Family Of Linear Maps Research Articles

V-universal Hopf algebras (co)acting on Ω-algebras

We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra structures over Hopf algebras, J. Noncommut. Geom., doi: 10.4171/JNCG/428.] bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of A, called [Formula: see text]-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study V-universal measuring coalgebras and V-universal comeasuring algebras between [Formula: see text]-algebras A and B, relative to a fixed subspace V of [Formula: see text]. By considering the case [Formula: see text], we derive the notion of a V-universal (co)acting bialgebra (and Hopf algebra) for a given algebra A. In particular, this leads to a refinement of the existence conditions for the Manin–Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the V-universal acting bi/Hopf algebra and the finite dual of the V-universal coacting bi/Hopf algebra under certain conditions on V in terms of the finite topology on [Formula: see text].

ROOTED TREE MAPS AND THE KAWASHIMA RELATIONS FOR MULTIPLE ZETA VALUES

Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the noncommutative polynomial algebra in two letters. These give a class of relations among multiple zeta values, which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.

Rankin-Cohen Brackets on Hilbert Modular forms and Special values of certain Dirichlet series

Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain Dirichlet series of Rankin type associated to Siegel cusp forms. This is a generalisation of the work due to Kohnen (Math Z 207:657–660, 1991) and Herrero (Ramanujan J 36:529–536, 2015) in the case of elliptic modular forms to the case of Siegel cusp forms which is also considered earlier by Lee (Complex Var Theory Appl 31:97–103, 1996) for a special case.

Rankin–Cohen brackets on Siegel modular forms and special values of certain Dirichlet series

Given a fixed Siegel cusp form of genus two, we consider a family of linear maps between the spaces of Siegel cusp forms of genus two by using the Rankin–Cohen brackets and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Siegel cusp forms of genus two constructed using this method involve special values of certain Dirichlet series of Rankin type associated to Siegel cusp forms. This is a generalisation of the work due to Kohnen (Math Z 207:657–660, 1991) and Herrero (Ramanujan J 36:529–536, 2015) in the case of elliptic modular forms to the case of Siegel cusp forms which is also considered earlier by Lee (Complex Var Theory Appl 31:97–103, 1996) for a special case.

Rankin–Cohen brackets on Jacobi forms and the adjoint of some linear maps

Given a fixed Jacobi cusp form, we consider a family of linear maps between the spaces of Jacobi cusp forms using the Rankin–Cohen brackets, and then we compute the adjoint maps of these linear maps with respect to the Petersson scalar product. The Fourier coefficients of the Jacobi cusp forms constructed using this method involve special values of certain Dirichlet series associated to Jacobi cusp forms. This is a generalization of the work due to Kohnen (Math Z, 207:657–660, 1991) and Herrero (Ramanujan J, 10.1007/s11139-013-9536-5, 2014) in case of elliptic modular forms to the case of Jacobi cusp forms which is also considered earlier by Sakata (Proc Japan Acad Ser A, Math Sci 74, 1998) for a special case.

Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

The pressure function $P(A, s)$ plays a fundamental role in the calculation of the dimension of "typical" self-affine sets, where $A=(A_1,\ldots, A_k)$ is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on $A$. As a consequence, we show that the dimension of "typical" self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.

New Tools for Investigating Positive Maps in Matrix Algebras

We provide a novel tool which may be used to construct new examples of positive maps in matrix algebras (or, equivalently, entanglement witnesses). It turns out that this can be used to prove positivity of several well known maps (such as reduction map, generalized reduction, Robertson map, and many others). Furthermore, we use it to construct a new family of linear maps and prove that they are positive, indecomposable and (nd)optimal.

Characterization of higher derivations on CSL algebras

Let AlgL be a CSL algebra. We say that a family of linear maps δ={δn,δn:AlgL→AlgL,n∈N} is higher derivable at Ω∈AlgL if ∑i+j=nδi(A)δj(B)=δn(Ω) for all A,B∈AlgL with AB=Ω. In this paper, a necessary and sufficient condition for a family of linear maps δ={δn,n∈N} on AlgL to be higher derivable at Ω∈AlgL is given. Moreover, we show that if there is a faithful projection P in L such that PΩP and (I−P)Ω(I−P) are a left or right separating point in PAlgLP and (I−P)AlgL(I−P) respectively, then a family of linear maps δ={δn,n∈N} on AlgL is higher derivable at Ω if and only if it is a higher derivation. In particular, if AlgL is an irreducible CDCSL algebra or a nest algebra, then a family of linear maps δ={δn,n∈N} on AlgL is higher derivable at Ω≠0 if and only if it is a higher derivation.

On the degrees of matrix coefficients of intertwining operators

Let G be a reductive algebraic group defined over a p-adic field F with residue field Fq and G = G(F ). Fix a special maximal compact subgroup K0 of G. For a maximal parabolic subgroup P = MU of G and a smooth irreducible representation π of M = M(F ) we consider the family of induced representations IP (π, s), s ∈ C, which extend the fixed K0-representation I K0 P∩K0(π|M∩K0), and the associated intertwining operators M(s) = MP |P (π, s) : IP (π, s)→ IP (π, s). For any open subgroup K of K0 the restriction M(s) : I0 P∩K0(π|M∩K0) K = IP (π, s) K → IP (π, s) = I K0 P∩K0 (π|M∩K0) of M(s) to the space of K-fixed vectors is a family of linear maps between finite-dimensional vector spaces which do not depend on s. It is well known that the matrix coefficients of the linear operators M(s) are rational functions of q−s, whose denominators can be controlled explicitly (e.g. [23, IV.1.1, IV.1.2]). In particular, their degrees are bounded independently of K and π. What can be said about the degrees of the numerators? In this note, we study the following conjecture, which should provide a bound of the correct order of magnitude. Let G′ be the derived group of G and set G′ = G′(F ). Note that K ′ 0 = K0 ∩ G′ is a special maximal compact subgroup of G′.

On the Representation of Shift-Varying Linear System Maps

We give an expression for the most general continuous linear input-output map that takes the space of (Lebesgue) integrable functions into itself. This expression is a function-space limit of an integral. As an application, a representation result is given for an important family of linear maps that take the space of bounded measurable functions into itself and satisfy a certain continuity condition, and it is noted that no such result holds for a general continuous linear maps that takes the space into itself and meets the continuity condition. Some related material concerning engineering education is also given.

Nonpositive evolutions in open system dynamics

The long-time evolution of a system in interaction with an external environment is usually described by a family of linear maps ${\ensuremath{\gamma}}_{t},$ generated by master equations of Block-Redfield type. These maps are, in general, nonpositive; a widely adopted cure for this physical inconsistency is to restrict the domain of definition of the dynamical maps to those states for which ${\ensuremath{\gamma}}_{t}$ remains positive. We show that this prescription has to be modified when two systems are immersed in the same environment and evolve with the factorized dynamics ${\ensuremath{\gamma}}_{t}\ensuremath{\bigotimes}{\ensuremath{\gamma}}_{t}$ starting from an entangled initial state.