quasi simple lie algebras of type a2 and in particular specialize on the case where the coordinates are not associative here the coordinates are cayley dickson algebras over laurent polynomial rings in v 3 variables which we call alternative tori in giving our classification we need to present much information on these alternative tori and the lie algebras coordinatized by them

representations of some quantum tori lie subalgebras in this paper we define the q analog virasoro like lie subalgebras in x a b c d the embedding formulas into x are introduced irreducible highest weight representations of aq bq and cq series of the q analog virasoro like lie algebras in terms of vertex operators are constructed we also construct the polynomial representations of the aq bq cq and dq series of the q analog virasoro like lie algebras

multiloop realization of extended affine lie algebras and lie tori an important theorem in the theory of infinite dimensional lie algebras states that any affine kac moody algebra can be realized that is to say constructed explicitly using loop algebras in this paper we consider the corresponding problem for a class of lie algebras called extended affine lie algebras ealas that generalize affine algebras ealas occur in families that are constructed from centreless lie tori so the realization problem for ealas reduces to the realization problem for centreless lie tori we show that all but one family of centreless lie tori can be realized using multiloop algebras in place of loop algebras we also obtain necessary and sufficient conditions for two centreless lie tori realized in this way to be isotopic a relation that corresponds to isomorphism of the corresponding families of ealas

integrable representations for toroidal extended affine lie algebras abstract let g be any untwisted affine kac moody algebra any fixed complex number and g the corresponding toroidal extended affine lie algebra of nullity two for any k tuple 1 k of weights of g and k tuple a a 1 a k of distinct non zero complex numbers we construct a class of modules v a for the extended affine lie algebra g we prove that the g module v a is completely reducible we also prove that the g module v a is integrable when all weights i in are dominant thus we obtain a new class of irreducible integrable weight modules for the toroidal extended affine lie algebra g

quantum tori and the structure of elliptic quasi simple lie algebras abstract we study and classify those tame irreducible elliptic quasi simple lie algebras which are simply laced and of rankl 3 the first step is to identify the core of such an algebra up to central isogeny by identifying the coordinates when the type isdorethe coordinates are laurent polynomials in variables while for typeathe coordinates can be any quantum torus in variables the next step is to study the universal central extension as well as the derivation algebra of the core these are related to the first connes cyclic homology group of the coordinates the final step is to use this information to give constructions of lie algebras which we then prove yield representatives of all isomorphism classes of the above types of algebras

extended affine weyl groups abstract in this paper we study the weyl groups of reduced extended affine root systems the root systems of extended affine lie algebras we start by describing the extended affine weyl group as a semidirect product of a finite weyl group and a heisenberg like normal subgroup this provides a unique expression for the weyl group elements in terms of some naturally arisen transformations which is crucial in the further study of extended affine weyl groups we use this to give a presentation called a presentation by conjugation for an important subclass of extended affine weyl groups using a new notion called the index which is an invariant of the extended affine root systems we show that one of the important features of finite and affine root systems related to weyl group holds for the class of extended affine root systems

representations of the lie algebra of derivations for quantum torus abstract let der c q be the derivation lie algebra of the quantum torus c q we construct a functor from gld modules to der c q modules which generalizes the constructions obtained by shen scientia sinica a 29 1986 570 larsson internat j modern phys a 7 1992 6493 and rao j algebra 182 1996 401 we also give a complete description of the structure of the der c q modules when all the entries of the quantum torus matrix q qij d d are roots of unity

locally extended affine lie algebras abstract we propose a new simplified definition of extended affine lie algebras ealas for short and also discuss a general version of extended affine lie algebras called locally extended affine lie algebras local ealas for short we prove a conjecture by v kac for local ealas it turns out that the root system of a local eala becomes a locally finite version of an extended affine root system several examples of new ealas and local ealas are introduced and finally we classify local ealas of nullity 0 and show the connection to locally finite split simple lie algebras

algebras of diffeomorphisms of the n torus the general algebra d n v of infinitesimal diffeomorphisms of the n torus s1 n involving generators lv m depending on a structure vector v cn and a vector valued index m zn is constructed several results are proved for this algebra special cases of the algebra were previously presented in ramos shrock e ramos and r e shrock int j mod phys a 4 4279 1989 the concept of the q rank of the space v of structure vectors denoted rq v is defined and certain related cartan matrices are introduced it is shown that d n v as an ungraded algebra is simple if and only if rq v n i e the q rank of v is maximal a classification under isomorphisms is given for the algebra and is shown to reduce to a classification of the cartan matrices the space of structure vectors is isomorphic to the unique cartan subalgebra a number of properties concerning the central extensions of these algebras are then proved for the case dim v 1 rq v n it is shown that the central extension is unique fo

vertex operators arising from the homogeneous realization for we use the underlying fock space for the homogeneous vertex operator representation of the affine lie algebra to construct a family of vertex operators as an application an irreducible module for an extended affine lie algebra of type a n 1 coordinatized by a quantum torus q of 2 variables or 3 variables is obtained moreover this module turns out to be a highest weight module which is an analog of the basic module for affine lie algebras

bcn graded lie algebras arising from fermionic representations abstract we use fermionic representations to obtain a class of bc n graded lie algebras coordinatized by quantum tori with nontrivial central extensions

multiple minimal nodal solutions for a quasilinear schrodinger equation with symmetric potential abstract we deal with the quasilinear schrodinger equation div u p 2 u a x 1 u p 2 u u q 2 u u w 1 p r n where 2 p n 0 and p q p n p n p the potential a 0 has a potential well and is invariant under an orthogonal involution of r n we apply variational methods to obtain for large existence of solutions which change sign exactly once we study the concentration behavior of these solutions as by taking q close p we also relate the number of solutions which change sign exactly once with the equivariant topology of the set where the potential a vanishes

derivations and central extensions of finitely generated graded lie algebras abstract some general results concerning derivations of finitely generated lie algebras are established these are employed in order to determine the derivations and central extensions of kac moody lie algebras

classification of division zn graded alternative algebras abstract the octonion torus or cayley torus appears as a coordinate algebra of extended affine lie algebras of type a 2 and f 4 a generalized version of the octonion torus called division z n graded alternative algebras is classified in this paper using the result we can complete the classification of division a 2 z n graded lie algebras up to central extensions which are a generalization of the cores of extended affine lie algebras of type a 2

vertex operator representations for tkk algebras abstract it is known that up to isomorphism there are only one lattice s and one non lattice semi lattice s in r 2 in s tan tkk algebras and vertex operator representations j algebra 211 1999 298 342 we obtained a vertex operator representation for the baby tkk algebra g t s the goal of the present paper is to study the tkk algebra g t s which includes the baby tkk algebra as a subalgebra and its vertex operator representations as a by product we also obtain two representations for the baby tkk algebra g t s one of the representations recovers the construction given in s tan tkk algebras and vertex operator representations j algebra 211 1999 298 342

vertex representations for n toroidal lie algebras and a generalization of the virasoro algebra vertex representations are obtained for toroidal lie algebras for any number of variables these representations afford representations of certainn variable generalizations of the virasoro algebra that are abelian extensions of the lie algebra of vector fields on a torus

lie algebras graded by finite root systems and the intersection matrix algebras of slodowy this paper is about toroidal lie algebras certain intersection matrix lie algebras defined by slodowy and their relationship to one another and to certain lie algebra analogues of steinberg groups the main result of the paper is the identification of the intersection matrix algebras arising from multiply affinized cartan matrices of types a d and e with certain steinberg lie algebras and toroidal lie algebras propositions 5 9 and 5 10 a major part of the paper studies and classifies lie algebras graded by finite root systems these become the princi pal tool in our analysis of intersection matrix algebras each lie algebra graded by a simply laced finite root system of rank 2 has attached to it an algebra which according to the type and rank is either commutative and associative only associative or alternative all these possibilities occur in our description of inter section matrix algebras let r be any associative algebra with identity not necessarily finite dimen sional over a field k of characteristic 0 for each positive integer n the associative algebra m r of n n matrices with coefficients in r forms a lie algebra over k under the commutator product we denote this lie algebra by ol r let eis be the i j matrix unit of m r and assume that n 2 the subalgebra e r of oi r generated by the elements reis r r i 4 j is an ideal of 91 r and is perfect i e it is its own derived algebra now any perfect lie algebra o has a universal central extension also perfect called a universal covering algebra u c a of o ga so in particular e r has a u c a that we will denote by t r we define 112 r by the exact sequence 0 1 0 f2 r t r e r 0 dedicated to our teacher maria j wonenburger both authors gratefully acknowledge the support of the natural sciences and engineering research council of canada

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices

irreducible modules over the derivation algebras of rational quantum tori abstract let g be the derivation lie algebra of a rational quantum torus c q where q q i j d d and all q i j are roots of unity in lin and tan 2004 lt a class of uniformly bounded irreducible weight modules over g were constructed which generalized the construction given by shen 1986 sh larsson 1992 l and eswara rao 1996 e these modules look mysterious because of their very unclear weight sets in this note we use very simple methods with a very short proof to re establish all even stronger results in lin and tan 2004 lt and make these mysterious modules crystal clear a necessary and sufficient condition for two such modules to be isomorphic is also given