application of generating function techniques to lie superalgebras the construction of generating functions for multiplicities of irreducible representations of lie superalgebras from generating functions for characters is examined and applied in order to obtain polynomial tensors and branching rules problems arising because of the existence of typical and atypical representations are discussed in detail the techniques are applied to osp 1 2 spl 1 2 osp 3 2 and osp 4 2

stagnation analysis of dgmres we study problems for which the iterative method dgmres for solving drazin inverse solution of singular linear systems of equations which makes no progress in its initial iterations particularly our tool for analysis is nonlinear system of equations the stagnation system that characterize this behavior for problems of dimension two and index one we can solve this system explicitly we partially extend this result to higher dimensions for a class of eigenvector matrices called extreme we also study the real matrices

irreducible characters of general linear superalgebra and super duality we develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra including all the finite dimensional modules by directly relating the problem to the classical kazhdan lusztig theory furthermore we prove that certain parabolic bgg categories over the general linear algebra and over the general linear superalgebra are equivalent we also verify a parabolic version of a conjecture of brundan on the irreducible characters in the bgg category of the general linear superalgebra

dgmres a gmres type algorithm for drazin inverse solution of singular non symmetric linear systems abstract in a recent work by the author linear algebra appl 298 1999 99 krylov subspace methods were derived for drazin inverse solution of consistent or inconsistent linear systems of the form ax b where a c n n is a singular and in general non hermitian matrix that has arbitrary index one of these methods modeled after the generalized conjugate residual method gcr and denoted dgcr is considered in the present work again it is shown that all of the approximations produced by dgcr exist and a gmres like algorithm denoted dgmres for its implementation is derived like gmres dgmres too is economical computationally and storagewise

a unified approach to krylov subspace methods for the drazin inverse solution of singular nonsymmetric linear systems abstract consider the linear system ax b where a c n n is a singular matrix in the present work we propose a general framework within which krylov subspace methods for drazin inverse solution of this system can be derived in a convenient way the krylov subspace methods known to us to date treat only the cases in which a is hermitian and its index ind a is unity necessarily in the present work a is not required to be hermitian it can have any type of spectrum and ind a is arbitrary we show that as is the case with nonsingular systems the krylov subspace methods developed here terminate in a finite number of steps that is at most n ind a for one of the methods derived here we also provide an analysis by which we are able to bound the errors the relevant bounds decreasing with increasing dimension of the krylov subspaces involved the results of this paper are applicable to consistent systems as well as to inconsistent ones an interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard krylov subspace approach to nonsingular systems indeed our approach here reduces to that relevant for nonsingular systems upon setting ind a 0 everywhere

technical communique upper bounds for the solution of the discrete algebraic lyapunov equation new upper bounds for the solution of the discrete algebraic lyapunov equation dale p apa t q are presented the only restriction on their applicability is that a be stable there are no restrictions on the singular values of a nor on the diagonalizability of a the new bounds relate the size of p to the radius of stability of a the upper bounds are computable when the large dimension of a make direct solution of the dale impossible the new bounds are shown to reflect the dependence of p on a better than previously known upper bounds

character formula for infinite dimensional unitarizable modules of the general linear superalgebra the fock space of m p bosonic and n q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra glm p n q its tensor powers decompose into direct sums of infinite dimensional irreducible highest weight glm p n q modules we obtain an explicit decomposition of any tensor power of this fock space into irreducibles and develop a character formula for the irreducible glm p n q modules arising in this way

representations of low rank orthosymplectic superalgebras by superfield techniques representations of lie superalgebras may be realised as functions superfields on graded manifolds suitably chosen onset spaces the technique is illustrated for osp 1 2 and osp 2 2 with a review of well known results and applied in the cases of osp 3 2 and osp 4 2 to construct some classes of irreducible representations

finite and infinite dimensional representations of the orthosymplectic superalgebra osp 3 2 the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the orthosymplectic superalgebra osp 3 2 for all cases the star or grade star conditions for the algebra are investigated only two finite dimensional representations are grade star representations if the representation space is required to be a graded hilbert space when the even part is so 3 sp 2 r su 2 su 1 1 an infinite class of infinite dimensional star representations is found one of them can be realized in terms of two valued functions of a complex variable this representation reduces to the sum of two metaplectic representations of sp 2 we show that it is precisely this metaplectic representation for osp 3 2 which gives the spin energy eigenstates for the one dimensional harmonic oscillator with spin 1 2 states

quasi finite modules for lie superalgebras of infinite rank we classify the quasi finite irreducible highest weight modules over the infinite rank lie superalgebras gl and d and determine the necessary and sufficient conditions for such modules to be unitarizable the unitarizable irreducible modules are constructed in terms of fock spaces of free quantum fields and explicit formulae for their formal characters are also obtained by investigating howe dualities between the infinite rank lie superalgebras and classical lie groups

lie superalgebras and the multiplet structure of the genetic code ii branching schemes continuing our attempt to explain the degeneracy of the genetic code using basic classical lie superalgebras we present the branching schemes for the typical codon representations typical 64 dimensional irreducible representations of basic classical lie superalgebras and find three schemes that do reproduce the degeneracies of the standard code based on the orthosymplectic algebra osp 5 2 and differing only in details of the symmetry breaking pattern during the last step

howe duality and combinatorial character formula for orthosymplectic lie superalgebras we study the howe dualities involving the reductive dual pairs o d spo 2m 2n and sp d osp 2m 2n on the super symmetric tensor of cd cm n we obtain complete decompositions of this space with respect to their respective joint actions we also use these dualities to derive a character formula for these irreducible representations of spo 2m 2n and osp 2m 2n that appear in these decompositions

kostant homology formulas for oscillator modules of lie superalgebras abstract we provide a systematic approach to obtain formulas for characters and kostant u homology groups of the oscillator modules of the finite dimensional general linear and ortho symplectic superalgebras via howe dualities for infinite dimensional lie algebras specializing these lie superalgebras to lie algebras we recover in a new way formulas for kostant homology groups of unitarizable highest weight representations of hermitian symmetric pairs in addition two new reductive dual pairs related to the above mentioned u homology computation are worked out

representations of osp 2 1 and the metaplectic representation shift operator techniques are used to treat the irreducible representations of the superalgebra osp 2 1 apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact su 2 algebra the case when the star conditions on the even part are those satisfied by the noncompact su 1 1 algebra is also treated in this case no gradestar representations arise and the star representations are found to consist of the direct sum of two discrete series representations of su 1 1 one of these representations can be realized in terms of functions of a single complex variable and turns out to be a simple example of a metaplectic representation

extension of the mathfrak o mathfrak s mathfrak p m n sim mathfrak s mathfrak o m n correspondence to the infinite dimensional chiral spinors and self dual tensors the spinor representations of the orthosymplectic lie superalgebras are considered and constructed these are infinite dimensional irreducible representations of which the superdimension coincides with the dimension of the spinor representation of next we consider the self dual tensor representations of and their generalizations these are also infinite dimensional and correspond to the highest irreducible component of the pth power of the spinor representation we determine the character of these representations and deduce a superdimension formula from this it follows that also for these representations the correspondence holds

finite dimensional representations of periplectic lie superalgebras abstract we study categories of finite dimensional modules over the periplectic lie superalgebras and obtain a bgg type reciprocity in particular we prove that these categories have only finitely many blocks we also compute the characters for irreducible modules over periplectic lie superalgebras of ranks 2 and 3 and obtain explicit description of the blocks for ranks 2 3 and 4

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

classification of all star irreps of gl m n all the finite dimensional star irreps of gl m n are classified in terms of their highest weights thereby completing the classification of all finite dimensional star irreps of the basic classical lie superalgebras the lowest weights of such irreps are determined explicitly and it is shown that the contravariant and covariant tensor irreps of gl m n are star irreps of type 1 and 2 respectively this explains the applicability of the young diagram method to these two classes of representations however it is shown that one parameter families of typical star irreps occur that are intrinsically different from the tensorial irreps

condition number of drazin inverse and their condition numbers of singular linear systems various normwise relative condition numbers that measure the sensitivity of drazin inverse and the solution of singular linear systems are characterized the sensitivity of condition number itself is then investigated finally upper bounds are derived for the sensitivity of structured perturbation

classification of all star and grade star irreps of gl n 1 a method for inducing nondegenerate forms on irreducible gl m n modules that implies some general results on star and grade star representations is investigated these results are applied to obtain a complete classification in terms of highest weights of the irreducible star and grade star representations of gl n 1 it is demonstrated that while gl n 1 admits a large class of star representations the irreducible grade star representations are comparatively rare moreover for n 2 all grade star irreducible representations are also star representations and for n 2 are atypical the superalgebra gl 2 1 proves to be a special case and admits a two parameter family of four dimensional typical grade star irreducible representations that are not star representations in particular typical grade star irreducible representations of gl n 1 exist only for n 1 2

representations of supergroups an explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations the rules of supersymmetrization are characterized by extended young supertableaux this constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all casimir operators can be calculated we have applied this approach so far to the supergroups su n m osp n 2m p n for which we have obtained all the representations constructible as direct products of the fundamental defining representations an argument is presented toward the irreducibility of all these representations

regular kac moody superalgebras and integrable highest weight modules we define regular kac moody superalgebras and classify them using integrable modules we give conditions for irreducible highest weight modules of regular kac moody superalgebras to be integrable this paper is a major part of the proof for the classification of finite growth contragredient lie superalgebras

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

a two step algorithm for solving singular linear systems with index one abstract in this paper a two step algorithm for solving singular linear systems is presented we compare this algorithm with a dgcr type algorithm a sidi a unified approach to krylov subspace methods for the drazin inverse solution of singular non symmetric linear systems linear algebra appl 298 1999 99 113 by numerical experiments an error analysis is given

with central charge n we study representations of the central extension of the lie algebra of differential operators on the circle the algebra we obtain complete and specialized character formulas for a large class of representations which we call primitive these include all quasi finite irreducible unitary representations we show that any primitive representation with central chargen has a canonical structure of an irreducible representation of the with the same central charge and that all irreducible representations of with central chargen arise in this way we also establish a duality between integral modules of and finite dimensional irreducible modules ofgl n and conjecture their fusion rules

character and dimension formulae for queer lie superalgebra closed formulae are constructed for the characters and dimensions of the finite dimensional simple modules of the queer lie superalgebra mathfrak q n this is achieved by refining brundan s algorithm for computing simple mathfrak q n characters

kostant s formula for kac moody lie algebras abstract this paper is a generalization of h garland and j lepowsky s paper of 1976 let g a be the kac moody lie algebra defined by a symmetrizable generalized cartan matrix a m the irreducible highest weight module of g a with a dominant integral highest weight kostant s homology and cohomology formulas for the lie algebra g a and the module m are proved without assuming that the reductive part of the parabolic subalgebra of g a is of finite type a resolution of m is constructed for any j the jth term of the resolution has a filtration such that all the factors of the filtration are generalized verma modules of the form vm w p p where w ranges over a certain subset of the weyl group of g a

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

condition number for the drazin inverse and the drazin inverse solution of singular linear system with their condition numbers in this paper we investigate the condition number of drazin inverse and drazin inverse solution of singular linear system ax b where a is a nxn rank deficient matrix and b a real vector of size n x a real vector let a and b be two positive real numbers when we consider the weighted frobenius norm aa bb p q f on the data we get the formula of condition number of the drazin inverse solution of singular linear system for the normwise condition number the sensitivity of the relative condition number itself is studied the componentwise perturbation is also investigated

gradient methods for computing the drazin inverse solution the problem of computing the vector a db in the cases when it provides a solution of a singular linear system is reduced to the problem of finding a least squares solution of an appropriate linear system later the presented ideas are extended to arbitrary linear systems as a consequence we present iterative methods for computing the vector a db in general the functionality of the exposed algorithms is based on a specific representation of the drazin inverse solution as well as the properties that we introduce convergence results of the proposed methods are also considered illustrative numerical examples are presented

finite dimensional half integer weight modules over queer lie superalgebras we give a new interpretation of representation theory of the finite dimensional half integer weight modules over the queer lie superalgebra mathfrak q n it is given in terms of the brundan s work on finite dimensional integer weight mathfrak q n modules by means of lusztig s canonical basis using this viewpoint we compute the characters of the finite dimensional half integer weight irreducible modules for a large class of irreducible modules whose highest weights are of special types i e totally connected or totally disconnected we derive closed form character formulas that are reminiscent of the kac wakimoto character formula for basic lie superalgebras

kac wakimoto character formula for ortho symplectic lie superalgebras abstract we classify finite dimensional tame modules over the ortho symplectic lie superalgebras for these modules we show that their characters are given by the kac wakimoto character formula thus establishing the kac wakimoto conjecture for the ortho symplectic lie superalgebras we further relate the kac wakimoto formula to the super jacobi polynomials of sergeev and veselov and show that these polynomials up to a sign give the super characters for these tame modules

robust performance assessment of feedback control systems the proper measure of closed loop performance variation in the presence of model plant mismatch is discussed in this paper a generalized closed loop error transfer function which is a special representation of the dual youla parameter and has a close relationship with the pointwise n gap metric is proposed as the suitable means of representing closed loop performance variation in case of plant perturbation and the closed loop performance variation measure is accordingly defined as its maximum singular value frequency by frequency it is shown that this measure is essential and informative in characterizing closed loop performance variation this measure is also shown to be readily applicable to on line closed loop performance assessment or monitoring even without the explicit model of the plant its variant defined as the h function which features the relative performance variation as well as generalized stability margin variation with respect to the nominal plant is also discussed

indices of representations of simple superalgebras indices and anomaly numbers for representations of basic classical lie superalgebras are defined and their explicit expressions are derived in terms of kac dynkin labels useful properties of indices and anomalies are determined and several examples are given a similar analysis is made for superindices and superanomalies and it is demonstrated how all these objects form a helpful tool in decomposing tensor products or in constructing branching rules for representations

dfom algorithm and error analysis for projection methods for solving singular linear system in this paper dfom algorithm is presented we compare dgmres algorithm with dfom algorithm by numerical experiment error analysis for projection method is also given

structure of atypical representations of the lie superalgebras sl m n in contrast to lie algebras lie superalgebras contain typical and atypical irreducible representations character formulae for typical representations of classical lie superalgebras have been known for a long time but the structure of atypical representations remained unsolved the author constructs the character formula for atypical representations of the special linear lie superalgebras sl m n

infinite dimensional lie superalgebras and hook schur functions making use of a howe duality involving the infinite dimensional lie superalgebra and the finite dimensional group gll of cw3 we derive a character formula for a certain class of irreducible quasi finite representations of in terms of hook schur functions we use the reduction procedure of to to derive a character formula for a certain class of level 1 highest weight irreducible representations of the affine lie superalgebra associated to the finite dimensional lie superalgebra these modules turn out to form the complete set of integrable modules of level 1 we also show that the characters of all integrable level 1 highest weight irreducible modules may be written as a sum of products of hook schur functions

frequency domain uncertainty and the graph topology a new metric on linear time invariant systems is defined this metric is no greater than the gap metric and is in fact the smallest metric for which a certain robust stabilization result holds unlike other known metrics which induce the graph topology it has a clear frequency response interpretation this allows questions regarding robustness in the face of parametric uncertainty to be considered in terms of this metric

integrable highest weight modules over affine superalgebras and appell s function we classify integrable irreducible highest weight representations of non twisted affine lie superalgebras we give a free field construction in the level 1 case the analysis of this construction shows in particular that in the simplest case of the sl 2 1 level 1 affine superalgebra the characters are expressed in terms of the appell elliptic function our results demonstrate that the representation theory of affine lie superalgebras is quite different from that of affine lie algebras

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

convergence properties of krylov subspace methods for singular linear systems with arbitrary index abstract krylov subspace methods have been recently considered to solve singular linear systems ax b in this paper we derive the necessary and sufficient conditions guaranteeing that a krylov subspace method converges to a vector adb px0 where ad is the drazin inverse of a and p is the projection p i a d a let k be the index of a we further show that a d b px 0 x 0 r a k 1 n a is a generalized least squares solution of ax b in r a k n a finally we present the convergence bounds for the quasi minimal residual algorithm qmr and transpose free quasi minimal residual algorithm tfqmr the index k of a in this paper can be arbitrary which extends to the main results of freund and hochbruck numer linear algebra appl 1 1994 403 420 that only considers the case k 1

representations of n 2 extended supergravity and unitarity conditions in osp n 4 the structure of unitary irreducible representations of the lie superalgebra osp 2 4 the algebra of n 2 extended supergravity is investigated in detail in particular four new classes of shortened multiplets are found and the complete unitarity conditions are given the shortened multiplets are shown to correspond to atypical infinite dimensional representations finally unitarity conditions for osp n 4 3 n 8 are constructed

perturbation analysis of singular linear systems with arbitrary index a new perturbation result for singular but consistent linear systems with coefficient matrix of arbitrary index however if the system is inconsistent we also derive a perturbation bound of the generalized linear least squares solution our treatment extends a classical perturbation result in numerical linear algebra

super duality and irreducible characters of ortho symplectic lie superalgebras we formulate and establish a super duality which connects parabolic categories o for the ortho symplectic lie superalgebras and classical lie algebras of bcd types this provides a complete and conceptual solution of the irreducible character problem for the ortho symplectic lie superalgebras in a parabolic category o which includes all finite dimensional irreducible modules in terms of classical kazhdan lusztig polynomials

regular subalgebras of lie superalgebras and extended dynkin diagrams using the method of extended cartan matrices and extended dynkin diagrams a classification of maximal regular semisimple subalgebras of the basic classical lie superalgebras is obtained especially in the case of exceptional lie superalgebras some curious inclusion relations are discovered

explicit construction of the exceptional superalgebras f 4 and g 3 a general method for constructing the extension of an ordinary lie algebra a0 to a superalgebra a0 a1 is given once one knows in which representation of a0 the odd generators a1 are explicit matrix representations for the superalgebras f 4 and g 3 and for ordinary algebras e8 f4 and g2 are presented

generalised verma modules for the orthosymplectic lie superalgebra ospk 2 abstract the composition factors with their multiplicities and the submodule lattices are determined for generalised verma modules over the orthosymplectic lie superalgebra osp k 2 the results enable us to obtain explicit formulae for the formal characters and dimensions of the finite dimensional irreducible modules applying these results we also compute the first and second cohomology groups of the lie superalgebra with coefficients in finite dimensional kac modules and irreducible modules

principal five dimensional subalgebras of lie superalgebras the analog of sl 2 for lie superalgebras is osp 1 2 a five dimensional superalgebra all basic classical lie superalgebras l that contain a principal five dimensional osp 1 2 subalgebra are classified moreover the decomposition of the standard representation and of the adjoint representation of l into irreducible components of the principal osp 1 2 subalgebra is given

the analysis of restart dgmres for solving singular linear systems abstract a surprising phenomenon concerning with restart dgmres a sidi dgmres a gmres type algorithm for drazin inverse solution of singular nonsymmetric linear systems linear algebra appl 335 2001 189 204 is presented that small values of the restart parameter may converge faster than larger values we take three examples where dgmres 2 converge while dgmres 3 stagnates to interpret the phenomenon two of these examples reveals that dgmres convergence can be extremely sensitive to small changes in the initial residual

irreducible representations of the exceptional lie superalgebras d 2 1 α the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the exceptional lie superalgebrasd 2 1 for all cases the star or grade star conditions for the algebra are investigated among the finite dimensional representations there are no star and only a few grade star representations but an infinite class of infinite dimensional star representations is found explicit expressions are given for the doublet representation of d 2 1 the one missing label problem d 2 1 su 2 su 2 su 2 is discussed in detail and solved explicitly

complete stagnation of gmres we study problems for which the iterative method gmres for solving linear systems of equations makes no progress in its initial iterations our tool for analysis is a nonlinear system of equations the stagnation system that characterizes this behavior we focus on complete stagnation for which there is no progress until the last iteration we give necessary and sufficient conditions for complete stagnation of systems involving unitary matrices and show that if a normal matrix completely stagnates then so does an entire family of nonnormal matrices with the same eigenvalues finally we show that there are real matrices for which complete stagnation occurs for certain complex right hand sides but not for real ones 2003 elsevier science inc all rights reserved

classification of generalized quantum statistics associated with the exceptional lie super algebras generalized quantum statistics gqs associated with a lie algebra or lie superalgebra extends the notion of para bose or para fermi statistics such gqs have been classified for all classical simple lie algebras and basic classical lie superalgebras in the current paper we finalize this classification for all exceptional lie algebras and superalgebras since the definition of gqs is closely related to a certain z grading of the lie super algebra g our classification reproduces some known z gradings of exceptional lie algebras for exceptional lie superalgebras such a classification of z gradings has not been given before

bernstein gelfand gelfand resolutions for basic classical lie superalgebras abstract we study kostant cohomology and bernstein gelfand gelfand resolutions for finite dimensional representations of basic classical lie superalgebras for each choice of parabolic subalgebra and representation of such a lie superalgebra there is a natural definition of the boundary and coboundary operators which define co homology of the nilradical of the parabolic subalgebra we prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of generalised verma modules every such a resolution is then given by modules induced by these homology groups we also prove that if these homology groups are completely reducible a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the kostant quabla operator which is equivalent with disjointness of the boundary and coboundary operators then we use these results to derive very explicit conditions under which bgg resolutions exist which are particularly useful for the superalgebras of type i for the unitarisable representations of gl m n and osp 2 2 n we derive conditions on the parabolic subalgebra under which the bgg resolutions exist we also apply the obtained theory to construct specific examples of bgg resolutions for osp m 2 n finally we state some results for typical modules of gl m n and osp 2 2 n

quasifinite representations of classical lie subalgebras of w1 we show that there are precisely two up to conjugation anti involutions of the algebra of differential operators on the circle preserving the principal gradation we classify the irreducible quasifinite highest weight representations of the central extension d of the lie subalgebra of this algebra fixed by and find the unitary ones we realize them in terms of highest weight representations of the central extension of the lie algebra of infinite matrices with finitely many non zero diagonals over the algebra c u um 1 and its classical lie subalgebras ofb canddtypes character formulas forpositive primitiverepresentations of d including all the unitary ones are obtained we also realize a class of primitive representations of d in terms of free fields and establish a number of duality results between these primitive representations and finite dimensional irreducible representations of finite dimensional lie groups and supergroups we show that the vacuum modulevcof d carries a vertex algebra structure and establish a relationship betweenvcforc 12z and w algebras

character formulae for ortho symplectic lie superalgebras osp n 2 abstract the characters of finite dimensional irreducible modules of the lie superalgebra osp n 2 are completely determined in terms of those of generalized verma modules as a by product the irreducible decomposition of the tensor of any finite dimensional irreducible module with the natural module is obtained

hilbert series howe duality and branching for classical groups an extension of the littlewood restriction rule is given that covers all pertinent parameters and simplifies to the original under littlewood s hypotheses two formulas are derived for the gelfand kirillov dimension of any unitary highest weight representation occurring in a dual pair setting one in terms of the dual pair index and the other in terms of the highest weight for a fixed dual pair setting all the irreducible highest weight representations which occur have the same gelfand kirillov dimension we define a class of unitary highest weight representations and show that each of these representations l has a hilbert series hl q of the form

survey paper from youla kucera to identification adaptive and nonlinear control some 20 years ago formulae were presented for the set of all linear time invariant controllers stabilizing a linear time invariant plant this paper traces the development of many ideas from these formulae covering linear h 2 and h control identification adaptive control and nonlinear systems

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets