quantum symmetric kac moody pairs the present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable kac moody algebras the resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras they give rise to triangular decompositions including a quantum analog of the iwasawa decomposition and they can be written explicitly in terms of generators and relations moreover their centers and their specializations are determined the constructions follow g letzter s theory of quantum symmetric pairs for semisimple lie algebras the main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable kac moody algebras due to kac and wang the resulting theory comprises various classes of examples which have previously appeared in the literature such as q onsager algebras and the twisted q yangians introduced by molev ragoucy and sorba

atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

automorphisms of borcherds superalgebras and fixed point subalgebras abstract let g be a borcherds superalgebra and let be a finite order automorphism of g satisfying certain conditions we prove that the fixed point subalgebra g is equal to a borcherds superalgebra we also compute the trace of on g from the twisted denominator identity of g and derive closed form formula for root multiplicities of g

maps between classifying spaces of kac moody groups abstract we classify up to homotopy the self maps of the classifying space of any non affine kac moody group of rank two

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

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

hom lie algebra structures on semi simple lie algebras abstract hom lie algebras can be considered as a deformation of lie algebras in this note we prove that the hom lie algebra structures on finite dimensional simple lie algebras are trivial we find when a finite dimensional semi simple lie algebra admits non trivial hom lie algebra structures and the isomorphic classes of non trivial hom lie algebras are determined

enumerations of the hamiltonian walks on a cubic sublattice a massively parallel supercomputer was used to exhaustively enumerate all of the hamiltonian walks for simple cubic sublattices of four different sizes up to 3 4 4 the behaviour of the logarithm of the number of walks was found to be linear in the number of vertices in the lattice the linear fit is shown to agree also with the asymptotic limit of the flory mean field theoretical estimate thus we suggest that the fit obtained yields the number of walks for any size fragment of the cubic lattice to logarithmic accuracy the significance of this result to the validity of polymer models is also discussed

a classification of involutive automorphisms of an affine kac moody lie algebra abstract in this thesis we classify the conjugacy classes of involutions in aut g where g is an affine kac moody lie algebra we distinguish between two kinds of involutions those which preserve the conjugacy class of a borel subalgebra and those which don t we give a complete and non redundant list of representatives of involutions of the first kind and we compute their fixed points sets we prove that any involution of the first kind has a conjugate which leaves invariant the components of the gauss decomposition g n h n we also give a complete list of representatives of the conjugacy classes of involutions of the second kind

passive localization of near field sources by path following a new algorithm for passively estimating the ranges and bearings of multiple narrow band sources using a uniform linear sensor array is presented the algorithm is computationally efficient and converges globally it minimizes the music cost function subject to geometrical constraints imposed by the curvature of the received wavefronts the estimation problem is reduced to one of solving a set of two coupled 2d polynomial equations the proposed algorithm solves this nonlinear problem using a modification of the path following or homotopy method for an array having m sensors the algorithm reduces the global 2d search over range and bearing to 2 m spl minus 1 independent 1d searches this imparts a high degree of parallelism that can be exploited to obtain source location estimates very efficiently

deformations of lie algebras using σ derivations in this article we develop an approach to deformations of the witt and virasoro algebras based on sigma derivations we show that sigma twisted jacobi type identity holds for generators of such deformations for the sigma twisted generalization of lie algebras modeled by this construction we develop a theory of central extensions we show that our approach can be used to construct new deformations of lie algebras and their central extensions which in particular include naturally the q deformations of the witt and virasoro algebras associated to q difference operators providing also corresponding q deformed jacobi identities c 2005 elsevier inc all rights reserved

dominant k theory and integrable highest weight representations of kac moody groups we give a topological interpretation of the highest weight representations of kac moody groups given the unitary form g of a kac moody group over c we define a version of equivariant k theory kg on the category of proper g cw complexes we then study kac moody groups of compact type in detail see section 2 for definitions in particular we show that the grothendieck group of integrable highest weight representations of a kac moody group g of compact type maps isomorphically onto k g eg where eg is the classifying space of proper g actions for the affine case this agrees very well with recent results of freed hopkins teleman we also explicitly compute kg eg for kac moody groups of extended compact type which includes the kac moody group e10

vogan diagrams of real forms of affine kac moody lie algebras abstract a vogan diagram is actually a dynkin diagram with some additional structure this paper develops theory of vogan diagrams for almost compact real forms of indecomposable nontwisted affine kac moody lie algebras here the equivalence classes of vogan diagrams are in one one correspondence with the isomorphism classes of almost compact real forms

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

commutative 2 cocycles on lie algebras abstract on lie algebras we study commutative 2 cocycles i e symmetric bilinear forms satisfying the usual cocycle equation we note their relationship with antiderivations and compute them for some classes of lie algebras including finite dimensional semisimple current and kac moody algebras

the number of smallest knots on the cubic lattice it has been shown that the smallest knots on the cubic lattice are all trefoils of length 24 in this paper we show that the number of such unrooted knots on the cubic lattice is 3496

generalized derivations of lie algebras abstract suppose l is a finite dimensional lie algebra with multiplication l l l let l denote the set of triples f f f with f f f hom l l such that f i l i l f f we consider the lie algebra gender l f hom l l f f f f f l well researched subalgebras of gender l include the derivation algebra der l f hom l l f f f l and the centroid c l f hom l l f 0 f l we now study the subalgebra qder l f hom l l f f f f l and the subspace qc l f hom l l f f 0 l in characteristic 2 gender l qder l qc l and we are concerned with the inclusions der l qder l and c l qc l qder l if z l 0 then c l qc l qder l and under reasonable conditions on lie algebras with toral cartan subalgebras we show qder l der l c l if l is a parabolic subalgebra of a simple lie algebra of rank 1 in characteristic 0 then we even have gender l ad l i l in general qc l is not closed under composition or lie bracket however if z l 0 then qc l is a commutative associative algebra and we describe conditions that force qc l c l or equivalently gender l qder l we show that in characteristic 0 gender l preserves the radical of l thus generalizing the classical result for der l we also discuss some applications of the main results to the study of functions f hom l l such that f or f i l defines a lie multiplication

unitary structure in representations of infinite dimensional groups and a convexity theorem in this paper we show that a kac moody algebra fl a associated to a symmetrizable generalized cartan matrix a carries a contravariant hermitian form which is positive definite on all root spaces we deduce that every integrable highest weight g a module l a carries a contravariant positivedefinite hermitian form this allows us to define the moment map and prove a generalization of the schur horn kosta nt heckman atiya h pressley convexity theorem the proofs are based on an identity which also gives estimates for the action of 9 a on fl a and l a we hope that the main idea behind the paper is apparent it is to use the interplay between the coadjoint and the highest weight representations we are grateful to v guillemin for an introduction to the moment map

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

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

on the universality of knot probability ratios let pn denote the number of self avoiding polygons of length n on a regular three dimensional lattice and let pn k be the number which have knot type k the probability that a random polygon of length n has knot type k is pn k pn and is known to decay exponentially with length sumners and whittington 1988 j phys a math gen 21 1689 94 pippenger 1989 discrete appl math 25 273 8 little is known rigorously about the asymptotics of pn k but there is substantial numerical evidence orlandini et al 1988 j phys a math gen 31 5953 67 marcone et al 2007 phys rev e 75 41105 rawdon et al 2008 macromolecules 41 4444 51 janse van rensburg and rechnitzer 2008 j phys a math theor 41 105002 that pn k grows as where nk is the number of prime components of the knot type k it is believed that the entropic exponent is universal while the exponential growth rate is independent of the knot type but varies with the lattice the amplitude ck depends on both the lattice and the knot type the above asymptotic form implies that the relative probability of a random polygon of length n having prime knot type k over prime knot type l is in the thermodynamic limit this probability ratio becomes an amplitude ratio it should be universal and depend only on the knot types k and l in this communication we examine the universality of these probability ratios for polygons in the simple cubic face centred cubic and body centred cubic lattices our results support the hypothesis that these are universal quantities for example we estimate that a long random polygon is approximately 28 times more likely to be a trefoil than be a figure eight independent of the underlying lattice giving an estimate of the intrinsic entropy associated with knot types in closed curves

representations of brauer category and categorification abstract we study representations of the locally unital and locally finite dimensional algebra b associated to the brauer category b 0 with defining parameter 0 over an algebraically closed field k with characteristic p 2 the grothendieck group k 0 b mod will be used to categorify the integrable highest weight sl k module v 0 1 2 with the fundamental weight 0 1 2 as its highest weight where b mod is a subcategory of b lfdmod in which each object has a finite flag and sl k is either sl or sl p depending on whether p 0 or 2 p as g modules c z k 0 b mod is isomorphic to v 0 1 2 where g is a lie subalgebra of sl k see definition 4 2 when p 0 standard b modules and projective covers of simple b modules correspond to monomial basis and so called quasi canonical basis of v 0 1 2 respectively

mismatch of near field bearing range spatial geometry in source localization by a uniform linear array many near field source localization algorithms intentionally simplifies the exact spatial geometry among the emitter and the sensors in order to speed up the signal processing involved for example the fresnel approximation is a second order taylor series approximation such intentional approximation introduces a systemic error in the algorithm s modeling of the actual objective reality from which the measured data arise a mismatch thus exists between the algorithm s presumptions versus the data it processes this modeling mismatch will introduce a systematic bias in the bearing range estimates of the near field source localization algorithm this bias is non random and adds towards the random estimation errors due to the additive and or multiplicative noises the open literature currently offers no rigorous mathematical analysis on this issue this proposed project aims to fill this literature gap by deriving explicit formulas of the degrading effects in three dimensional source localization due to approximating the source sensor geometry by any order of the taylor s series expansion

isomorphisms of kac moody groups which preserve bounded subgroups a subgroup of a kac moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs in this paper we study the isomorphisms between kac moody groups over arbitrary fields of cardinality at least 4 which preserve the set of bounded subgroups we show that such an isomorphism between two such kac moody groups induces an isomorphism between the respective twin root data of these groups as a consequence we obtain the solution of the isomorphism problem for kac moody groups over finite fields of cardinality at least 4

on the topology of kac moody groups we study the topology of spaces related to kac moody groups given a kac moody group over mathbb c let text k denote the unitary form with maximal torus mathrm t having normalizer mathrm n mathrm t in this article we study the cohomology of the flag manifold text k mathrm t as a module over the nil hecke algebra as well as the co homology of text k as a hopf algebra in particular if mathbb f has positive characteristic we show that text h text k mathbb f is a finitely generated algebra and that text h text k mathbb f is finitely generated only if text k is a compact lie group we also study the stable homotopy type of the classifying space text bk and show that it is a retract of the classifying space text bn t of mathrm n mathrm t we illustrate our results with the example of rank two kac moody groups

knot complexity and related observables from path integrals for semiflexible polymers we analyse earlier obtained numerical results for the knot complexity and the topological persistence length i e the minimal number of steps on the lattice needed for the knot to be formed with help of well known path integrals for semiflexible chains in addition we provide an estimate for the number of different knots having knot diagrams with exactly n crossings analytical results obtained are in satisfactory agreement with the available numerical data

low dimensional cohomology of current lie algebras and analogs of the riemann tensor for loop manifolds abstract we obtain formulas for the first and second cohomology groups of a general current lie algebra with coefficients in the current module and apply them to compute structure functions for manifolds of loops with values in compact hermitian symmetric spaces

isomorphisms of kac moody groups we give the solution of the isomorphism problem for kac moody groups over algebraically closed fields of any characteristic in particular we prove a conjecture of kac and peterson and compute the automorphism group of a kac moody group over an algebraically closed field of characteristic zero

a modified path following algorithm using a known algebraic path a new computationally efficient path following algorithm is presented for source localization that uses a uniform linear sensor array an algebraic relation is demonstrated between the bearing of a source under a plane wave assumption and the actual source bearing and range this relation can be used as a path to follow to the peak of the 2 d music spectrum as a result in the case of an array that has m sensors with n sources this new algorithm can reduce the number of searches to n independent 1 d searches following the known algebraic path

hom lie structures on kac moody algebras abstract we describe hom lie structures on affine kac moody and related lie algebras and discuss the question when they form a jordan algebra

bounds for the minimum step number of knots in the simple cubic lattice knots are found in dna as well as in proteins and they have been shown to be good tools for structural analysis of these molecules an important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot here we address this problem by characterizing both analytically and numerically the minimum length also called minimum step number needed to form a particular knot in the simple cubic lattice our analytical work is based on improvement of a method introduced by diao to enumerate conformations of a given knot type for a fixed length this method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 which is 24 to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34 respectively using an independent method based on the bfacf algorithm we provide a complete list of numerical estimates upper bounds of the minimum step numbers for prime knots up to ten crossings which are improvements over current published numerical results we enumerate all minimum lattice knots of a given type and partition them into classes defined by bfacf type 0 moves

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

almost sure identifiability of multidimensional harmonic retrieval two dimensional 2 d and more generally multidimensional harmonic retrieval is of interest in a variety of applications including transmitter localization and joint time and frequency offset estimation in wireless communications the associated identifiability problem is key in understanding the fundamental limitations of parametric methods in terms of the number of harmonics that can be resolved for a given sample size consider a mixture of 2 d exponentials each parameterized by amplitude phase and decay rate plus frequency in each dimension suppose that i equispaced samples are taken along one dimension and likewise j along the other dimension we prove that if the number of exponentials is less than or equal to roughly ij 4 then assuming sampling at the nyquist rate or above the parameterization is almost surely identifiable this is significant because the best previously known achievable bound was roughly i j 2 for example consider i j 32 our result yields 256 versus 32 identifiable exponentials we also generalize the result to n dimensions proving that the number of exponentials that can be resolved is proportional to total sample size

satake diagrams of affine kac moody algebras satake diagrams of affine kac moody algebras untwisted and twisted are obtained from their dynkin diagrams these diagrams give a classification of restricted root systems associated with these algebras in the case of simple lie algebras these root systems and satake diagrams correspond to symmetric spaces which have recently found many physical applications in quantum integrable systems quantum transport problems random matrix theories etc we hope these types of root systems may have similar applications in theoretical physics in future and may correspond to symmetric spaces analogue of affine kac moody algebras if they exist

orbits invariants and representations associated to involutions of reductive groups by an involution of an algebraic group or lie algebra we mean an automorphism 0 such that 0 2 is the identity automorphism let 0 be an involution of a complex reductive lie algebra q and let p denote the 1 eigenspace of 0 on q get g be the adjoint group of q and let k g glog go kostant and rallis 10 have made a detailed study of the action of k on p in particular they obtain very precise information concerning the orbits of k on p and the representation of k on c p the algebra of polynomial functions on p in this paper we consider a global analogue of the problem studied by kostant and rallis let g be a reductive affine algebraic group over an algebraically closed field k of characteristic 2 let 0 be an involution of g and let k gegi0 g g if g is semisimple and the base field is c then g k is the complexification of a riemannian symmetric space we study the action of k and of groups closely related to k by left translation on the coset space g k for technical reasons it is convenient to study a slightly different but equivalent problem let r g g be defined by r g g0 g i then z g p is a closed subvariety of g furthermore z is constant on left cosets of g modulo k and induces an isomorphism o g k p we have o k g k k o g k for k e k and geg thus p is stable under conjugation by k and a is a k isomorphism of affine k varieties we consider the action of k on p by conjugation let x e g have jordan decomposition x x x then we show that x e p if and only if both x and x are in p thus we have a jordan decomposition for elements of p this will play a key role in our results a torus s of g is o anisotropic if o s s 1 for every ses it is known that any two maximal 0 anisotropic tori of g are k let a be a maximal 0 anisotropic torus of g the following result characterizes semisimple elements of p