on double centralizer properties between quantum groups and ariki koike algebras abstract in this paper the double centralizer properties for tensor spaces as bimodules over quantum groups and ariki koike algebras as well as their relations with shoji s algebra h h see j algebra 226 2000 818 856 are studied we proved that under the separation condition see j reine angew math 513 1999 53 69 the natural homomorphism from the ariki koike algebra to the endomorphism algebra of tensor space as module over quantum group is surjective we also show that the natural homomorphism from shoji s algebra h h to that endomorphism algebra is always surjective and h h is actually isomorphic to a direct sum of some matrix algebras over some hecke algebras of type a

schur weyl quasi duality and co triangular hopf quasigroups schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic and plays a central role in many parts of algebraic lie theory in this paper we will introduce the notion of schur weyl quasi duality and study it for this generally we consider a braided vector space v c and its braided lie algebra endk v then we can construct its braided enveloping algebra u endk v which is a connected braided c cocommutative hopf algebra let h be a triangular hopf quasigroup with bijective antipode and b be a cotriangular hopf quasigroup with bijective antipode let v be any finite dimensional vector space in the category lq h r b of generalized long quasimodules we show that u endkv h b ksn v n is a schur weyl quasi duality under suitable conditions

asymptotic profile in selection mutation equations gauss versus cauchy distributions abstract in this paper we study the asymptotic large time behaviour of a selection mutation competition model for a population structured with respect to a phenotypic trait when the rate of mutation is very small we assume that the reproduction is asexual and that the mutations can be described by a linear integral operator we are interested in the interplay between the time variable t and the rate e of mutations we show that depending on 0 the limit e 0 with t e can lead to population number densities which are either gaussian like when is small or cauchy like when is large

precision and reliability of periodically and quasiperiodically driven integrate and fire neurons neurons in the brain communicate via trains of all or none electric events known as spikes how the brain encodes information using spikes char22 the neural code char22 remains elusive here the robustness against noise of stimulus induced neural spike trains is studied in terms of attractors and bifurcations the dynamics of model neurons converges after a transient onto an attractor yielding a reproducible sequence of spike times at a bifurcation point the spike times on the attractor change discontinuously when a parameter is varied reliability the stability of the attractor against noise is reduced when the neuron operates close to a bifurcation point we determined using analytical spike time maps the attractor and bifurcation structure of an integrate and fire model neuron driven by a periodic or a quasiperiodic piecewise constant current and investigated the stability of attractors against noise the integrate and fire model neuron became mode locked to the periodic current with a rational winding number p q and produced p spikes per q cycles there were q attractors p q mode locking regions formed arnold tongues in the model reliability was the highest during 1 1 mode locking when there was only one attractor as was also observed in recent experiments the quasiperiodically driven neuron mode locked to either one of the two drive periods or to a linear combination of both of them mode locking regions were organized in arnold tongues and reliability was again highest when there was only one attractor these results show that neuronal reliability in response to the rhythmic drive generated by synchronized networks of neurons is profoundly influenced by the location of the arnold tongues in parameter space

projections and yetter drinfel d modules over hopf co quasigroups abstract in this paper we introduce the notions of yetter drinfel d module and strong projection over a hopf quasigroup h if the antipode of h is invertible we show that the category of left left yetter drinfel d modules yd h h is braided and there exists a categorical equivalence between the category of hopf quasigroups in yd h h and the category of strong hopf algebra projections associated to h

explicit solutions for replicator mutator equations extinction versus acceleration we consider a class of nonlocal reaction diffusion problems referred to as replicator mutator equations in evolutionary genetics by using explicit changes of unknown function we show that they are equivalent to the heat equation and therefore compute their solution explicitly based on this we then prove that in the case of beneficial mutations in asexual populations solutions dramatically depend on the tails of the initial data they can be global become extinct in finite time or even be defined for no positive time in the first case we prove that solutions are accelerating and in many cases converge for large times to some universal gaussian profile this sheds light on the biological relevance of such models

braided lie algebras and bicovariant differential calculi over co quasitriangular hopf algebras we show that if g is the quantum tangent space or quantum lie algebra in the sense of woronowicz of a bicovariant first order differential calculus over a co quasitriangular hopf algebra a r then a certain extension of it is a braided lie algebra in the category of a comodules this is used to show that the woronowicz quantum universal enveloping algebra u g is a bialgebra in the braided category of a comodules we show that this algebra is quadratic when the calculus is inner examples with this unexpected property include finite groups and quantum groups with their standard differential calculi we also find a quantum lie functor for co quasitriangular hopf algebras which has properties analogous to the classical one this functor gives trivial results on standard quantum groups oq g but reasonable ones on examples closer to the classical case such as the cotriangular jordanian deformations in addition we show that split braided lie algebras define generalized lie algebras in a different sense of deforming the adjoint representation we construct these and their enveloping algebras for oq sl n recovering the witten algebra for n 2

mathematical properties of a class of integro differential models from population genetics we propose a mathematical analysis of an integro differential model arising in population genetics the model describes the dynamics of fitness distribution in an asexual population under the effect of mutation and selection these two processes are represented by two nonlocal terms first we prove the existence and uniqueness of the solution and we derive asymptotic estimates of the distribution as the fitness tends to pm infty based on these asymptotic estimates we show that the cumulant generating function of the distribution is well defined and satisfies a linear nonlocal transport equation that we solve explicitly this explicit formula allows us to characterize the dependence of the long time behavior of the distribution with respect to the mutation kernel on the one hand if the kernel contains some beneficial mutations the distribution diverges which is reminiscent of the results of alfaro and carles siam j appl math 74 2014 pp 1919 1934 who analyzed a mutator replicator equat

double centralizer properties dominant dimension and tilting modules double centralizer properties play a central role in many parts of algebraic lie theory soergel s double centralizer theorem relates the principal block of the bernstein gelfand gelfand category o of a semisimple complex lie algebra with the coinvariant algebra i e the cohomology algebra of the corresponding flag manifold schur weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic or via the quantized version in nondefining characteristic in this paper we exhibit algebraic structures behind these double centralizer properties we show that the finite dimensional algebras relevant in this context have dominant dimension at least two with respect to some projective injective or tilting modules general arguments which combine methods from ring theory qf 3 rings and dominant dimension with tools from representation theory approximations tilting modules then imply the validity of these double centralizer properties as well as new ones in contrast to the traditional proofs e g by the fundamental theorems of invariant theory no computations are necessary

hopf quasigroups and the algebraic 7 sphere we introduce the notions of hopf quasigroup and hopf coquasigroup h generalising the classical notion of an inverse property quasigroup g expressed respectively as a quasigroup algebra kg and an algebraic quasigroup k g we prove basic results as for hopf algebras such as anti co multiplicativity of the antipode s h h that s2 id if h is commutative or cocommutative and a theory of crossed co products we also introduce the notion of a moufang hopf co quasigroup and show that the coordinate algebras k s2n 1 of the parallelizable spheres are algebraic quasigroups commutative hopf coquasigroups in our formulation and moufang we make use of the description of composition algebras such as the octonions via a cochain f introduced in 2 we construct an example k s7 z23 of a hopf coquasigroup which is noncommutative and nontrivially moufang we use hopf coquasigroup methods to study differential geometry on k s7 including a short algebraic proof that s7 is parallelizable looking at combinations of left and right invariant vector fields on k s7 we provide a new description of the structure constants of the lie algebra g2 in terms of the structure constants f of the octonions in the concluding section we give a new description of the q deformation quantum group cq s3 regarded trivially as a moufang hopf coquasigroup trivially since it is in fact a hopf algebra but now in terms of f built up via the cayley dickson process

stochastic resonance in an autonomous system with a nonuniform limit cycle in a recent numerical study gang et al phys rev lett 71 807 1993 presented the first example of stochastic resonance in an autonomous system they considered a two variable model in which a limit cycle is born as some parameter is varied their numerical experiments revealed various noise induced effects including a noise induced shift in the frequency of the limit cycle oscillations and noise induced oscillations in the absence of a deterministic limit cycle we show that both of these effects are simple consequences of the nonuniformity of the motion along the limit cycle

infinite dimensional schur weyl duality and the coxeter laplace operator we extend the classical schur weyl duality between representations of the groups sl n mathbb c and mathfrak s n to the case of sl n mathbb c and the infinite symmetric group mathfrak s mathbb n our construction is based on a dynamic or inductive scheme of schur weyl dualities it leads to a new class of representations of the infinite symmetric group which has not appeared earlier we describe these representations and in particular find their spectral types with respect to the gelfand tsetlin algebra the main example of such a representation acts in an incomplete infinite tensor product as an important application we consider the weak limit of the so called coxeter laplace operator which is essentially the hamiltonian of the xxx heisenberg model in these representations

connected braided hopf algebras abstract we prove a braided version of kostant cartier milnor moore theorem the category of connected cocommutative 2 id braided hopf algebras over a field of zero characteristic is equivalent via the enveloping construction to the category of generalized lie algebras this statement includes an embedding of any generalized lie algebra into its universal enveloping algebra the isomorphism h u prim h and primitive generation results the embedding theorem may be derived from the poincare birkhoff witt theorem for quadratic algebras of koszul type remark c we also provide a direct proof that uses neither koszul cohomologies nor the algebraic deformation theory we consider the primitive generation problem for subalgebras biideals and homomorphic images of connected braided hopf algebras in much more general context when the braiding is not necessary involutive and even it is not necessary invertible

a new approach to braided monoidal categories let h be a hopf quasigroup over k possessing an adjoint quasiaction then we first show that if m is any right h comodule and n is any right h quasimodule such that n m2 idn m where n m n m m n is a favourable map then we have h k as a generalization of pareigis theorem j pure appl algebra 155 91 2001 then we mainly give a new approach to a braided monoidal category by generalizing one of schauenburg s main results schauenburg p j algebra 169 874 890 1994 in the setting of hopf quasigroups introduced by klim and majid j algebra 323 3067 3110 2010 this yields new sources of braidings which provide solutions to the yang baxter equation playing an important role in various areas of mathematics let h be a hopf quasigroup over k possessing an adjoint quasiaction then we first show that if m is any right h comodule and n is any right h quasimodule such that n m2 idn m where n m n m m n is a favourable map then we have h k as a generalization of pareigis theorem j pure appl algebra 155 91 2001 then we mainly give a new approach to a braided monoidal category by generalizing one of schauenburg s main results schauenburg p j algebra 169 874 890 1994 in the setting of hopf quasigroups introduced by klim and majid j algebra 323 3067 3110 2010 this yields new sources of braidings which provide solutions to the yang baxter equation playing an important role in various areas of mathematics

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

on the antipode of a quasitriangular hopf algebra let a r be a quasitriangular hopf algebra with antipode s in the category of vector spaces over a field k in this paper we show that s2 is inner which means s is bijective and when a is finite dimensional there is a grouplike element h e a such that s4 a h r for all a e a suppose that a is any hopf algebra with antipode s over the field k proposition 1 of section 1 gives a sufficient condition for 3 to be inner which we apply to quasitriangular hopf algebras in section 2 its statement and proof are based on calculations found in 2 pp 66671 generally s2 need not be inner suppose that a is finite dimensional if s2 were inner then all ideals of a would be invariant under s2 and hence all subcoalgebras of the dual hopf algebra a would be invariant under s where s s is the antipode of a there are examples 5 over algebraically closed lields in which the latter is not the case section 2 begins with a discussion of properties of quasit angular hopf algebras a r over the held k used in this paper write r cr c r ea a and set u cs r r the main result of section 2 is that u is invertible and s2 a uau for all a e a this result was established in z exercise 7 3 61 under the hypothesis that s is bijective if a is any finite dimensional hopf algebra with antipode s over the field k there are distinguished grouplike elements g e a and tl e a which relate left and right integrals and which together describe s4 in section 3 we consider s4 when a r is a unite dimensional quasit angular hopf algebra let h iiu where u c s r r l is defined as above and u s u i by relating h to g and 1 we are able to show that h is a grouplike element and that s4 a hah for all a e a when a is unimodular we prove that h g thus the product vu does not depend on r in the unimodular case we assume that the reader has some familiarity with the elementary aspects of the theory of hopf algebras a recommended reference is 6

synchronization of randomly driven nonlinear oscillators when nonlinear oscillators with stable limit cycles are subject to periodic forces these oscillators may become entrained or mode locked to the driving force remarkably a similar phenomenon occurs when the nonlinear oscillators are driven by a random force in particular when nonlinear oscillators with different initial conditions are strongly driven with the same random force their fluctuating behavior may reliably converge to an identical synchronized response analytical estimates are derived for the conditions rates and structural stability for synchronization of a broad class of randomly driven nonlinear oscillators which suggest different experimental procedures for assessing the nonlinear response of biological chemical and physical oscillators to fluctuating inputs

evaluation of entrainment of a nonlinear neural oscillator to white noise the lyapunov exponent for a one dimensional neural oscillator model the theta neuron is computed for white noise forcing using the steady state solution to the associated fokker planck equation the latter is mildly singular due to the nature of the multiplicative input in agreement with previous results with similar models the exponent is negative for all forcing amplitudes but here it is shown to be small relative to that for periodic drive in a range of forcing strengths thus the synchronization of an ensemble of independent neurons receiving common but random input can be slow moreover this implies that aperiodic input may be suboptimal in some contexts for preserving the reliability of fine spike timing a potentially important component of the neural code

algebras hyperalgebras nonassociative bialgebras and loops abstract sabinin algebras are a broad generalization of lie algebras that include lie malcev and bol algebras as very particular examples we present a construction of a universal enveloping algebra for sabinin algebras and the corresponding poincare birkhoff witt theorem a nonassociative counterpart of hopf algebras is also introduced and a version of the milnor moore theorem is proved loop algebras and universal enveloping algebras of sabinin algebras are natural examples of these nonassociative hopf algebras identities of loops move to identities of nonassociative hopf algebras by a linearizing process in this way nonassociative algebras and hopf algebras interlace smoothly