fake lie groups with maximal tori iv this is a continuation of n s 1 2 3 which wc refer to as i ii and iii in this series of papers fake lie groups and adic fake lie groups are studied we will recall some of the basic definitions further notations and terminology may be found in i ii and li1 an adic fake lie group of type g g a compact connected lie group is a finite loop space x such that the classifying space is in the adic genus of bg i e bg and bx are equivalent after localisation at or any p completion x is a fake lie group of type g if x is in the genus of bg i e x and bg are p local equivalent for every prime genusoa bg denotes the adic genus and genus bg the genus of bg a maximal torus for a adic fake lie group x of type g is a map j r btx bx tx a torus such that the homotopy fiber o f f r has the homotopy type of a finite cw complex and such that rank tx rank x rank g the weyl group wx is defined to be the set of homotopy classes of homotopy equivalences q btx btx such that ft 7 q 7 in i 3 7 3 6 it is proved that wx wg as abstract groups one of the main goals was and is to determine all fake lie groups allowing a maximal torus in ii and iii this is done for simply connected lie groups and unitary groups here we will extend these results to the general case

proof of wahl s conjecture on surjectivity of the gaussian map for flag varieties introduction let x be a smooth projective variety over c with ample line bundles t and 2 on x we can form their external tensor product to get a line bundle t fxi 2 on x x x let 5d be the ideal sheaf of the diagonal d c x x x wahl defines a map which he calls as the gaussian map i21 2 h x x x d 0 1 2 h x lc 4 0 l c 0 2 which is induced from the canonical projection id d d by identifying the cxxxkkd y od module d k supported in d with the sheaf of 1 forms ql on d x wahl has extensively studied this map in w1 w2 w3 and w4 and he made the following conjecture in w41

borel de siebenthal pairs global weyl modules and stanley reisner rings we develop the theory of integrable representations for an arbitrary standard maximal parabolic subalgebra of an affine lie algebra we see that such subalgebras can be thought of as arising in a natural way from a borel de siebenthal pair of semisimple lie algebras we see that although there are similarities with the representation theory of the standard maximal parabolic subalgebra there are also very interesting and non trivial differences including the fact that there are examples of non trivial global weyl modules which are irreducible and finite dimensional we also give a presentation of the endomorphism ring of the global weyl module although these are no longer polynomial algebras we see that for certain parabolics these algebras are stanley reisner rings which are both koszul and cohen macaulay

designing the optimal bit balancing energetic cost speed and reliability we consider the challenge of operating a reliable bit that can be rapidly erased we find that both erasing and reliability times are non monotonic in the underlying friction leading to a trade of

higher homotopy commutativity of h spaces and the permuto associahedra in this paper we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an an space to give the definition we use polyhedra called the permuto associahedra which are constructed by kapranov we also show that if a connected ap space has the finitely generated mod p cohomology for a prime p and the multiplication of it is homotopy commutative of the p th order then it has the mod p homotopy type of a finite product of eilenberg mac lane spaces k z 1 s k z 2 s and r z p i 1 s for i 1

optimal cycles for low dissipation heat engines we consider the optimization of a finite time carnot engine characterized by small dissipations we bound the power with a simple inequality and show that the optimal strategy is to perform small cycles around a given working point which can be thus chosen optimally remarkably this optimal point is independent of the figure of merit combining power and efficiency that is being maximized furthermore for a general class of dissipative dynamics the maximal power output becomes proportional to the heat capacity of the working substance since the heat capacity can scale supraextensively with the number of constituents of the engine this enables us to design optimal many body carnot engines reaching maximum efficiency at finite power per constituent in the thermodynamic limit

deconstructing hopf spaces we characterize hopf spaces with finitely generated cohomology as an algebra over the steenrod algebra we deconstruct the original space into an h space y with finite mod p cohomology and a finite number of p torsion eilenberg mac lane spaces we give a precise description of homotopy commutative h spaces in this setting

endomorphisms of the cohomology of complex grassmannians for any complex grassmann manifold c we classify all endomorphisms of the rational cohomology ring of c which are nonzero on dimension 2 some applications of this result are given

the classification of p compact groups for p odd a p compact group as defined by dwyer and wilkerson is a purely homotopically defined p local analog of a compact lie group it has long been the hope and later the conjecture that these objects should have a classification similar to the classification of compact lie groups in this paper we finish the proof of this conjecture for p an odd prime proving that there is a one to one correspondence between connected p compact groups and finite reflection groups over the p adic integers we do this by providing the last and rather intricate piece namely that the exceptional compact lie groups are uniquely determined as p compact groups by their weyl groups seen as finite reflection groups over the p adic integers our approach in fact gives a largely self contained proof of the entire classification theorem for p odd

finite loop spaces are manifolds one of the motivating questions for surgery theory was whether every finite h space is homotopy equivalent to a lie group this question was answered in the negative by hilton and roitberg s discovery of some counterexamples 18 however the problem remained whether every finite h space is homotopy equivalent to a closed smooth manifold this question is still open but in case the h space admits a classifying space we have the following theorem

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

finite h spaces and algebras over the steenrod algebra a correction we are grateful to k ishiguro for pointing out a regrettable fallacy which vitiates a passage of our paper 1 starting with p 141 line 3 on the other hand and finishing with p 141 line 21 thus i wi is prime to p we wish to replace this passage with the following argument let h and k be the graded fields of fractions of h and k respectively we recall from 1 8 that k is integral over h it follows that foreachx oink wecanfindy oink suchthatxyeh it is now easy to choose elements ui ui2 me in k which form a base for k over t in particular the elements u u2 ye u generate over h a free module m whose poincare series has the form f t k t where k t is a polynomial with k l m by 1 8 again k is finitely generated as a module over h and so choosing a common denominator a 0 in h we can arrange that k is contained in m or equivalently ak c m from the inclusions

finite time landauer principle we study the thermodynamic cost associated with the erasure of one bit of information over a finite amount of time we present a general framework for minimizing the average work required when full control of a system s microstates is possible in addition to exact numerical results we find simple bounds proportional to the variance of the microscopic distribution associated with the state of the bit in the short time limit we get a closed expression for the minimum average amount of work needed to erase a bit the average work associated with the optimal protocol can be up to a factor of 4 smaller relative to protocols constrained to end in local equilibrium assessing prior experimental and numerical results based on heuristic protocols we find that our bounds often dissipate an order of magnitude less energy

the ring of sections of a complete symmetric variety we study the ring of sections a x of a complete symmetric variety x that is of the wonderful completion of g h where g is an adjoint semisimple group and h is the fixed subgroup for an involutorial automorphism of g we find generators for pic x we generalize the prv conjecture to complete symmetric varieties and construct a standard monomial theory for a x that is compatible with g orbit closures in x this gives a degeneration result and the rational singularityness for a x 2003 elsevier science usa all rights reserved

a note on affine toda couplings the author proves the clebsch gordan property for the couplings of an untwisted affine toda field theory utilizing dorey s 1991 conjecture

embeddings of semisimple complex lie groups and cohomological components of modules let g g be an embedding of semisimple complex lie groups b b a pair of nested borel subgroups and g b g b the associated embedding of flag manifolds let o be an equivariant invertible sheaf on g b and o o be its restriction to g bg b consider the g equivariant pullback h g b o h g b o the borel weil bott theorem and schur s lemma imply that is either surjective or zero if is nonzero the image of the dual map is a g irreducible component in a g irreducible module called a cohomological component we establish a necessary and sufficient condition for nonvanishing of also we prove a theorem on the structure of the set of pairs of dominant weights with v v cohomological here v v and v denote the respective highest weight modules simplified specializations are formulated for regular and diagonal embeddings in particular we give an alternative proof of a recent theorem of dimitrov and roth beyond the regular and diagonal cases we study equivariantly embedded rational curves and we also show that the generators of the algebra of ad invariant polynomials on a semisimple lie algebra can be obtained as cohomological components our methods rely on kostant s theory of lie algebra cohomology

the uniqueness of plethystic factorisation we prove that a plethysm product of two schur functions can be factorised uniquely and classify homogeneous and indecomposable plethysm products

optimal finite time erasure of a classical bit information erasure inevitably leads to the generation of heat minimizing this dissipation will be crucial for developing small scale information processing systems but little is known about the optimal procedures required we have obtained closed form expressions for maximally efficient erasure cycles for deletion of a classical bit of information stored by the position of a particle diffusing in a double well potential we find that the extra heat generated beyond the landauer bound is proportional to the square of the hellinger distance between the initial and final states divided by the cycle duration which quantifies how far out of equilibrium the system is driven finally we demonstrate close agreement between the exact optimal cycle and the protocol found using a linear response framework

self maps of flag manifolds rationally a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology two algebraic results for cohomology endomorphisms then serve a to determine those flag manifolds which have nontrivial self maps that factor through a complex projective space and b for a special class of flag manifolds to classify the self maps of their rationalizations up to homotopy 0 introduction the homotopy classes of self maps of a complex projective space are classified by an integral degree and we show that for rational homotopy there is an analogous result for a much larger class of homogeneous spaces these spaces are quotients of a unitary group by a closed connected subgroup of maximal rank and just as for projective spaces the degree of a self map is determined by its effect on a two dimensional cohomology class the first step is to observe that homotopy classes of endomorphisms of the minimal model for the rational homotopy type of such a space correspond bijectively with endomorphisms of the rational cohomology algebra see theorem 1 1 the second step is to classify the cohomology endomorphisms such a classification was given in gh1 for the grassmann manifold of complex p planes in complex n space for n greater than 2p2 and it follows that the rational homotopy classes of self maps are indeed classified by a rational degree here in theorems 1 3 and 1 4 we give the analogous result for the flag manifolds m 1 p q u 1 p q u 1 x u p x u q the cohomology endomorphisms fall into a finite number often one of families each of which is classified by a rational degree some of these families are represented by the endomorphisms induced by conjugation in u 1 p q by elements of the normalizer of u 1 x u p x u q but if 1 p and q are distinct we get only the identity in this way the rest are represented by idempotents which we call projective endomorphisms because they factor through the cohomology of complex projective space such projective endomorphisms for any flag manifold are in turn classified in theorem 1 3 by factorizations over z of the polynomial

vector bundles over classifying spaces of compact lie groups the completion theorem of atiyah and segal as says that the complex k theory group k bg of the classifying space of any compact lie group g is isomorphic to r g the representation ring completed with respect to its augmentation ideal however the group k bg bg z bu does not directly contain information about vector bundles over the infinite dimensional complex bg itself the purpose of this paper is to compare the grothendieck group of vector bundles over bg which we denote k bg with both k bg and r g the main result is an algebraic description of k bg in terms of the representation rings of certain subgroups of g as one consequence we show that of the natural maps

equidimensionality of convolution morphisms and applications to saturation problems fix a split connected reductive group g over a field k and a positive integer r for any r tuple of dominant coweights i of g we consider the restriction m of the r fold convolution morphism of mirkovic vilonen to the twisted product of affine schubert varieties corresponding to we show that if all the coweights i are minuscule then the fibers of m are equidimensional varieties with dimension the largest allowed by the semi smallness of m we derive various consequences the equivalence of the non vanishing of hecke and representation ring structure constants and a saturation property for these structure constants when the coweights i are sums of minuscule coweights this complements the saturation results of knutson tao and kapovich leeb millson we give a new proof of the p r v conjecture in the sums of minuscules setting finally we generalize and reprove a result of spaltenstein pertaining to equidimensionality of certain partial springer resolutions of the nilpotent cone for gln

projective normality of the wonderful compactification of semisimple adjoint groups we prove that if g is a semisimple adjoint group over an algebraically closed field of arbitrary characteristic x is the wonderful compactification of g and widehat g is a simply connected covering of g then for any widehat g times widehat g linearised very ample line bundle l over x the cone over x given by l is normal

fibrations of classifying spaces we investigate fibrations of the form z y x where two of the three spaces are classifying spaces of compact connected lie groups we obtain certain finiteness conditions on the third space which make it also a classifying space our results allow to express some of the basic notions in group theory in terms of homotopy theory i e in terms of classifying spaces as an application we prove that every retract of the classifying space of a compact connected lie group is again a classifying space

equivariant stable homotopy and sullivan s conjecture let g be a p group and let x be a g complex let eg denote a contractible space on which g acts freely by the homotopy fixed point set of x we mean the fixed point set f eg x g where f eg x denotes the function space of all maps eg x equipped with a g action by 9f 9f9 1 if we let denote the one point space with trivial g action we may also consider the g space f x it is canonically g homeomorphic to x the g map e g induces a map x g f x g in 19 d sullivan proposed the following

homotopy nilpotency in p regular loop spaces we consider the problem how far from being homotopy commutative is a loop space having the homotopy type of the p completion of a product of finite numbers of spheres we determine the homotopy nilpotency of those loop spaces as an answer to this problem

on the irreducibility of irreducible characters of simple lie algebras we establish an irreducibility property for the characters of finite di mensional irreducible representations of simple lie algebras or simple algebraic groups over the complex numbers i e that the characters of irreducible represen tations are irreducible after dividing out by generalized weyl denominator type factors for sl r the irreducibility result is the following let a1 a2 ar 1 0 be the highest weight of an irreducible rational representation vof sl r as sume that the integers a1 r 1 a2 r 2 ar 1 1 are relatively prime then the character of vis strongly irreducible in the following sense for any natural number d the function g d g sl r c is irreducible in the ring of regular functions of sl r c

maps between p completed classifying spaces ii we investigate maps between p completed classifying spaces of compact connected lie groups let g and g be two connected compact lie groups for a space x let xp be a p completion of x if p does not divide the order of the weyl group of g we give descriptions of the set of homotopy classes bg p bg p in terms of k theory and in terms of admissible maps of adams and mahmud

non markovianity and negative entropy production rates entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems its positivity can be ensured for a wide class of setups but the entropy production rate can become negative sometimes this is often taken as an indicator of non markovianity we make this link precise by showing under which conditions a negative entropy production rate implies non markovianity and when it does not for a system coupled to a single heat bath this can be established within a unified language for two setups i the dynamics resulting from a coarse grained description of a markovian master equation and ii the classical hamiltonian dynamics of a system coupled to a bath the quantum version of the latter result is shown not to hold despite the fact that the integrated thermodynamic description is formally equivalent to the classical case the instantaneous fixed point of a non markovian dynamics plays an important role in our study our key contribution is to provide a consistent theoretical framework to study the finite time thermodynamics of a large class of dynamics with a precise link to its non markovianity

minimal dissipation in processes far from equilibrium a central goal of thermodynamics is to identify optimal processes during which the least amount of energy is dissipated into the environment generally even for simple systems such as the parametric harmonic oscillator optimal control strategies are mathematically involved and contain peculiar and counterintuitive features we show that optimal driving protocols determined by means of linear response theory exhibit the same step and ensuremath delta peak like structures that were previously found from solving the full optimal control problem however our method is significantly less involved since only a minimum of a quadratic form has to be determined in addition our findings suggest that optimal protocols from linear response theory are applicable far outside their actual range of validity

optimal performance of periodically driven stochastic heat engines under limited control we consider the performance of periodically driven stochastic heat engines in the linear response regime reaching the theoretical bounds for efficiency and efficiency at maximum power typically requires full control over the design and the driving of the system we develop a framework which allows us to quantify the role that limited control over the system has on the performance specifically we show that optimizing the driving entering the work extraction for a given temperature protocol leads to a universal one parameter dependence for both maximum efficiency and maximum power as a function of efficiency in particular we show that reaching carnot efficiency and hence curzon ahlborn efficiency at maximum power requires to have control over the amplitude of the full hamiltonian of the system since the kinetic energy cannot be controlled by an external parameter heat engines based on underdamped dynamics can typically not reach carnot efficiency we illustrate our general theory with a paradigmatic case study of a heat engine consisting of an underdamped charged particle in a modulated two dimensional harmonic trap in the presence of a magnetic field

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

optimization induced by stability and the role of limited control near a steady state a relationship between stability and self optimization is found for weakly dissipative heat devices the effect of limited control on operation variables around an steady state is such that after instabilities the paths toward relaxation are given by trajectories stemming from restitution forces which improve the system thermodynamic performance power output efficiency and entropy generation statistics over random trajectories for many cycles shows this behavior as well two types of dynamics are analyzed one where an stability basin appears and another one where the system is globally stable under both dynamics there is an induced trend in the control variables space due to stability in the energetic space this behavior translates into a preference for better thermodynamic states and thus stability could favor self optimization under limited control this is analyzed from the multiobjective optimization perspective as a result the statistical behavior of the system is strongly influenced by the pareto front the set of points with the best compromise between several objective functions and the stability basin additionally endoreversible and irreversible behaviors appear as very relevant limits the first one is an upper bound in energetic performance connected with the pareto front and the second one represents an attractor for the stochastic trajectories

kronecker products minuscule representations and polynomial identities the special role of the highest long and highest short roots in the derivation of extremum properties of the weights of irreducible representations of semisimple lie algebras is pointed out these properties are used to give an intrinsic and unifying reformulation as well as a new proof of klimyk s theorem on kronecker products ukrain mat z 18 19 1966 this new form of klimyk s theorem reveals the special position of the minuscule representations in kronecker products as an immediate consequence the explicit formula for the kronecker product of an arbitrary representation with a minuscule representation is obtained explicit expressions for the weights of the minuscule representations are given new derivations of theorems of dynkin trudy mosk obsch 1 39 1952 and feingold proc am math soc 70 109 1978 on kronecker products as well as the necessary condition for a kronecker product to decompose in two irreducible components are obtained the proofs are based on a theorem of parth

the z theorem for compact lie groups keywords categories of finite p subgroups z theorem compact lie group control of fusion p toral subgroups mod p cohomology reference ctg article 1991 001doi 10 1007 bf01445193 record created on 2008 12 16 modified on 2017 05 12

unique decomposition of tensor products of irreducible representations of simple algebraic groups we show that a tensor product of irreducible finite dimensional representations of a simple lie algebra over a field of characteristic zero determines the individual constituents uniquely this is analogous to the uniqueness of prime factorisation of natural numbers

rational equivalences between classifying spaces the paper contains a homotopy classification of rational equivalences between classifying spaces of compact connected lie groups with an application to genus sets of such spaces

maximum work extraction and implementation costs for nonequilibrium maxwell s demons we determine the maximum amount of work extractable in finite time by a demon performing continuous measurements on a quadratic hamiltonian system subjected to thermal fluctuations in terms of the information extracted from the system the maximum work demon is found to apply a high gain continuous feedback involving a kalman bucy estimate of the system state and operates in nonequilibrium a simple and concrete electrical implementation of the feedback protocol is proposed which allows for analytic expressions of the flows of energy entropy and information inside the demon this let us show that any implementation of the demon must necessarily include an external power source which we prove both from classical thermodynamics arguments and from a version of landauer s memory erasure argument extended to nonequilibrium linear systems

homotopy fixed point methods for lie groups and finite loop spaces a loop space x is by definition a triple x bx e in which x is a space bx is a connected pointed space and e x qbx is a homotopy equivalence from x to the space qbx of based loops in bx we will say that a loop space x is finite if the integral homology h x z is finitely generated as a graded abelian group i e if x appears at least homologically to be a finite complex in this paper we prove the following theorem

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

on adic genus and lambda rings sufficient conditions on a space are given which guarantee that the k theory ring is an invariant of the adic genus an immediate consequence of this result about adic genus is that for any positive integer n the power series ring z x 1 x n admits uncountably many pairwise non isomorphic a ring structures

rational homotopy of the homotopy fixed point sets of circle actions in this paper when g is the circle s 1 and m is a g space we study the rational homotopy type of the fixed point set m g the homotopy fixed point set m hg and the natural injection m g m hg c 2009 elsevier inc all rights reserved

the cos λ and sin λ transforms as intertwining operators between generalized principal series representations of sl n 1 k abstract in this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple lie groups by showing that the cos and sin transforms on the grassmann manifolds gr p k su n 1 k s u p k u n 1 p k are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup p p of sl n 1 k the index p indicates the dependence of the parabolic on p the general results of knapp and stein 1971 1980 20 21 and vogan and wallach 1990 44 then show that both transforms have meromorphic extension to c and are invertible for generic c furthermore known methods from representation theory combined with a selberg type integral allow us to determine the k spectrum of those operators

the classification of 2 compact groups we prove that any connected 2 compact group is classified by its 2 adic root datum and in particular the exotic 2 compact group di 4 constructed by dwyer wilkerson is the only simple 2 compact group not arising as the 2 completion of a compact connected lie group combined with our earlier work with moller and viruel for p odd this establishes the full classification of p compact groups stating that up to isomorphism there is a one to one correspondence between connected p compact groups and root data over the p adic integers as a consequence we prove the maximal torus conjecture giving a one to one correspondence between compact lie groups and finite loop spaces admitting a maximal torus our proof is a general induction on the dimension of the group which works for all primes it refines the andersen grodal moller viruel methods by incorporating the theory of root data over the p adic integers as developed by dwyer wilkerson and the authors furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by jackowski mcclure oliver in the early 1990s less

on the mod p cohomology of bpu p we study the mod p cohomology of the classifying space of the projective unitary group pu p we first prove that conjectures due to j f adams and kono and yagita 1993 about the structure of the mod p cohomology of the classifying space of connected compact lie groups hold in the case of pu p finally we prove that the classifying space of the projective unitary group pu p is determined by its mod p cohomology as an unstable algebra over the steenrod algebra for p 3 completing previous work by dwyer miller and wilkerson 1992 and broto and viruel 1998 for the cases p 2 3

strongly multiplicity free modules for lie algebras and quantum groups abstract let u be either the universal enveloping algebra of a complex semisimple lie algebra g or its drinfel d jimbo quantisation over the field c z of rational functions in the indeterminate z we define the notion of strongly multiplicity free smf for a finite dimensional u module v and prove that for such modules the endomorphism algebras end u v r are generic in the sense that in the classical unquantised case they are quotients of kohno s infinitesimal braid algebra t r while in the quantum case they are quotients of the group ring c z b r of the r string braid group b r in the classical case the generators are generalisations of the quadratic casimir operator c of u while in the quantum case they arise from r matrices which may be thought of as square roots of a quantum analogue of c in a completion of u r this unifies many known results and brings some new cases into their context these include the irreducible 7 dimensional module in type g 2 and arbitrary irreducibles for sl 2 the work leads naturally to questions concerning non semisimple deformations of the relevant endomorphism algebras which arise when the ground rings are varied

conjugation properties of tensor product multiplicities it was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product of two irreducible representations of a simple lie algebra is invariant under conjugation of one of them at a given level this also applies to the fusion multiplicities of affine algebras here we show that in the case of su 3 the lists of multiplicities in the tensor products and are identical up to permutations this latter property does not hold in general for other lie algebras we conjecture that the same property should hold for the fusion product of the affine algebra of su 3 at finite levels but this is not investigated in the present paper

entropy generation and unified optimization of carnot like and low dissipation refrigerators the connection between carnot like and low dissipation refrigerators is proposed by means of their entropy generation and the optimization of two unified compromise based figures of merit their optimization shows that only a limited set of heat transfer laws in the carnot like model are compatible with the results stemming from the low dissipation approximation even though there is an agreement of the related physical spaces of variables a comparison between two operation regimes and relations among entropy generation efficiency cooling power and power input are obtained with emphasis on the role of dissipation symmetries the results extend previous findings for heat engines at maximum power conditions

on stable maps from eilenberg maclane spaces to classifying spaces 0 introduction in recent years there has been a general interest in the problem of understanding the homotopy type of function spaces map x y where x is an infinite complex the special case where x is the classifying space of a finite group gives the celebrated segal s conjecture 4 when y lim fqnn qs and sullivan s conjecture 10 when y is a finite complex one would like to know whether the affirmative solution to these conjectures can also be applied to the study of map x y for other infinite complexes x the philosophy behind this approach to the problem has already manifested itself in 5 6 7 and 8 among others to give interesting results in this note we will utilize the solution to segal s conjecture to compute stable maps from eilenberg maclane spaces to classifying spaces as an application we will also describe the homotopy type of map x y at a prime p where x k 1t n wt finite n 2 and y q g g 0 u sp the original motivation behind this paper was due to anderson hodgkin 1 there they proved vanishing results on the topological k theory of higher eilenberg maclane spaces using atyiah s theorem on the k theory of the classifying spaces of compact lie groups we shall prove that the same result holds for stable cohomotopy groups the author is especially grateful to professor gunnar carlsson for pointing out 1 and how segal s conjecture comes into play

optimal protocols for slowly driven quantum systems the design of efficient quantum information processing will rely on optimal nonequilibrium transitions of driven quantum systems building on a recently developed geometric framework for computing optimal protocols for classical systems driven in finite time we construct a general framework for optimizing the average information entropy for driven quantum systems geodesics on the parameter manifold endowed with a positive semidefinite metric correspond to protocols that minimize the average information entropy production in finite time we use this framework to explicitly compute the optimal entropy production for a simple two state quantum system coupled to a heat bath of bosonic oscillators which has applications to quantum annealing

connected finite loop spaces with maximal tori finite loop spaces are a generalization of compact lie groups however they do not enjoy all of the nice properties of compact lie groups for example having a maximal torus is a quite distinguished property actually an old conjecture due to wilkerson says said that every connected finite loop space with a maximal torus is equivalent to a compact connected lie group we give some more evidence for this conjecture by showing that the associated action of the weyl group on the maximal torus always represents the weyl group as a crystallographic group we also develop the notion of normalizers of maximal tori for connected finite loop space and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal tori is equivalent to a compact connected lie group if it has the right normalizer of the maximal torus actually in the cases under consideration the information about the weyl group is sufficient to give the answer all this is done by first studying the analoguous local problems