on additive transformations preserving a multiplicative matrix function abstract let m n k be the ring of all n n matrices over a division ring k and f be a multiplicative matrix function from m n k to a multiplicative abelian group with zero g 0 f ab f a f b a b m n k we call an additive transformation on m n k preserves a multiplicative matrix function f if f a f a a m n k in this paper we characterize all additive surjective transformations on m n k over any division ring k chk 2 that leave a non trivial multiplicative matrix function invariant applications to several related preservers are considered

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

method of quantum characters in equivariant quantization let g be a reductive lie group g its lie algebra and m a g manifold suppose h m is a h g equivariant quantization of the function algebra m on m we develop a method of building h g equivariant quantization on g orbits in m as quotients of h m we are concerned with those quantizations that may be simultaneously represented as subalgebras in h g and quotients of h m it turns out that they are in one to one correspondence with characters of the algebra h m we specialize our approach to the situation g gl n m end n and h m the so called reflection equation algebra associated with the representation of h g on n for this particular case we present in an explicit form all possible quantizations of this type they cover symmetric and bisymmetric orbits we build a two parameter deformation family and obtain as a limit case the g equivariant quantization of the kirillov kostant souriau bracket on symmetric orbits

nonlinear maps preserving numerical radius of indefinite skew products of operators let h be a complex hilbert space of dimension greater than 2 and j b h be an invertible self adjoint operator denote by a j 1a j the indefinite conjugate of a b h with respect to j and denote by w a the numerical radius of a let w and v be subsets of b h which contain all rank one operators and let w v be a surjective map we show that satisfies w ab w a b and w a b w a b for all a b w if and only if there exist scalars i 1 1 i 1 2 unitary or conjugate unitary operators u v on h satisfying u u 1i v v 2i and a functional w c with a 1 such that a a uav for all a w satisfies w ab a w a b a for all a b w if and only if either there exist 1 1 a unitary or conjugate unitary operator u on h satisfying u u i and a functional w c with a 1 such that a a uau for all a w or there exist a nonzero real number b a unitary or conjugate unitary operator u on h satisfying u ju bj 1 and a functional w c with a 1 such that a a ua u for all a w

maps preserving the local spectrum of jordan product of matrices abstract let m n c be the algebra of all n n complex matrices and fix a nonzero vector x 0 c n we show that a map from m n c into itself satisfies t s s t x 0 t s s t x 0 t s m n c if and only if there exists an invertible matrix a m n c such that a x 0 x 0 and t a t a 1 for all t m n c

ring isomorphisms and linear or additive maps preserving zero products on nest algebras it is shown that every ring isomorphism between norm closed standard subalgebras of nest algebras acting on infinite dimensional banach spaces is either a spatial isomorphism or a spatial conjugate isomorphism this is then used to characterize the linear or additive maps between nest algebras which preserve zero products

preservers of spectral radius numerical radius or spectral norm of the sum on nonnegative matrices abstract let m n be the set of entrywise nonnegative n n matrices denote by r a the spectral radius perron root of a m n characterization is obtained for maps f m n m n such that r f a f b r a b for all a b m n in particular it is shown that such a map has the form a s 1 as or a s 1 a tr s for some s m n with exactly one positive entry in each row and each column moreover the same conclusion holds if the spectral radius is replaced by the spectrum or the peripheral spectrum similar results are obtained for maps on the set of n n nonnegative symmetric matrices furthermore the proofs are extended to obtain analogous results when spectral radius is replaced by the numerical range or the spectral norm in the case of the numerical radius a full description of preservers of the sum is also obtained but in this case it turns out that the standard forms do not describe all such preservers

de rham complex for quantized irreducible flag manifolds it is shown that quantized irreducible flag manifolds possess a canonical q analogue of the de rham complex generalizing the well known situation for the standard podle quantum sphere this analogue is obtained as the universal differential calculus of a distinguished first order differential calculus the corresponding differential d can be written as a sum of differentials and the universal differential calculus corresponding to the first order differential calculi d and are given in terms of generators and relations relations to well known quantized exterior algebras are established the dimensions of the homogeneous components are shown to be the same as in the classical case the existence of a volume form is proven

induced representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n we construct the induced holomorphic representations of the multiparameter hopf superalgebras uuq gl m n and uuq sl m n the first superalgebra we constructed earlier as the dual of the multiparameter quantum deformation of the supergroup gl m n the second superalgebra is a hopf subalgebra of the first for a special choice of the parameters the representations are labelled by m n integer numbers respectively m n 1 complex numbers and act in the space of formal power series of m n m n 1 2 non commuting variables of which mn are odd and the rest are even these variables generate a q deformation of a flag supermanifold of the supergroup gl m n respectively sl m n

quantization of borderline levi conjugacy classes of orthogonal groups we construct an equivariant quantization of a special family of levi conjugacy classes of the complex orthogonal group so n whose stabilizer contains a cartesian factor so 2 so p 1 p n p n mod 2

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

on classical adjoint commuting mappings between matrix algebras abstract let f be a field and let m and n be integers with m n 3 let m n denote the algebra of n n matrices over f in this note we characterize mappings m n m m that satisfy one of the following conditions 1 f 2 or f n 1 and adj a b adj a b for all a b m n and f with i n 0 2 is surjective and adj a b adj a b for every a b m n here adj a denotes the classical adjoint of the matrix a and i n is the identity matrix of order n we give examples showing the indispensability of the assumption i n 0 in our results

unitary similarity invariant function preservers of skew products of operators abstract let b h denote the banach algebra of all bounded linear operators on a complex hilbert space h with dim h 3 and let a and b be subsets of b h which contain all rank one operators suppose f is a unitary invariant norm the pseudo spectra the pseudo spectral radius the c numerical range or the c numerical radius for some finite rank operator c the structure is determined for surjective maps a b satisfying f a b f a b for all a b a to establish the proofs some general results are obtained for functions f f 1 h 0 0 where f 1 h is the set of rank one operators in b h satisfying a f u a u f a for a complex unit a f 1 h and unitary u b h b for any rank one operator x f 1 h the map t f t x on 0 is strictly increasing and c the set f x x f 1 h and x 1 attains its maximum and minimum

equivalence classes of vogan diagrams abstract a vogan diagram is a dynkin diagram with an involution and the vertices fixed by the involution may be painted they represent real simple lie algebras and two diagrams are said to be equivalent if they represent the same lie algebra in this article we classify the equivalence classes of all vogan diagrams in doing so we find that the underlying dynkin diagrams have certain properties in graph painting we show that this combinatorial property provides an easy classification for most of the simply laced dynkin diagrams

schur product of matrices and numerical radius range preserving maps abstract let f a be the numerical range or the numerical radius of a square matrix a denote by a b the schur product of two matrices a and b characterizations are given for mappings on square matrices satisfying f a b f a b for all matrices a and b analogous results are obtained for mappings on hermitian matrices

the locally finite part of the dual coalgebra of quantized irreducible flag manifolds for quantized irreducible flag manifolds the locally finite part of the dual coalgebra is shown to coincide with a natural quotient coalgebra overline u of u q mathfrak g on the way the coradical filtration of overline u is determined a graded version of the duality between overline u and the quantized coordinate ring is established this leads to a natural construction of several examples of quantized vector spaces as an application covariant first order differential calculi on quantized irreducible flag manifolds are classified

quantum group covariant anti symmetrizers tensors vielbein hodge map and laplacian glq n and soq n covariant deformations of the completely symmetric antisymmetric projectors with an arbitrary number of indices are explicitly constructed as polynomials in the braid matrices the precise relation between the completely antisymmetric projectors and the completely antisymmetric tensor is determined adopting the glq n and soq n covariant differential calculi on the corresponding quantum group covariant noncommutative spaces we introduce a generalized notion of vielbein basis or frame based on differential operator valued 1 forms we then give a thorough definition of a soq n covariant bilinear hodge map acting on the bimodule of differential forms on introduce the exterior coderivative and show that the laplacian acts on differential forms exactly as in the undeformed case namely it acts on each component as it does on functions

dynamical yang baxter equation and quantum vector bundles we develop a categorical approach to the dynamical yang baxter equation dybe for arbitrary hopf algebras in particular we introduce the notion of a dynamical extension of a monoidal category which provides a natural environment for quantum dynamical r matrices dynamical twists etc in this context we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits we build a dynamical twist for any pair of a reductive lie algebra and its levi subalgebra using this twist we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive lie groups

nonlinear invertibility preserving maps on matrix algebras abstract let m n denote the algebra of all n n matrices over the complex field c we prove that if the maps m n m n have the property that for every x y m n and c we have that x y m n is invertible if and only if x y m n is invertible and one of the maps or is either surjective or continuous there exist then invertible matrices u v m n such that either x x u x v x m n or x x u x t v x m n

on idempotent states on quantum groups idempotent states on a compact quantum group are shown to yield group like projections in the multiplier algebra of the dual discrete quantum group this allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the haar state on a finite quantum hypergroup a natural order structure on the set of idempotent states is also studied and some examples discussed

classical adjoint commuting mappings on hermitian and symmetric matrices let m and n be integers such that m n 3 and let f and k be fields which possess involutions of f and of k respectively let hnf be the k linear space n n hermitian matrices over f in this note we address the general description of mappings satisfying one of the following conditions 1 hn f hm k with either k 2 or f k 3 and f and k of characteristic 2 if and are the identity maps and is surjective satisfying adj a b adj a b for every a b hn f 2 hn f hm f with either f 2 or f n 1 and adj a b adj a b for every a b hn k and f 3 hn f hm k is additive with adj a adja for every a hn f here f a f a a and k a k a a are the fixed fields with respect to the involutions of f and of k respectively and adja denotes the classical adjoint of the matrix a

on determinant preserver problems abstract let m n and t n be the vector spaces of n n matrices and upper triangular matrices over a field f with some cardinality and characteristic restrictions respectively we characterise transformations on these two spaces separately which satisfy one of the following conditions 1 det a b det a b for all a b and 2 is surjective and det a b det a b for all a b and two specific 3 is additive and preserves determinant

a poisson structure on compact symmetric spaces we present some basic results on a natural poisson structure on any compact symmetric space the symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold

a formal inverse to the cayley hamilton theorem abstract we prove that a q algebra r with formal trace can be realized as n n matrices if and only if it satisfies the cayley hamilton identity of degree n

asymptotic similarity preserving additive maps on b x abstract let x be an infinite dimensional complex banach space and denote b x the algebra of all bounded linear operators acting on x we show that an additive surjective map on b x preserves asymptotic similarity in both directions if and only if there exist a nonzero scalar c an invertible bounded linear or conjugate linear operator a and an asymptotic similarity invariant additive functional on b x such that either t cata 1 t i for all t or t cat a 1 t i for all t in the case that x has infinite multiplicity especially if x is an infinite dimensional hilbert space above asymptotic similarity invariant additive functional is always zero

linear maps transforming the higher numerical ranges abstract let k 1 n the k numerical range of a m n is the set w k a tr x ax k x is n k x x i k and the k numerical radius of a is the quantity w k a max z z w k a suppose k 1 k 1 n and n c n k min k n k it is shown that there is a linear map m n m n satisfying w k a w k a for all a m n if and only if n n k k or n n k n k is a positive integer moreover if such a linear map exists then there are unitary matrix u m n and nonnegative integers p q with p q n n such that has the form a u a a p a t a t q u or a u a a p a t a t q u where m n m n has the form a tr a i n n k a k linear maps m n m n satisfying w k a w k a for all a m n are also studied furthermore results are extended to triangular matrices

maps on matrix algebras preserving idempotents abstract let m n be the algebra of all n n complex matrices and p n the set of all idempotents in m n suppose m n m n is a surjective map satisfying a b p n if and only if a b p n a b m n c then either is of the form a tat 1 a m n or is of the form a ta t t 1 a m n where t m n is a nonsingular matrix

generalized gel fand invariants and characteristic identities for quantum groups the generalized gel fand invariants for an arbitrary quantum group are explicitly constructed and their eigenvalues in any irreducible representation are computed these invariants enable one to develop characteristic identities for the quantum group and as a natural application these identities are used to construct projection operators for tensor products of representations to illustrate the general theory the quantum group uq gl m is studied in detail

equivariant quantization of poisson homogeneous spaces and kostant s problem we find a partial solution to the longstanding problem of kostant concerning description of the so called locally finite endomorphisms of highest weight irreducible modules the solution is obtained by means of its reduction to a far reaching extension of the quantization problem while the classical quantization problem consists in finding product deformations of the commutative algebras of functions we consider the case when the initial object is already a noncommutative algebra the algebra of functions within q calculus

non linear numerical radius isometries on atomic nest algebras and diagonal algebras abstract a nonlinear map between operator algebras is said to be a numerical radius isometry if w t s w t s for all t s in its domain algebra where w t stands for the numerical radius of t let n and m be two atomic nests on complex hilbert spaces h and k respectively denote alg n the nest algebra associated with n and d n alg n alg n the diagonal algebra we give a thorough classification of weakly continuous numerical radius isometries from alg n onto alg m and a thorough classification of numerical radius isometries from d n onto d m

embeddable quantum homogeneous spaces abstract we discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups on the von neumann algebra level we recall an interesting duality for such objects studied earlier by m izumi r longo s popa for compact kac algebras and by m enock in the general case of locally compact quantum groups a definition of a quantum homogeneous space is proposed along the lines of the pioneering work of vaes on induction and imprimitivity for locally compact quantum groups the concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces among various examples we single out the quantum analog of the quotient of the cartesian product of a quantum group with itself by the diagonal subgroup analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles the former turns out to be an interesting application of the duality mentioned above

explicit quantization of dynamical r matrices for finite dimensional semisimple lie algebras 1 1 classical r matrices in the early eighties belavin and drinfeld bd classified nonskewsymmetric classical r matrices for simple lie algebras it turned out that such r matrices up to isomorphism and twisting by elements from the exterior square of the cartan subalgebra are classified by combinatorial objects which are now called belavin drinfeld triples by definition a belavin drinfeld triple for a simple lie algebra g is a triple 1 2 t where 1 2 are subsets of the dynkin diagram of g and t 1 2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition if 1 then there exists k such that t k 1 1 but t 1 the r matrix corresponding to such a triple is given by a certain explicit formula these results generalize in a straightforward way to semisimple lie algebras in s the third author generalized the work of belavin and drinfeld and classified classical nonskewsymmetric dynamical r matrices for simple lie algebras it turns out that they have an even simpler classification up to gauge transformations they are classified by generalized belavin drinfeld triples which are defined as the usual belavin drinfeld triples but without any nilpotency condition the dynamical rmatrix corresponding to such a triple is given by a certain explicit formula as before these results can be generalized to semisimple lie algebras

on the solvability of a class of second kind integral equations on unbounded domains we consider integral equations of the form x x k x y z y y dy in operator form kz where is some subset ofrn n 1 the functionsk z and are assumed known withz l and y the space of bounded continuous functions on the function yis to be determined the class of domains and kernelskconsidered includes the case rnandk x y x y with l1 rn in which case ifzis the characteristic function of some setg the integral equation is one of wiener hopf type the main theorems proved using arguments derived from collectively compact operator theory are conditions on a setw l which ensure that ifi kzis injective for allz wtheni kzis also surjective and moreover the inverse operators i kz 1onyare bounded uniformly inz these general theorems are used to recover classical results on wiener hopf integral operators of21and19 and generalisations of these results and are applied to analyse the lippmann schwinger integral equation

determinant preserving maps on matrix algebras abstract let m n be the algebra of all n n complex matrices if m n m n is a surjective mapping satisfying det a b det a b a b m n c then either is of the form a man a m n or is of the form a ma t n a m n where m n m n are nonsingular matrices with det mn 1

quantum zonal spherical functions and macdonald polynomials abstract a unified theory of quantum symmetric pairs is applied to q special functions previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions here a distinguished family of such functions invariant under the weyl group associated to the restricted roots is shown to be a family of macdonald polynomials as conjectured by koornwinder and macdonald our results place earlier work for lie algebras of classical type in a general context and extend to the exceptional cases

symplectic structures associated to lie poisson groups the lie poisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered for the natural poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described thus the construction of the kirillov symplectic form is generalized for lie poisson groups

numerical solutions of integral equations on the half line numerical approximation schemes of quadrature type are investigated for integral equations of the form x s int limits 0 infty k s t x t dt y s in this paper the hypotheses will ensure that the integral operator is compact in an appropriate banach space a sequel will deal with wiener hopf integral equations

schur multiplicative maps preserving unitarily invariant norms characterizations are obtained for maps on real or complex matrices which preserve both the schur hadamard product and a given unitarily invariant norm

unital and multiplicatively spectrum preserving surjections between semi simple commutative banach algebras are linear and multiplicative abstract let t be a surjective map from a unital semi simple commutative banach algebra a onto a unital commutative banach algebra b suppose that t preserves the unit element and the spectrum f g of the product of any two elements f and g in a coincides with the spectrum t f t g then b is semi simple and t is an isomorphism the condition that t is surjective is essential an example of a non linear and non multiplicative unital map from a commutative c algebra into itself such that t f t g f g holds for every f g are given we also show an example of a surjective unital map from a commutative c algebra onto itself which is neither linear nor multiplicative such that t f t g f g holds for every f g

quantum conjugacy classes of simple matrix groups let g be a simple complex classical group and mathfrak g its lie algebra let mathcal u hbar mathfrak g be the drinfeld jimbo quantization of the universal enveloping algebra mathcal u mathfrak g we construct an explicit mathcal u hbar mathfrak g equivariant quantization of conjugacy classes of g with levi subgroups as the stabilizers

quantized flag manifolds and irreducible representations we study irreducible representations of a certain quantization of the algebra of polynomial functions on a generalized flag manifold regarded as a real manifold all irreducible representations are classified for a subclass of flag manifolds containing in particular the irreducible compact hermitian symmetric spaces for this subclass it is shown that the irreducible representations are parametrized by the symplectic leaves of the underlying poisson bracket we also discuss the relation between the quantized flag manifolds studied in this paper and the quantum flag manifolds studied by soibel man lakshimibai reshetikhin juraco s ovicek and korogodsky

braided momentum in the q poincare group the q poincare group of m schlieker et al z phys c 53 79 1992 is shown to have the structure of a semidirect product and coproduct b soq 1 3 where b is a braided quantum group structure on the q minkowski space of four momentum with braided coproduct p p 1 1 p here the necessary b is not a usual kind of quantum group but one with braid statistics similar braided vectors and covectors v r v r exist for a general r matrix the abstract structure of the q lorentz group is also studied

symmetric pairs for quantized enveloping algebras abstract let be an involution of a semisimple lie algebra g let g denote the fixed lie subalgebra and assume the cartan subalgebra of g has been chosen in a suitable way we construct a quantum analog of u g which can be characterized as the unique subalgebra of the quantized enveloping algebra of g which is a maximal right coideal that specializes to u g

non levi closed conjugacy classes of sp q 2 n we construct an explicit quantization of semi simple conjugacy classes of the complex symplectic group sp 2n with non levi isotropy subgroups through an operator realization on highest weight modules over the quantum group u q bigl mathfrak sp 2n bigr

maps leaving functional values of operator products invariant abstract let a be a standard operator algebra on a complex hilbert space h of dimension greater than 2 by invariants of certain functional values of operator products we characterize some surjective maps on a furthermore several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as for example k numerical radius k 1 operator norm ky fan k norm schatten p norm 1 p and so on

additive preservers of numerical range abstract we characterize all additive mappings preserving classical numerical range on various subalgebras of m n the subalgebra of all diagonal matrices d n the subalgebra of all upper triangular matrices t n and subalgebras of block diagonal and block triangular matrices characterizations obtained show that in some cases m n for example all such preservers are linear on the other hand we show that the group of all additive preservers on t n contains some additional mappings

on transformation partly preserving a nonsingular bilinear function abstract let f be an arbitrary field and let vn f be an n dimensional vector space over f we say that a transformation on vn f partly preserves a nonsingular bilinear function f vn f vn f f if there exists a basis e1 e2 en for vn f and a map e1 e2 en vn f such that f ej f ej vn f j 1 2 n let mm n f be the vector space of m n matrices and let mn f be the vector space of n n matrices over f we prove that a transformation on vn f is an invertible linear transformation if and only if partly preserves a nonsingular bilinear function then we characterize transformations on mn f which satisfy det a det a and tr a b tr ab finally we characterize the transformation groups wm n f a paq a mm n f where p glm f q gln f ppt em qqt en on vector space mm n f by as few invariants as possible

quantized algebras of functions on homogeneous spaces with poisson stabilizers let g be a simply connected semisimple compact lie group with standard poisson structure k a closed poisson lie subgroup 0 q 1 we study a quantization c gq kq of the algebra of continuous functions on g k using results of soibelman and dijkhuizen stokman we classify the irreducible representations of c gq kq and obtain a composition series for c gq kq we describe closures of the symplectic leaves of g k refining the well known description in the case of flag manifolds in terms of the bruhat order we then show that the same rules describe the topology on the spectrum of c gq kq next we show that the family of c algebras c gq kq 0 q 1 has a canonical structure of a continuous field of c algebras and provides a strict deformation quantization of the poisson algebra mathbb c g k finally extending a result of nagy we show that c gq kq is canonically kk equivalent to c g k

approximate solution of second kind integral equations on infinite cylindrical surfaces the paper considers second kind integral equations of the form x g x fs k x y 0 y ds y abbreviated 0 g ko in which s is an infinite cylindrical surface of arbitrary smooth cross section the truncated equation abbreviated 4 a eag ka4 a obtained by replacing s by sa a closed bounded surface of class c2 the boundary of a section of the interior of s of length 2a is also discussed conditions on k are obtained in particular implying that k commutes with the operation of translation in the direction of the cylinder axis which ensure that i k is invertible that i ka is invertible and i ka is uniformly bounded for all sufficiently large a and that ka converges to k in an appropriate sense as a oo uniform stability and convergence results for a piecewise constant boundary element collocation method for the truncated equations are also obtained a boundary integral equation which models three dimensional acoustic scattering from an infinite rigid cylinder illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem and convergence of a particular collocation method

non levi closed conjugacy classes of soq n we construct explicit quantization of semisimple conjugacy classes of the complex orthogonal group so n with non levi isotropy subgroups through an operator realization on highest weight modules of the quantum group uqso n

multiplicative preservers on semigroups of matrices we characterize multiplicative maps on semigroups of square matrices satisfying p p for matrix sets p such as rank k idempotent matrices totally nonnegative matrices p0 matrices m0 matrices positive semidefinite matrices hermitian matrices normal matrices and contractions we also characterize multiplicative maps satisfying g x x for various functions g on square matrices such as the spectrum spectral radius numerical range numerical radius and matrix norms

a criterion for an ideal to be induced 1 1 let g be a complex semisimple lie algebra in the study of the primitive spectrum prim u g of the enveloping algebra u g of g an important question which remains open is to determine those primitive ideals which are completely prime one such family is provided by those ideals which are annihilators of modules induced from a one dimensional module of a parabolic subalgebra p for the purposes of this article we call such ideals induced with respect to p

nonlinear determinant preserving maps on matrix algebras abstract let f be a field having at least n 2 1 distinct elements and denote by m n the space of all n n matrices over f we prove that if and are maps from m n into itself one of them being surjective such that det x y det x y for all x y m n there exist then x 0 m n and u v m n satisfying det u v 1 such that either x u x x 0 v and x u x x 0 v for all x m n or x u x x 0 t v and x u x x 0 t v for all x m n

c numerical radius isometries on matrix algebras and triangular matrix algebras abstract let c c 1 c n t r n and m n be the set of n n complex matrices for any a m n define the c numerical range and the c numerical radius of a by w c a i 1 n c i a x i x i x 1 x n is an orthonormal set in c n and w c a max z z w c a respectively let t n be the set of n n upper triangular matrices when w c is a norm on m n mappings t on m n or t n satisfying w c t a t b w c a b for all a b are characterized as an intermediate step we also characterize additive c numerical range preservers on m n or t n

quantum flag manifolds quantum symmetric spaces and their associated universal k matrices abstract let u be a connected simply connected compact lie group with complexification g let u and g be the associated lie algebras let be the dynkin diagram of g with underlying set i and let u q u be the associated quantized universal enveloping algebra of u for some 0 q distinct from 1 let o q u be the coquasitriangular quantized function hopf algebra of u whose drinfeld double o q g r we view as the quantized function algebra of g considered as a real algebraic group we show how the datum of an involution of and a invariant function i r can be used to deform o q g r into a algebra o q id g r by a modification of the drinfeld double construction we then show how by a generalized theory of universal k matrices a specific subalgebra of o q id g r admits homomorphisms into both u q u and o q u the images being coideal subalgebras of respectively u q u and o q u we illustrate the theory by showing that two main classes of examples arise by such coideals namely quantum flag manifolds and quantum symmetric spaces except possibly for certain exceptional cases in the former case this connects to work of the first author and neshveyev while for the latter case we heavily rely on recent results of balagovic and kolb

permanent polya problem for additive surjective maps abstract let m n f denote the set of square matrices of size n over a field f with characteristics different from two we say that the map f m n f m n f is additive if f a b f a f b for all a b m n f the main goal of this paper is to prove that for n 2 there are no additive surjective maps t m n f m n f such that per t a det a for all a m n f also we show that an arbitrary additive surjective map t m n f m n f which preserves permanent is linear and thus can be completely characterized

examples of braided groups and braided matrices matrix braided groups are developed as an analog of the coordinate functions on a group or supergroup the 1 in the super case is replaced by braid statistics there are braided group analogs of all the classical simple lie groups as well as braided matrix groups and braided matrices b r for every regular solution r of the quantum yang baxter equations a direct verification of b r is provided and some of the simplest examples are computed in detail

on a class of determinant preserving maps for finite von neumann algebras abstract let r be a finite von neumann algebra with a faithful tracial state and let denote the associated fuglede kadison determinant in this paper we characterize all unital bijective maps on the set of invertible positive elements in r which satisfy a b a b we show that any such map originates from a preserving jordan automorphism of r either automorphism or anti automorphism in the more restrictive case of finite factors in establishing the aforementioned result we make crucial use of the solutions to the equation a b a b in the set of invertible positive operators in r to this end we give a new proof of the inequality a b a b using a generalized version of the hadamard determinant inequality and conclude that equality holds for invertible b if and only if a is a nonnegative scalar multiple of b

immanant preserving and immanant converting maps we generalize a result in g dolinar p semrl determinant preserving maps on matrix algebras linear algebra appl 348 2002 189 192 proving that if and are arbitrary irreducible complex characters of sn and t mn c mn c is a surjective map satisfying d t a t b d a b for all a b mn c and all c then t is linear our main theorem combined with known results on linear preservers converters of immanants allows us to characterize the pairs for which such maps exist and in the cases they exist to obtain the respective description

the center of the reflection equation algebra via quantum minors abstract we give simple formulas for the elements c k appearing in a quantum cayley hamilton formula for the reflection equation algebra re algebra associated to the quantum group u q gl n answering a question of kolb and stokman the c k s are certain canonical generators of the center of the re algebra and hence of u q gl n itself they have been described by reshetikhin using graphical calculus by nazarov tarasov using quantum yangians and by gurevich pyatov and saponov using quantum schur functions however no explicit formulas for these elements were previously known as byproducts we prove a quantum girard newton identity relating the c k s to the so called quantum power traces and we give a new presentation for the quantum group u q gl n as a localization of the re algebra along certain principal minors

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

maps preserving numerical radius and cross norms of operator products let h be a complex hilbert space with dim h 3 and x1d4b2 x1d4b1 h be subsets that contain all rank one idempotents let f stand for numerical radius w or a cross operator norm n all the general surjective maps x1d4b2 x1d4b1 satisfying f ab f a b resp f aba f a b a for all a b x1d4b2 are characterized

algebraic groups with a commuting pair of involutions and semisimple symmetric spaces let g be a connected reductive algebraic group defined over an algebraically closed field f of characteristic not 2 denote the lie algebra of g by 9 in this paper we shall classify the isomorphism classes of ordered pairs of commuting involutorial automorphisms of g this is shown to be independent of the characteristic of f and can be applied to describe all semisimple locally symmetric spaces together with their line structure involutorial automorphisms of g occur in several places in the literature cartan has already shown that for f c the isomorphism classes of involutorial automorphisms of g correspond bijectively to the isomorphism classes of real semisimple lie algebras which correspond in their turn to the isomorphism classes of riemannian symmetric spaces see helgason 111 if one lifts this involution to the group g then the present work gives a characteristic free description of these isomorphism classes in a similar manner we can show that semisimple locally symmetric spaces correspond to pairs of commuting involutorial automorphisms of g namely let go a be a semisimple locally symmetric pair i e go is a real semisimple lie algebra and rr e aut g an involution then by a result of berger 2 there exists a cartan involution 8 of go such that 00 ea if we denote the complexilication of go by g then o and 8 induce a pair of commuting involutions of g conversely if c 8 e aut g are commuting involutions then c and 8 determine two locally semisimple symmetric pairs for if u is a o and o stable compact real form with conjugation r then ger 1 ge and g f3 i ger are semisimple locally symmetric pairs where

maps on b h preserving involution abstract let b h be the algebra of all bounded linear operators on a complex hilbert space h we use the form of idempotence preserving maps to claracterize the form of involutivity preserving maps

the multilinear algebra of jose dias da silva and the portuguese school of mathematics abstract in portugal multilinear algebra has a strong group of researchers that achieved in the last years some impressive results in this area this paper surveys the results obtained by this group and the current state of knowledge on the different problems

algebraic structures related to reflection equations quadratic algebras related to the reflection equations are introduced they are quantum group comodule algebras the quantum group fq gl 2 is taken as the example the properties of the algebras centre representation realizations real forms fusion procedure etc as well as the generalizations are discussed

q difference intertwining operators for uq sl n general setting and the case n 3 we construct representations pi r of the quantum algebra uq sl n labelled by n 1 complex numbers ri and acting in the space of formal power series of n n 1 2 non commuting variables these variables generate a flag manifold of the matrix quantum group slq n which is dual to uq sl n the conditions for reducibility of pi r and the procedure for the construction of the q difference intertwining operators are given the representations and q difference intertwining operators are given in the most explicit form for n 3

noncommutative kahler structures on quantum homogeneous spaces abstract building on the theory of noncommutative complex structures the notion of a noncommutative kahler structure is introduced in the quantum homogeneous space case many of the fundamental results of classical kahler geometry are shown to follow from the existence of such a structure allowing for the definition of noncommutative lefschetz hodge kahler dirac and laplace operators quantum projective space endowed with its heckenberger kolb calculus is taken as the motivating example the general theory is then used to show that the calculus has cohomology groups of at least classical dimension

quantization of classical dynamical r matrices with nonabelian base we construct some classes of dynamical r matrices over a nonabelian base and quantize some of them by constructing dynamical pseudo twists in the sense of xu this way we obtain quantizations of r matrices obtained in earlier work of the second author with schiffmann and varchenko a part of our construction may be viewed as a generalization of the donin mudrov nonabelian fusion construction we apply these results to the construction of equivariant star products on poisson homogeneous spaces which include some homogeneous spaces introduced by de concini