Local analytic structure in certain commutative topological algebras

The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the k \mathfrak {k} -finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.

Norming the Algebra of Formal Power Series in n Indeterminates

A representation of solution of stochastic differential equations

Potentials for some tensor algebras

It is well known that the complete norm topology on a Banach algebra is not unique in general, though semisimplicity is sufficient (but not necessary) for uniqueness. In this note we consider a class of topological algebras of formal power series which have unique Fréchet space topology. The structure of these algebras in the Banach algebra case will be considered in a later paper.

On the generalized spectra of topological algebras

On functional representations of topological algebras

In [ 2 ] Bieri and Strebel introduced a geometric invariant for finitely generated abstract metabelian groups that determines which groups are finitely presented. For a valuable survey of their results, see [ 6 ]; we recall the definition briefly in Section 4. We shall introduce a similar invariant for pro- p groups. Let [ ] be the algebraic closure of [ ] p and U be the formal power series algebra [ ][lobrk ] T [robrk ], with group of units U × . Let Q be a finitely generated abelian pro- p group. We write ℤ p [lobrk ] Q [robrk ] for the completed group algebra of Q over ℤ p . Let T ( Q ) be the abelian group Hom( Q , U × ) of continuous homomorphisms from Q to U × . We write 1 for the trivial homomorphism. Each v ∈ T ( Q ) extends to a unique continuous algebra homomorphism vmacr; from ℤ p [lobrk ] Q [robrk ] to U .

Using quantum differential operators, we construct a super representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) -submodule isomorphic to the regular representation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) . In this way, we obtain a presentation of U υ ( g l m | n ) U_{\boldsymbol {\upsilon }}(\mathfrak {gl}_{m|n}) by a basis together with explicit multiplication formulas of the basis elements by generators.

Chapter 9 - Concrete Categories

The most important examples of coarse spaces arise from proper metrics on spaces or from metrisable compactifications. In both cases, the bounded subsets are precisely the relatively compact ones. On the other hand, one of the characteristic properties a coarse map required to have is formulated by the bounded subsets. Motivated from the above two facts and in order to be able to develop the notion of locally properness which has been introduced by Viet-Trung Luu for discrete spaces, we introduce a new notion of compatibility between the coarse structure and the topology when a coarse space also carries a topology. The spirit of this is to let the local part of the theory govern by the topology of spaces. Having established this we are able to introduce some basic notions such as pull-back and push-forward coarse structures, and products and coproducts of coarse spaces which also are carrying a topology with the required compatibility between the topology and the coarse structure. We use the notion of basepoint projection introduced by Paul Mitchener and Thomas Schick to develop a notion of pointed coarse spaces which leads us to a new notion of collapsing from a coarse point of view. This enables us to introduce some essential notions such as coarse quotient spaces, coarse spaces obtained by coarse collapsing and coarse spaces obtained by coarse attaching via a coarse map. We introduce a new notion which in a sense is the analogue for coarse geometry of locally compactness for topology and investigate some of its properties. Then, we develop basic notions in the coarse homotopy theory including some constructions such as coarse smash product, coarse suspensions and coarse mapping cone needed to develop coarse homotopy theory and we prove some of their properties. The coarse homotopy groups are introduced next and then we develop an exact sequence of coarse homotopy groups. We also give a more complete exposition on the coarse CW-complexes broad enough to provide an appropriate foundation in order to carry over more of the tools from algebraic topology into coarse geometry. Having established that we prove a coarse version of the theorem of J.H.C. Whitehead which allows certain aspects of coarse homotopy classification. Next, we pursue a big step forward and calculate the coarse homotopy groups of the standard coarse spheres. More precisely, we prove $\pi^{crs}_k(S^n_{\preal}) \cong \pi_k(S^n)$ for all $k \leq n$. This has some intense applications, namely, this enables us to carry over some important theorems from algebraic topology concerning the coarse homotopy groups of coarse CW-complexes when they are $\preal$-spaces. Then, as a one result of these theorems, we introduce the coarse Eilenberg-Maclane spaces.

Let A and B be two algebras over the complex field C. A sequence {Fn}0g"gra (resp.{F"}0gn<a,) of linear operators from A into B is a higher derivation of rank m (resp.infinite rank) if, for each n=0,1, , m (resp.n=0,1, -) and any x,y eA, We consider the continuity of such {F"} when A and B are commutative topological algebras with complete metrizable topology.Some applications are given to algebras of formal power series.

Purpose of the article: development of a methodology for the application of methods for analyzing big data based on topological constructions in relation to behavioral analytics systems to ensure corporate and cyber-physical security. Method: the technique is based on the algebraic theory of persistent homology. Along with algebraic topology, embedology (Takens-Mane embedding theory) and the theory of metric spaces are used. Result: the necessary concepts of algebraic topology are given, which underlie the analysis of user / entity behavior profiles: Vietoris-Rips simplicial complex, filtering by a set of cloud points, homology groups, persistence modules, topological characteristics and dependencies. At the first stage of the technique, the time series that describe the time-varying behavior of the user / entity are transformed into a cloud of points in the topological space. For this transformation, the methods of the Takens-Mane embedding theory and the algorithm of the method of false neighbors are used. At the subsequent stages of the methodology for the base and current point clouds, topological dependencies, diagrams (persistence, bar codes) characterizing the base and current behavior profiles, respectively, are built. At the final stage, the deviation of the current behavior profile from the baseline is revealed. To estimate the deviation, the Wasserstein, Chebyshev, bottleneck metrics and scaling based on the generalized Harrington desirability function are used. The results of practical testing of the proposed method of applying topological algorithms to the data of the monitoring system for the work of corporate network users with information resources are presented

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the setting of homotopy type theory, each type can be interpreted as a space. Moreover, given an element of a type, i.e. a point in the corresponding space, one can define another type which encodes the space of loops based at this point. In particular, when the type we started with is a groupoid, this loop space is always a group. Conversely, to every group we can associate a type (more precisely, a pointed connected groupoid) whose loop space is this group: this operation is called delooping. The generic procedures for constructing such deloopings of groups (based on torsors, or on descriptions of Eilenberg-MacLane spaces as higher inductive types) are unfortunately equipped with elimination principles which do not directly allow eliminating to untruncated types, and are thus difficult to work with in practice. Here, we construct deloopings of the cyclic groups Zm which are cellular, and thus do not suffer from this shortcoming. In order to do so, we provide type-theoretic implementations of lens spaces, which constitute an important family of spaces in algebraic topology. Our definition is based on the computation of an iterative join of suitable maps from the circle to an arbitrary delooping of Zm. In some sense, this work generalizes the construction of real projective spaces by Buchholtz and Rijke, which handles the case m = 2, although the general setting requires more involved tools. Finally, we use this construction to also provide cellular descriptions of dihedral groups, and explain how we can hope to use those to compute the cohomology and higher actions of such groups.