Cyclic cohomology

A morphism Lie algebra is a triple (g, h, φ) consisting of two Lie algebras g, h and a Lie algebra homomorphism φ : g → h. We define representations and cohomology of morphism Lie algebras. As applications of our cohomology, we study some aspects of deformations, abelian extensions of morphism Lie algebras and classify skeletal morphism sh Lie algebras. Finally, we consider the cohomology of morphism Lie groups and find a relation with the cohomology of morphism Lie algebras. 2010 MSC classification: 17B55, 17B56.

In this thesis we construct explicit formulae for characteristic classes in Noncommutative geometry. The general framework for the construction of characteristic classes in Noncommutative geometry is provided by Connes' theory of cycles and their ...

- A deep link between quantization and (non-commutative) geometry is uncovered via a careful usage of modular theory.
- It is premature to assume that space-time geometry in quantum gravity becomes relevant only at the emergent macroscopic level:
* At the covariant quantum level, there is still some viable option for a-posteriori derived space-time (non-commutative) geometries, induced by states (see arXiv:1007.4094v1).
* The ideas involved are strongly motivated by algebraic quantum field theory, suitably freed from its old reliance on background manifolds.
* The mathematical techniques rely on an interplay between Tomita-Takesaki modular theory and derived categorical non-commutative geometry that seems to hint at further deep modifications of the notions of space and geometry, waiting to be further explored.

An interested subject in homology theory, is the operators theory. Famous operators are those of Adams and Steenrod operators. Some results of such operations are in the works of Gouda & Elhamdadi [1] on Hochschild and cyclic homology. In this article, we are interested in Steenrod operator in Dihedral and Reflexive homology on commutative Banach algebra

We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.

Additive manufacturing by laser fusion on metal oxides powder bed such as e.g. alumina (Al2O3) or aluminium titanate (Al2T iO5) has developed considerably in the last few years and allows today the production of a wide range of complex objects. The mathematical problem considered is to control the temperature inside some part Ω of a powder layer. This phenomenon is governed by a parabolic initial boundary value problem with a heat source corresponding to the laser trajectory on some part of the boundary ∂Ω. The main questions concern the optimization of the trajectories scanned by the laser on the boundary ∂Ω according to given criteria: imposing that during the thermal process the temperature reaches a melting value in the structure to be built, a desired temperature distribution at the end of the thermal process, minimizing the thermal gradients, all this in the shortest possible thermal treatment time. To achieve this goal, we start by proving the existence of an optimal control, followed by first order necessary optimality conditions. Finally, we establish a second order sufficient optimality condition. Keywords. Optimal control problems, parabolic equations, heat equations with moving source, trajectories, time of thermal treatment, cost functionals, existence of an optimal control, adjoint problem, first order necessary optimality conditions, second order sufficient optimality conditions.

Partial actions of groups on C * -algebras and the closely related actions and coactions of Hopf algebras received much attention over the last decades. They arise naturally as restrictions of their global counterparts to non-invariant subalgebras, and the ambient eveloping global (co)actions have proven useful for the study of associated crossed products. In this article, we introduce the partial coactions of C * -bialgebras, focussing on C * -quantum groups, and prove existence of an enveloping global coaction under mild technical assumptions. We also show that partial coactions of the function algebra of a discrete group correspond to partial actions on direct summands of a C * -algebra, and relate partial coactions of a compact or its dual discrete C * -quantum group to partial coactions or partial actions of the dense Hopf subalgebra. As a fundamental example, we associate to every discrete C * -quantum group a quantum Bernoulli shift.

This paper defines a formal approach to learning from examples described by labelled graphs. We propose a formal model based upon lattice theory and in particular with the use of Galois lattice. We enlarge the domain of formal concept analysis, by the use of the Galois ...

We prove that if G is a countable, discrete group having infinite, normal subgroups with the relative property (T), then the Bernoulli shift action of G on Π g∈G (X 0 , µ 0) g , for (X 0 , µ 0) an arbitrary probability space, has first cohomology group isomorphic to the character group of G.

I. M. Gelfand and D. B. Fuks have studied the cohomology of the Lie algebra of vector fields on a manifold. In this article, we generalize their main tools to compute the Leibniz cohomology, by extending the two spectral sequences associated to the diagonal and the order filtration. In particular, we determine some new generators for the diagonal Leibniz cohomology of the Lie algebra of vector fields on the circle.

We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter δ(M) which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or, alternatively, by stable norms of certain cohomology classes. In case of the 2-torus we obtain a positive lower bound for all Riemannian metrics and all nontrivial spin structures. For higher genus g the estimate is given by |λ| ≥ 2 √ π (2g + 1) area(M) − 1 δ(M). The corresponding estimate also holds for the L 2-spectrum of the Dirac operator on a noncompact complete surface of finite area. As a corollary we get positive lower bounds on the Willmore integral for all 2-tori embedded in R 3 .

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of the structural quantum group. On the level of $K_{0}$-groups of vector bundles, we realize the induced map by the pullback of explicit matrix idempotents. We also show how to extend our result to the case when the quantum-group representation is infinitely dimensional, and then apply it to the Ehresmann-Schauenburg quantum groupoid. Finally, we construct quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles.

We construct, in this paper, a generalization of the Dennis trace (for matrices) to the case of the supermatrices over an arbitrary (not necessarily commutative) superalgebra with unit. By analogy with the ungraded case, we show how it is possible to use this map to construct an isomorphism from the Hochschild homology of the superalgebra to the Hochschild homology of the supermatrix algebra.

We determine the periodic cyclic homology of the Iwahori-Hecke algebras Hq, for q ∈ C * not a "proper root of unity." (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a "weakly spectrum preserving" morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan-Lusztig and Lusztig show that, for the indicated values of q, there exists a weakly spectrum preserving morphism φq : Hq → J, to a fixed finite type algebra J. This proves that φq induces an isomorphism in periodic cyclic homology and, in particular, that all algebras Hq have the same periodic cyclic homology, for the indicated values of q. The periodic cyclic homology groups of the algebra H 1 can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group. Contents 12 4. A comparison theorem 15 5. Periodic cyclic homology of Iwahori-Hecke algebras 19 References 22 P.

Recent sampling theorems allow for the recovery of operators with bandlimited Kohn-Nirenberg symbols from their response to a single discretely supported identifier signal. The available results are inherently non-local. For example, we show that in order to recover a bandlimited operator precisely, the identifier cannot decay in time nor in frequency. Moreover, a concept of local and discrete representation is missing from the theory. In this paper, we develop tools that address these shortcomings. We show that to obtain a local approximation of an operator, it is sufficient to test the operator on a truncated and mollified delta train, that is, on a compactly supported Schwarz class function. To compute the operator numerically, discrete measurements can be obtained from the response function which are localized in the sense that a local selection of the values yields a local approximation of the operator. Central to our analysis is to conceptualize the meaning of localization for operators with bandlimited Kohn-Nirenberg symbol.

The original Calderón problem consists in recovering the potential (or the conductivity) from the knowledge of the related Neumann to Dirichlet map (or Dirichlet to Neumann map). Here, we first perturb the medium by injecting small-scaled and highly heterogeneous particles. Such particles can be bubbles or droplets in acoustics or nanoparticles in electromagnetism. They are distributed, periodically for instance, in the whole domain where we want to do reconstruction. Under critical scales between the size and contrast, these particles resonate at specific frequencies that can be well computed. Using incident frequencies that are close to such resonances, we show that (1) the corresponding Neumann to Dirichlet map of the composite converges to the one of the homogenized medium. In addition, the equivalent coefficient, which consist in the sum of the original potential and the effective coefficient, is negative valued with a controlable amplitude. (2) as the equivalent coefficient is negative valued, then we can linearize the corresponding Neumann to Dirichlet map using the effective coefficient's amplitude. (3) from the linearized Neumann to Dirichlet map, we reconstruct the original potential using explicit complex geometrical optics solutions (CGOs).

We describe a Jordan-algebraic version of E. Lieb convexity inequalities. A joint convexity of Jordan-algebraic version of quantum entropy is proven. A version of noncommutative Bernstein inequality is proven as an application of one of convexity inequalities. A spectral theory on semi-simple complex algebras is used as a tool to prove the convexity results. Possible applications to optimization and statistics are indicated.

These are the notes of two series of talks about Givental's proof of the mirror conjecture for projective complete intersections, given by the authors at the Universita' di Roma "La Sapienza" and at the Scuola Normale Superiore in Pisa in Spring and Autumn 1997.

If q is a p th root of unity there exists a quasi-coassocίative truncated quantum group algebra whose indecomposable representations are the physical representations of i7 g (s/2)» whose coproduct yields the truncated tensor product of physical representations of 6^(5/2), and whose β-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras ("quasi-quantum group algebras") S?*. We describe a space 3^ o f "functions on the quasi quantum plane," i.e. of polynomials in noncommuting complex coordinate functions z a9 on which multiplication operators Z a and the elements of $?* can act, so that z a will transform according to some representation τ^ of &*. can be made into a quasi-associative graded algebra ^-0 ^^ on which n>0 elements of &* act as generalized derivations. In the special case of the truncated U q (sl 2) algebra we show that the subspaces ^^ of monomials in z a of n th degree vanish for n > p-1, and that ^^ carries the 2 J + 1 dimensional irreducible representation of 5^* if n = 2J, J-0, ^, ... , ^(p-2). Assuming that the representation rf of the quasi-quantum group algebra gives rise to an R-maiήx with two eigenvalues, we develop a quasi-associative differential calculus on 3? T. This implies construction of an exterior differentiation, a graded algebra A^-0 A n^τ of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under S^*-valued gauge transformations.

We investigate classes of uniform pseudodifferential operators on Riemannian manifolds with bounded geometry. We prove that operators belonging to the classes are bounded in function spaces of Hardy-Sobolev-Besov type defined on the manifold. The classes contain the powers of the Laplace-Beltrami operators so we get the fractional lift property for the function spaces.

The Riordan group, along with its constituent elements, Riordan arrays, has been a tool for combinatorial exploration since its inception in 1991. More recently, this group has made an appearance in the area of mathematical physics, where it can be used as a toy model in the theory of the renormalization of scalar fields. In this context, its Hopf algebra nature is of importance. In this note, we explain these notions. Power series play a fundamental role in this discussion.

In this paper we analyze and numerically solve a problem related to the optimal location of green zones in metropolitan areas in order to mitigate the urban heat island effect. So, we consider a microscale climate model and analyze the problem within the framework of optimal control theory of partial differential equations. Finally we compute its numerical solution using the finite element method, with the help of the interior point algorithm IPOPT, interfaced with the FreeFem++ software package.

We characterize the C ⋆-algebras for which openness of projections in their second duals is preserved under Murray-von Neumann equivalence. They are precisely the extensions of the annihilator C ⋆-algebras by the commutative C ⋆-algebras. We also show that the annihilator C ⋆-algebras are precisely the C ⋆-algebras for which all projections in their second duals are open.

Let G ⊂ GL(V) be a linear Lie group with Lie algebra g and let A(g) G be the subalgebra of G-invariant elements of the associative supercommutative algebra A(g) = S(g *) ⊗ Λ(V *). To any G-structure π : P → M with a connection ω we associate a homomorphism µ ω : A(g) G → Ω(M). The differential forms µ ω (f) for f ∈ A(g) G which are associated to the G-structure π can be used to construct Lagrangians. If ω has no torsion the differential forms µ ω (f) are closed and define characteristic classes of a G-structure. The induced homomorphism µ ′ ω : A(g) G → H * (M) does not depend on the choice of the torsionfree connection ω and it is the natural generalization of the Chern Weil homomorphism.

An action of a Lie algebra g on a manifold M is just a Lie algebra homomorphism ζ : g → X(M). We define orbits for such an action. In general the space of orbits M/g is not a manifold and even has a bad topology. Nevertheless for a g-manifold with equidimensional orbits we treat such notions as connection, curvature, covariant differentiation, Bianchi identity, parallel transport, basic differential forms, basic cohomology, and characteristic classes, which generalize the corresponding notions for principal G-bundles. As one of the applications, we derive a sufficient condition for the projection M → M/g to be a bundle associated to a principal bundle.

Let G ⊂ GL(V) be a linear Lie group with Lie algebra g and let A(g) G be the subalgebra of G-invariant elements of the associative supercommutative algebra A(g) = S(g *) ⊗ Λ(V *). To any G-structure π : P → M with a connection ω we associate a homomorphism µ ω : A(g) G → Ω(M). The differential forms µ ω (f) for f ∈ A(g) G which are associated to the G-structure π can be used to construct Lagrangians. If ω has no torsion the differential forms µ ω (f) are closed and define characteristic classes of a G-structure. The induced homomorphism µ ′ ω : A(g) G → H * (M) does not depend on the choice of the torsionfree connection ω and it is the natural generalization of the Chern Weil homomorphism.

We consider the Schrödinger evolution on graphs, i.e., solutions to the equation $\partial _t u(t,\alpha ) = i\sum _{\beta \in \mathcal {A}}L(\alpha ,\beta )u(t,\beta )$, where $\mathcal {A}$ is the set of vertices of the graph and the matrix $(L(\alpha ,\beta ))_{\alpha ,\beta \in \mathcal {A}}$ describes interaction between the vertices, in particular two vertices α and β are connected if $L(\alpha ,\beta )\neq 0$. We assume that the graph has a “web-like” structure, i.e., it consists of an inner part, formed by a finite number of vertices, and some threads attach to it.We prove that such a solution $u(t,\alpha )$ cannot decay too fast along one thread at two different times, unless it vanishes at this thread.We also give a characterization of the dimension of the vector space formed by all the solutions of $\partial _t u(t,\alpha ) = i\sum _{\beta \in \mathcal {A}}L(\alpha ,\beta )u(t,\beta )$, when $\mathcal {A}$ is a finite set, in terms of the number of the different eigenvalu...

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, 1963) have well-understood geometric realizations, and here we deal with homotopy types represented by double groupoids satisfying a natural 'filling condition'. Any such double groupoid characteristically has associated to it 'homotopy groups', which are defined using only its algebraic structure. Thus arises the notion of 'weak equivalence' between such double groupoids, and a corresponding 'homotopy category' is defined. Our main result in the paper states that the geometric realization functor induces an equivalence between the homotopy category of double groupoids with filling condition and the category of homotopy 2-types (that is, the homotopy category of all topological spaces with the property that the n th homotopy group at any base point vanishes for n ≥ 3). A quasi-inverse functor is explicitly given by means of a new 'homotopy double groupoid' construction for topological spaces.

We study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Moser’s deformation arguments for groupoids, we obtain several rigidity and normal form results.

By means of an auxiliary coarser topology we study certain order properties of inductive tensor algebras over nuclear LF-spaces. § I. Introduction, Notations and Statement of Results A locally convex *-algebra Jl\_ <^~\ is a locally convex space equipped with a separately continuous multiplication and a continuous involution The separate continuity of the multiplication means that for all the maps f^g X f and f*&fx Q from Jl [^] to J[ \_£~\ are continuous. It is also assumed that Jl has a unit satisfying 1 = 1*. An order structure is introduced on Jl h = {f^.Jl :f=f*} 9 the real subspace of hermitian elements of JK* by defining the cone of positive elements, Jl+, to be the closure of the set The cone <Jl^ determines a transitive and reflexive partial order on <JL hj and we write f>Q whenever f-g^Ji +. When this order is antisymmetric the cone <JL+ is called proper. Evidently <Jl + is proper if and only if J? + n-JU = {0}.

In this paper, we present a purely algebraic formulation of higher gauge theory and gauged sigma models based on the abstract theory of graded commutative algebras and their morphisms. The formulation incorporates naturally BRST symmetry and is also suitable for AKSZ type constructions. It is also shown that for a full-fledged BV formulation including ghost degrees of freedom, higher gauge and gauged sigma model fields must be viewed as internal smooth functions on the shifted tangent bundle of a space time manifold valued in a shifted L 8-algebroid encoding symmetry. The relationship to other formulations where the L 8-algebroid arises from a higher Lie groupoid by Lie differentiation is highlighted.

In this technical paper, we present a new formulation of higher parallel transport in strict higher gauge theory required for the rigorous construction of Wilson lines and surfaces. Our approach is based on an original notion of Lie crossed module cocycle and cocycle 1-and 2-gauge transformation with a non standard double category theoretic interpretation. We show its equivalence to earlier formulations.

Topological integrals appear frequently in Lagrangian field theories. On manifolds without boundary, they can be treated in the framework of (absolute) (co)homology using the formalism of Cheeger-Simons differential characters. String and D-brane theory involve field theoretic models on worldvolumes with boundary. On manifolds with boundary, the proper treatment of topological integrals requires a generalization of the usual differential topological set up and leads naturally to relative (co)homology and relative Cheeger-Simons differential characters. In this paper, we present a construction of relative Cheeger-Simons differential characters which is computable in principle and which contains the ordinary Cheeger-Simons differential characters as a particular case.

In the current paper we address the problem of classification of cocycles over an irrational rotation. We use the renormalization group approach. A cocycle means a Cr-mapping u: "IF-, SU(2, C) (r >= 2). We fix an irrational rotation z:T~T. Two cocycles u, v:T-~SU(2, ~) are considered equivalent (or cohomologous) if there is a continuous map h: T~SU(2, ~) such that u. h o z = h. v. By definition, our problem is to classify cocycles up to this equivalence relation. By introducing a suitable renormalization map we are able to define a notion of a fiber rotation number for a class of cocycles which are in the basin of the attractor of the renormalization map. The attractor itself is built of algebraic Anosov maps on I? 2. We present a number of results and conjecuters resulting from this approach. We show how this approach sheds some light upon the problem of classifying linear ODE with almost-periodic, skew-hermitian coefficient matrix.

Dana P. Williams raised in [Proc. Am. Math. Soc., Ser. B, 2016] the following question: Must a second countable, locally compact, transitive groupoid have open range map? This paper gives a negative answer to that question. Although a second countable, locally compact transitive groupoid G may fail to have open range map, we prove that we can replace its topology with a topology which is also second countable, locally compact, and with respect to which G is a topological groupoid whose range map is open. Moreover, the two topologies generate the same Borel structure and coincide on the fibres of G..

Let (M, F) be a foliated manifold. We prove that there is a canonical isomorphism between the complex of base-like forms Ω * b (M, F) of the foliation and the "De Rham complex" of the space of leaves M/F when considered as a "diffeological" quotient. Consequently, the two corresponding cohomology groups H * b (M, F) and H * (M/F) are isomorphic. As an application we give a quick proof for the topological invariance of the base-like cohomology of Lie foliations (and Riemannian foliations) on compact manifolds.

By relating the set of branch points B(f) of a Fredholm mapping f to linearized bifurcation, we show, among other things, that under mild local assumptions at a single point, the set B(f) is sufficiently large to separate the domain of the mapping. In the variational case, we will also provide estimates from below for the number of connected components of the complement of B(f).

We consider a sheaf of exterior algebras on a simplicial poset S and introduce a notion of homological characteristic function. Two objects are associated with these data: a graded sheaf I and a graded cosheaf p Π. When S is a homology manifold, we prove the isomorphism H n´1´p pS; Iq-H p pS; p Πq which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence E 2 p,q-H n´1´p pS; U n´1`q b Iq ñ H p`q pS; p Πq, where U˚is the local homology stack on S. This spectral sequence, in turn, extends Zeeman's spectral sequence in interpretation of McCrory. We apply these results to toric topology. Let X be an orientable manifold with locally standard action of a compact torus and acyclic proper faces of the orbit space. A principal torus bundle Y is associated with X and the orbit type filtration on X is covered by a topological filtration on Y. Then the second pages of homological spectral sequences associated with these two filtrations are isomorphic in many positions.

We consider a sheaf of exterior algebras on a simplicial poset S and introduce a notion of homological characteristic function. Two objects are associated with these data: a graded sheaf I and a graded cosheaf p Π. When S is a homology manifold, we prove the isomorphism H n´1´p pS; Iq-H p pS; p Πq which can be considered as an extension of the Poincare duality. In general, there is a spectral sequence E 2 p,q-H n´1´p pS; U n´1`q b Iq ñ H p`q pS; p Πq, where U˚is the local homology stack on S. This spectral sequence, in turn, extends Zeeman's spectral sequence in interpretation of McCrory. We apply these results to toric topology. Let X be an orientable manifold with locally standard action of a compact torus and acyclic proper faces of the orbit space. A principal torus bundle Y is associated with X and the orbit type filtration on X is covered by a topological filtration on Y. Then the second pages of homological spectral sequences associated with these two filtrations are isomorphic in many positions.

Dana P. Williams raised in [Proc. Am. Math. Soc., Ser. B, 2016] the following question: Must a second countable, locally compact, transitive groupoid have open range map? This paper gives a negative answer to that question. Although a second countable, locally compact transitive groupoid G may fail to have open range map, we prove that we can replace its topology with a topology which is also second countable, locally compact, and with respect to which G is a topological groupoid whose range map is open. Moreover, the two topologies generate the same Borel structure and coincide on the fibres of G..

In the present article, we define the concept of isoclinism for n-Hom-Lie algebras and investigate some of its properties. Also, we introduce the factor sets on n-Hom-Lie algebras. As a result, it is shown that the equivalency between isoclinism and isomorphism of two finite-dimensional n-Hom-Lie algebras just depends on whether one of them be regular.