the classification of n dimensional non lie malcev algebras with n 4 dimensional annihilator abstract in this paper we give a complete classification of all n dimensional non lie malcev algebras with n 4 dimensional annihilator over an algebraically closed field of characteristic 0 we also show that such algebras are special

manifolds of g2 holonomy from the n 4 sigma model using the two dimensional 2d n 4 sigma model with u 1 r gauge symmetry and introducing the ade cartan matrices as gauge matrix charges we build toric hyper kahler eight real dimensional manifolds x8 dividing by one toric geometry circle action of x8 manifolds we present examples describing quotients x7 x8 u 1 of g2 holonomy in particular for the ar cartan matrix the quotient space is a cone on an s2 bundle over r intersecting wcp2 1 2 1 weighted projective spaces according to the ar dynkin diagram

superstrings and topological strings at large n we embed the large n chern simons topological string duality in ordinary superstrings this corresponds to a large n duality between generalized gauge systems with n 1 supersymmetry in four dimensions and superstrings propagating on noncompact calabi yau manifolds with certain fluxes turned on we also show that in a particular limit of the n 1 gauge theory system certain superpotential terms in the n 1 system including deformations if spacetime is noncommutative are captured to all orders in 1 n by the amplitudes of noncritical bosonic strings propagating on a circle with self dual radius we also consider d brane anti d brane system wrapped over vanishing cycles of compact calabi yau manifolds and argue that at large n they induce a shift in the background to a topologically distinct calabi yau which we identify as the ground state system of the brane anti brane system

characterizing nilpotent leibniz algebras by a new bound on their second homologies abstract let l be a finite dimensional nilpotent leibniz algebra such that dim l n and dim l 2 m 0 in this paper we prove dim h l 2 l n m 2 n m m 2 where h l 2 l is the second leibniz homology of l as a consequence for a non abelian nilpotent leibniz algebra l we find that s l n 1 2 1 dim h l 2 l 0 furthermore we determine all finite dimensional nilpotent leibniz algebras with s l less than or equal to three

an m theory flop as a large n duality we show how a recently proposed large n duality in the context of type iia strings with n 1 supersymmetry in 4 dimensions can be derived from purely geometric considerations by embedding type iia strings in m theory the phase structure of m theory on g2 holonomy manifolds and an s3 flop are the key ingredients in this derivation

all loop topological string amplitudes from chern simons theory we demonstrate the equivalence of all loop closed topological string amplitudes on toric local calabi yau threefolds with computations of certain knot invariants for chern simons theory we use this equivalence to compute the topological string amplitudes in certain cases to very high degree and to all genera in particular we explicitly compute the topological string amplitudes for open image in new window 2 up to degree 12 and open image in new window 1 open image in new window 1 up to total degree 10 to all genera this also leads to certain novel large n dualities in the context of ordinary superstrings involving duals of type ii superstrings on local calabi yau three folds without any fluxes

six dimensional nilpotent lie algebras abstract we give a full classification of six dimensional nilpotent lie algebras over an arbitrary field including fields that are not algebraically closed and fields of characteristic 2 to achieve the classification we use the action of the automorphism group on the second cohomology space as isomorphism types of nilpotent lie algebras correspond to orbits of subspaces under this action in some cases these orbits are determined using geometric invariants such as the gram determinant or the arf invariant as a byproduct we completely determine for a four dimensional vector space v the orbits of gl v on the set of two dimensional subspaces of v v

flat compact hermite lorentz manifolds in dimension 4 we give a classification up to finite cover of flat compact complete hermite lorentz manifolds up to complex dimension 4

the isomorphism problem for universal enveloping algebras of nilpotent lie algebras in this paper we study the isomorphism problem for the universal enveloping algebras of nilpotent lie algebras we prove that if the characteristic of the underlying field is not 2 or 3 then the isomorphism type of a nilpotent lie algebra of dimension at most 6 is determined by the isomorphism type of its universal enveloping algebra examples show that the restriction on the characteristic is necessary

m theory on g2 manifolds and the method of p q brane webs using a reformulation of the method of p q webs we study the four dimensional n 1 quiver theories from m theory on seven dimensional manifolds with g2 holonomy we first construct such manifolds as u 1 quotients of eight dimensional toric hyper kahler manifolds using n 4 supersymmetric sigma models we show that these geometries in general are given by real cones on s2 bundles over complex two dimensional toric varieties v2 cr 2 c r then we discuss the connection between the physics content of m theory on such g2 manifolds and the method of p q webs motivated by a result of acharya and witten preprint hep th 0109152 we reformulate the method of p q webs and reconsider the derivation of the gauge theories using toric geometry mori vectors of v2 and brane charge constraints for we find that the gauge group is given by g u w1n u w2n u w3n this is required by the anomaly cancellation condition

on noncommutative orbifolds of k3 surfaces using the algebraic geometry method of berenstein and leigh for the construction of the toroidal orbifold t2 t2 t2 z2 z2 with discrete torsion and considering local k3 surfaces we present noncommutative aspects of the orbifolds of product of k3 surfaces in this way the ordinary complex deformation of k3 can be identified with the resolution of stringy singularities by noncommutative algebras using crossed products we give representations and make some comments regarding the fractionation of branes illustrating examples are presented

classification of minimal algebras over any field up to dimension 6 we give a classification of minimal algebras generated in degree 1 defined over any field k of characteristic different from 2 up to dimension 6 this recovers the classification of nilpotent lie algebras over k up to dimension 6 in the case of a field k of characteristic zero we obtain the classification of nilmanifolds of dimension less than or equal to 6 up to k homotopy type finally we determine which rational homotopy types of such nilmanifolds carry a symplectic structure

on local calabi yau supermanifolds and their mirrors we use local mirror symmetry to study a class of local calabi yau supermanifolds with bosonic sub variety vb having a vanishing first chern class solving the usual super cy condition requiring the equality of the total u 1 gauge charges of bosons b and the ghost like fields f one bqb fqf as bqb 0 and fqf 0 several examples are studied and explicit results are given for local ar supergeometries a comment on purely fermionic super cy manifolds corresponding to the special case where qb 0 b and fqf 0 is also made

computing local zeta functions of groups algebras and modules we develop a practical method for computing local zeta functions of groups algebras and modules in fortunate cases using our method we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six we also determine the generic local subalgebra zeta functions associated with gl2 q finally we introduce and compute examples of graded subobject zeta functions

on nilpotent lie algebras of small breadth abstract a lie algebra l is said to be of breadth k if the maximal dimension of the images of left multiplication by elements of the algebra is k in this paper we give characterization of finite dimensional nilpotent lie algebras of breadth less than or equal to two furthermore using these characterizations we determined the isomorphism classes of these algebras

on limit multiplicites of discrete series representations in spaces of automorphic forms of course the groups f are then arithmetic subgroups of g in this situation it seems natural to assume that as n the spectral decomposition of l2 f g will approximate that of g in particular let 3 be a discrete series representation of g if f is a subgroup of g let re f 6 be the multiplicity of 3 in the representation of g by right translations on l2 f g choose a haar measure dg on g if f is arithmetic let v f g be the finite volume of f g for the measure induced by dg let d 3 be the formal degree of 3 for the measure dg

the groups with order p7 for odd prime p abstract we determine product presentations for the nilpotent lie rings with order p 7 where p 7 is prime and then use the baker campbell hausdorff formula to construct power commutator presentations for the corresponding groups the number of such groups is a polynomial depending on p whose leading term is 3 p 5 we complete the determination of groups with order p 7 for p 3 5 using the p group generation algorithm we provide access to the resulting presentations for the groups via a database distributed with computer algebra systems

on geometric transitions in string compactifications we reconsider the study of the geometric transitions and brane flux dualities in various dimensions first we give toric interpretations of the topology changing transitions in the calabi yau conifold and the spin 7 manifold the latter for instance can be viewed as three intersecting calabi yau conifolds according to the toric graph orbifolds of such geometries are given in terms of del pezzo complex surfaces second we propose a four dimensional f theory interpretation of type iib geometric transitions on the calabi yau conifold this gives a dual description of the m theory flop in terms of toric mirror symmetry in two dimensions we study the geometric transition in a singular spin 7 manifold constructed as a cone on su 3 u 1 in particular we discuss brane flux duality in such a compactification in both type iia and type iib superstrings these examples preserve one supercharge and so have supersymmetry in two dimensions then an interpretation in terms of f theory is given

topological representation zeta functions of unipotent groups inspired by work surrounding igusa s local zeta function we introduce topological representation zeta functions of unipotent algebraic groups over number fields these group theoretic invariants capture common features of established p adic representation zeta functions associated with pro p groups derived from unipotent groups we investigate fundamental properties of the topological zeta functions considered here we also develop a method for computing them under non degeneracy assumptions as an application among other things we obtain a complete classification of topological representation zeta functions of unipotent algebraic groups of dimension at most six

classification of 6 dimensional nilpotent lie algebras over fields of characteristic not 2 abstract nilpotent 6 dimensional lie algebras over any field of characteristic not 2 are classified the proof of this classification is essentially constructive for a given 6 dimensional nilpotent lie algebra l following the steps of the proof it is possible to find a lie algebra m that occurs in the classification and an isomorphism l m in the proof a method due to skjelbred and sund is used along with a method based on grobner bases to find isomorphisms

f theory duals of m theory on g2 manifolds from mirror symmetry using mirror pairs m3 w3 in type ii superstring compactifications on calabi yau threefolds we study geometrically f theory duals of m theory on seven manifolds with g2 holonomy we first develop a way of obtaining landau ginzburg lg calabi yau threefolds w3 embedded in four complex dimensional toric varieties mirror to the sigma model on toric calabi yau threefolds m3 this method gives directly the right dimension without introducing non dynamical variables then using toric geometry tools we discuss the duality between m theory on s1 m3 z2 with g2 holonomy and f theory on elliptically fibred calabi yau fourfolds with su 4 holonomy containing w3 mirror manifolds illustrative examples are presented

sign changes in harmonic analysis on reductive groups let g be a connected reductive group over a field f in this note the author constructs an element e g of the brauer group of f the square of this element is trivial for a local field e g may be regarded as an element of i and is needed for harmonic analysis on reductive groups over that field for a global field there is a product formula the purpose of this note is to provide a cohomological interpretation of the sign changes in harmonic analysis on reductive groups over local fields that are caused by inner twistings the best understood example occurs in the work of shelstad 8 on inner twistings of real groups theorem 6 3 of that paper gives the following character identity 1 q g q g x here g is a connected reductive group over r g is its quasi split inner form x and x are stabilized characters of matched tempered l packets of g r and g r and y and y are matched regular semisimple elements of g r and g r the sign change in the character identity is q g q g where q g is one half of the dimension of the symmetric space attached to g more precisely attached to the simply connected cover of the derived group of g although q g may be only half integral the difference q g q g is always integral it is reasonable to expect an analogous character identity for groups g over a nonarchimedean local field f we can determine what the sign change must be by considering the steinberg characters of g f and g f since these should be related by the character identity let r g denote the f rank of the derived group of g the steinberg character of g f has value i r g on the elliptic regular set of g f and hence the sign change in the character identity must be l r g r g there are two groups for which the character identity has been proved the multiplicative group of a central division algebra of dimension d2 over f for d 2 see proposition 15 5 on p 484 of 4 and d 3 see theorem 1 of 1 later in this paper we will use a cohomological construction to define a sign e g 1 for any connected reductive group g over a local field let g denote a quasi split inner form of g we will show that e g l r g r g when the base received by the editors may 10 1982 1980 mathematics subject classification primary 22e50 partially supported by the national science foundation under grant mcs 78 02331 1983 american mathematical society 0002 9947 82 0000 11 93 03 25

groups and nilpotent lie rings whose order is the sixth power of a prime abstract we prove that there are 3p2 39p 344 24gcd p 1 3 11gcd p 1 4 2gcd p 1 5 isomorphism types of groups and nilpotent lie rings with order p6 for every prime p 5 we establish the result and power commutator presentations for the groups in various ways the most novel method constructs product presentations for nilpotent lie rings with order p6 and then uses the baker campbell hausdorff formula to construct power commutator presentations for the corresponding groups public access to the group presentations is provided via a database distributed with computer algebra systems

the classification of 5 dimensional p nilpotent restricted lie algebras over perfect fields i abstract we first adapt a method due to skjelbred sund to classify p nilpotent restricted lie algebras it turns out that any p nilpotent restricted lie algebra of dimension n can be constructed as a central extension of a p nilpotent restricted lie algebra of dimension n 1 we apply these techniques to classify all p nilpotent restricted lie algebras of dimension 5 over a perfect field of characteristic p 5

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

mirror symmetry and landau ginzburg calabi yau superpotentials in f theory compactifications we study the landau ginzburg lg theory mirror to two dimensional 2d n 2 gauged linear sigma models on toric calabi yau manifolds we derive and solve new constraint equations for lg elliptic calabi yau superpotentials depending on the physical data of dual linear sigma models in the calabi yau threefold case we consider two examples first we give the mirror symmetry of the canonical line bundle over the hirzebruch surfaces fn second we find a special geometry with the affine so 8 lie algebra toric data extending the geometry of elliptically fibred k3 this geometry leads to a pure n 1 six dimensional so 8 gauge model from the f theory compactification for calabi yau fourfolds we give a new algebraic realization for ade hypersurfaces

quantum screening operators and canonical q de rham cocycles in this paper a close connection is established between certain cohomology spaces of representations of the quantum affine algebra and a twisted q de rham jackson aomoto cohomology of configuration spaces using the quantum screening operators

nc calabi yau orbifolds in toric varieties with discrete torsion using the algebraic geometric approach of berenstein et al hep th 005087 and hep th 009209 and methods of toric geometry we study non commutative nc orbifolds of calabi yau hypersurfaces in toric varieties with discrete torsion we first develop a new way of getting complex d mirror calabi yau hypersurfaces h d in toric manifolds m d 1 with a c r action and analyse the general group of the discrete isometries of h d then we build a general class of d complex dimensional nc mirror calabi yau orbifolds where the non commutativity parameters are solved in terms of discrete torsion and toric geometry data of m d 1 in which the original calabi yau hypersurfaces are embedded next we work out a generalization of the nc algebra for generic d dimensional nc calabi yau manifolds and give various representations depending on different choices of the calabi yau toric geometry data we also study fractional d branes at orbifold points we refine and extend the result for nc t 2 x t 2 x t 2 z 2 x z 2 to higher dimensional torii orbifolds in terms of clifford algebra

on noncommutative multi solitons solitons in noncommutative scalar field theories possess many stringy features in particular multi solitons in these theories behave like multiple d branes a useful moduli space description for multi solitons exists which reproduces stringy features of d brane moduli spaces in a simple way via properties of special projection operators in particular they are equivalent to a hilbert scheme resolution of symmetric product spaces