Search

In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Kotov and Strobl's construction of an $L_\infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_\infty$-algebras, and a functor further to that of $L_\infty$-algebras., 32 pages, comments are welcome

We characterize the role played by memory when linear time-invariant systems interact. This study is of interest as the phenomenon that occurs in this setting is arguably the same phenomenon that governs some cascading failure and contagion effects in interconnected systems. We aim to later extend solutions presented in this paper to problems in other desired settings. The characterization relies on basic methods in homological algebra, and is reminiscent of the rank-nullity theorem of linear algebra. Interconnection of systems is first expressed as an exact sequence, then loss of memory causes a loss of exactness, and finally exactness is recovered through specific algebraic invariants of the systems that encode the role of memory. We thus introduce a new invariant, termed lag , of linear time-invariant systems and characterize the role of memory in terms of the lag. We discuss properties of the lag, and prove several results regarding the characterization.

We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra., Comment: Typos corrected To appear in Proc. Amer. Math. Soc

This paper begins with the basic concepts of soft module. Later, we introduce inverse system in the category of intutionistic fuzzy soft modules and prove that its limit exists in this category. Generally, limit of inverse system of exact sequences of intutionistic fuzzy soft modules is not exact. Then we define the notion which is first derived functor of the inverse limit functor. Finally, using methods of homology algebra, we prove that the inverse system limit of exact sequence of intutionistic fuzzy soft modules is exact.

In the framework of Banach differential geometry we investigate the fiber-wise linear Poisson structures as well as the Lie groupoid and Lie algebroid structures which are defined in the canonical way by the structure of a W ∗ -algebra (von Neumann algebra) M . The main role in this theory is played by the complex Banach–Lie groupoid G ( M ) ⇉ L ( M ) of partially invertible elements of M over the lattice L ( M ) of orthogonal projections of M . The Atiyah sequence and the predual Atiyah sequence corresponding to this groupoid are investigated from the point of view of Banach Poisson geometry. In particular we show that the predual Atiyah sequence fits in a short exact sequence of complex Banach sub-Poisson V B -groupoids with G ( M ) ⇉ L ( M ) as the side groupoid.

We introduce a non-abelian exterior product of two crossed modules of Leibniz algebras and investigate its relation to the low-dimensional Leibniz homology. Later this non-abelian exterior product is applied to the construction of an eight term exact sequence in Leibniz homology. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the second Lie and Leibniz homologies of a Lie algebra.

We define a “compactification” of the representation ring of the linear group scheme over Specℤ, in the spirit of Arakelov geometry. We show that it is a λ-ring which is canonically isomorphic to a localized polynomial ring and that it plays a universal role with respect to natural operations on theK 0-theory of hermitian bundles defined by Gillet-Soule. As a byproduct, we prove that the natural pre-λ-ring structure of theK 0-theory of hermitian bundles is a λ-ring structure. This last result plays a key role in the proof of the main results of [18] and [12].

We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector spaces. Motivated by this example we define the "Homology functor" (we prove it is homological) for any co-Frobenius algebra, with coefficients in $H$-comodules, that recover usual homology of a complex when $H=k[{\mathbb Z}]\#k[x]/x^2$. Another easy example of co-Frobenius Hopf algebra gives the category of "mixed complexes" and we see (by computing an example) that this homology theory differs from cyclic homology, although there exists a long exact sequence analogous to the SBI-sequence. Finally, because we have a tensor triangulated category, its $K_0$ is a ring, and we prove a "last part of a localization exact sequence" for $K_0$ that allows us to compute -or describe- $K_0$ of some family of examples, giving light of what kind of rings can be categorified using this techniques., Comment: reference added, characterization of $\mathcal H_0$ improved, typos corrected

In this paper, we present a new module for the Kenzo system which makes it possible to compute the effective homotopy of the total space of a fibration, using the well-known long exact sequence of homotopy of a fibration defined by Jean-Pierre Serre. The programs are written in Common Lisp and require the implementation of new classes and functions corresponding to the definitions of setoid group (SG) and effective setoid group (ESG). Moreover, we have included a new module for working with finitely generated abelian groups, choosing the representation of a free presentation by means of a matrix in canonical form. These tools are then used to implement the long exact homotopy sequence of a fibration. We illustrate with examples some applications of our results.

© 2015, Tbilisi Centre for Mathematical Sciences. We construct a geometric analog of the analytic surgery group of Higson and Roe for the assembly mapping for free actions of a group with values in a Banach algebra completion of the group algebra. We prove that the geometrically defined group, in analogy with the analytic surgery group, fits into a six term exact sequence with the assembly mapping and also discuss mappings with domain the geometric group. In particular, given two finite dimensional unitary representations of the same rank, we define a map in the spirit of η-type invariants from the geometric group (with respect to assembly for the full group C ∗ -algebra) to the real numbers.

This is a narrative rather than a survey of the research relevant to the subject of the title, done basically at IMECC, Unicamp, from 1982 to 2005. Many people, including students, as well as the first author, contributed. The second author compiled the material in his master’s dissertation (Cerna, On the geometry and topology of the commutator of unit quaternions. Master’s Dissertation, IMECC, Unicamp, 2016) and he is not responsible for imprecisions, for forgetting to give credit where due, etc. We consider known a little basic homotopy, like the exact sequence of a fibration, a few basic homotopy groups of spheres, and the Bott periodicity for Spin, U, and Sp, and also basic definitions of Riemannian geometry.

We investigate the relative and Tate cohomology theories with respect to Ding modules and complexes, consider their relations with classical and Gorenstein cohomology theories. As an application, the Avramov-Martsinkovsky type exact sequence of Ding modules is obtained.

Let F be a p-adic field. In this paper, we continue the work of the first author and give a new realization of the tame local Langlands correspondence for PGSp(4, F) that is analogous to the construction of the local Langlands correspondence for real groups.

We introduce the notion of pointed category and the related definition of exact sequence. We show how the four basic elementary algebraic facts described in the introduction can be translated in purely categorical terms, that is in a diagrammatic formulation.

A subgroup \(H\) of an Abelian group \(G\) is called fully inert if \((\phi H + H)/H\) is finite for every \(\phi \in \mathrm{End}(G)\). Fully inert subgroups of free Abelian groups are characterized. It is proved that \(H\) is fully inert in the free group \(G\) if and only if it is commensurable with \(n G\) for some \(n \ge 0\), that is, \((H + nG)/H\) and \((H + nG)/nG\) are both finite. From this fact we derive a more structural characterization of fully inert subgroups \(H\) of free groups \(G\), in terms of the Ulm–Kaplansky invariants of \(G/H\) and the Hill–Megibben invariants of the exact sequence \(0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0\).

In the theory of canonical induction formulae for Mackey functors, Boltje [4] demonstrated that the plus constructions, together with the mark morphism, are useful for the study of canonical versions of induction theorems analogous to those in representation theory of finite groups. In this paper, we present a short exact sequence for the plus constructions derived from Cauchy–Frobenius lemma, and apply it to the proof of Boltje's integrality result for canonical induction formulae. The methods appearing in Boltje's theory, combined with the Dress construction for Mackey functors, are applicable to induction theorems on representations of the twisted quantum double of a finite group. As a sequel to such a research, we describe canonical versions of two induction theorems whose origins are Artin's induction theorem and Brauer's induction theorem on C -characters of a finite group.

We use Chern–Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every \(\bar{\partial }\partial \)-exact real form of the type \((k,k)\) on an \(n\)-dimensional complex manifold \(X\) arises as a difference of the Chern character forms of trivial Hermitian vector bundles with canonical connections, and that modulo \(\mathrm {Im}\,\partial +\mathrm {Im}\,\bar{\partial }\) every real form of type \((k,k)\), \(k

We compute the K-groups of the -algebra of bounded operators generated by the Boutet de Monvel operators with classical SG-symbols of order (0,0) and type 0 on , as defined by Schrohe, Kapanadze and Schulze. In order to adapt the techniques used in Melo, Nest, Schick and Schrohe's work on the K-theory of Boutet de Monvel's algebra on compact manifolds, we regard the symbols as functions defined on the radial compactifications of and . This allows us to give useful descriptions of the kernel and the image of the continuous extension of the boundary principal symbol map, which defines a -algebra homomorphism. We are then able to compute the K-groups of the algebra using the standard K-theory six-term cyclic exact sequence associated to that homomorphism.

In this paper, we introduce the concept of capability for crossed modules of Lie algebras, which is a generalization of capability in Lie algebras and groups. By using a special central ideal of a crossed module, we give a sufficient condition for the capability of a crossed module of Lie algebras. Also, we will extend the five-term exact sequence on homology of crossed modules of Lie algebras one term further and study the connection between the capability of crossed modules and this sequence. Finally, we study the relation between the capability and the center of a cover of a crossed module.

In a semi-abelian category, we give a categorical construction of the push forward of an internal pre-crossed module, generalizing the pushout of a short exact sequence in abelian categories. The main properties of the push forward are discussed. A simplified version is given for action accessible categories, providing examples in the categories of rings and Lie algebras. We show that push forwards can be used to obtain the crossed module version of the comprehensive factorization for internal groupoids.

We study the following rigidity problem in symplectic geometry: can one displace a Lagrangian submanifold from a hypersurface? We relate this to the Arnold Chord Conjecture, and introduce a refined question about the existence of relative leaf-wise intersection points, which are the Lagrangian-theoretic analogue of the notion of leaf-wise intersection points defined by Moser (Acta. Math. 141(1–2):17–34, 1978). Our tool is Lagrangian Rabinowitz Floer homology, which we define first for Liouville domains and exact Lagrangian submanifolds with Legendrian boundary. We then extend this to the ‘virtually contact’ setting. By means of an Abbondandolo–Schwarz short exact sequence we compute the Lagrangian Rabinowitz Floer homology of certain regular level sets of Tonelli Hamiltonians of sufficiently high energy in twisted cotangent bundles, where the Lagrangians are conormal bundles. We deduce that in this situation a generic Hamiltonian diffeomorphism has infinitely many relative leaf-wise intersection points., Geometriae Dedicata, 171 (1), ISSN:0046-5755, ISSN:1572-9168

Let I be a split radical ideal of a ring R. In this article, the exact sequence 1 → K 2(R, I) → U R (I) → V(R, I) → 1 is given by using the method of extension of groups, where U R (I) is determined by generators and relations. The results of Maazen and Stienstra on the presentation for relative K 2 group of split radical pairs are extended and amplified.

In this monograph, we extend S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore we establish an explicit description of an isomorphism by A. Neeman and V. Retakh, which links $\mathrm{Ext}$-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, we show that our construction behaves well with respect to structure preserving functors between exact monoidal categories. We use our main result to conclude, that both the Lie bracket and the squaring map in Hochschild cohomology are invariants under Morita equivalence. For quasi-triangular bialgebras, we further determine a significant part of the Lie bracket's kernel, and thereby prove a conjecture by L. Menichi. Along the way, we introduce $n$-extension closed and entirely extension closed subcategories of abelian categories, and study some of their properties., Comment: Modified version of author's PhD thesis (Bielefeld University, December 2013). 159 pages. --- Final version, to appear in "Memoirs of the American Mathematical Society". Corrected a mistake in Section 6 (the main results are not affected) and made minor changes according to the suggestions of the referees

This chapter introduces the C\(^*\)-algebras of bulk, half-space and boundary observables, together with their canonical representations which generate the physical models for topological insulators presented in Chap. 2. Then the exact sequence connecting these algebras is discussed. In particular, it is shown to be isomorphic to the Pimsner-Voiculescu exact sequence. This chapter also introduces the non-commutative analysis tools and the smooth sub-algebras to be used in the remainder of the book.

We show the existence of group-theoretic sections of the “etale-by-geometrically abelian” quotient of the arithmetic fundamental group of hyperbolic curves over \(p\)-adic local fields relative to a proper and flat model which are non-geometric, i.e., which do not arise from rational points.

We classify certain extensions of A𝕋-algebras using the six-term exact sequence in K-theory together with the Elliott invariants of the ideal and quotient. We also give certain necessary and sufficient conditions for such extension algebras being A𝕋-algebras.

The problem of the representation of an action of a Lie–Rinehart algebra on a Lie 𝖠-algebra by means of a homomorphism of Lie–Rinehart algebras is studied. An eight-term exact sequence associated to an epimorphism of Lie–Rinehart algebras for the cohomology of Lie–Rinehart algebras developed by Casas, Ladra and Pirashvili is obtained. This sequence is applied to study the obstruction theory of Lie–Rinehart algebra extensions.

Let R be a Noetherian commutative ring with dim RD d and let l be an ideal of R. For an integer n such that 2n dC 3, we define a relative Euler class group E n .R; lI R/. Using this group, in analogy to homology sequence of the K0-group, we construct an exact sequence E n .R; lI R/ E. p2/ ! E n .RI R/ E./ ! E n .R= lI R= l/; called the homology sequence of the Euler class group. The excision theorem in K-theory has a corresponding theorem for the Euler class group. An application is that for polynomial and Laurent polynomial rings, we get short split exact sequences 0! E n .RTtU;.t/I RTtU/ E. p2/ ! E n .RTtUI RTtU/ E./ ! E n .RI R/! 0

For a field $F$ and an integer $d\geq 1$, we consider the universal associative $F$-algebra $A$ generated by two sets of $d+1$ mutually orthogonal idempotents. We display four bases for the $F$-vector space $A$ that we find attractive. We determine how these bases are related to each other. We describe how the multiplication in $A$ looks with respect to our bases. Using our bases we obtain an infinite nested sequence of 2-sided ideals for $A$. Using our bases we obtain an infinite exact sequence involving a certain $F$-linear map $\partial: A \to A$. We obtain several results concerning the kernel of $\partial$; for instance this kernel is a subalgebra of $A$ that is free of rank $d$., 20 pages

Let □ = l 1 (ℬ) be the semigroup algebra of ℬ, the bicyclic semigroup. We give a resolution of l ∞ (ℬ) which simplifies the computation of the cohomology of l 1 (ℬ) dual bimodules. We apply this to the dual module l ∞ (ℬ) and show that the simplicial cohomology groups ℋ n (□, □′) vanish for n ≥ 2. Using the Connes–Tzygan exact sequence, these results are used to show that the cyclic cohomology groups ℋ□ n (□, □′) vanish when n is odd and are one-dimensional when n is even (n ≥ 2).

In this work, connections between partially continuous and singular Alexander-Spanier cohomologies and usual and continuous homologies are found. Conditions are obtained under which, for given partially continuous cohomologies, the Meyer-Vietoris sequence and suspension axiom hold.

The Witt group of a hyperelliptic curve over a field of characteristic different from two was determined by Parimala and Sujatha. Here, analogous results are obtained for the unramified Witt group in characteristic two using the analogue of Milnor's exact sequence for the Witt group of rational function fields developed earlier by the authors. In the elliptic case, if F is perfect and points of order two are rational, a generator and relation structure for the Witt group is given.

We prove a decomposition lemma for elementary tensors and present an analogous result of a (DN)-(Ω) splitting theorem for the theory of tensoring an exact sequence of Frechet spaces with a (DF)-space. This result is free of nuclearity and hilbertisability assumptions and can be applied in a natural way to vector-valued linear partial differential operators.

We introduce the Picard group of corings. We extend the well-known exact sequence from algebras and coalgebras over fields to corings. We extend the Aut-Pic property to corings and we give some new examples of corings having this property. Finally, we give the corresponding exact sequences for the category of entwined modules over an entwining structure, the category of Doi-Koppinen-Hopf modules over a Doi-Koppinen structure, and the category of graded modules by a $G$-set, where $G$ is a group.

We introduce and examine a special class of cube complexes. We show that special cube-complexes virtually admit local isometries to the standard 2-complexes of naturally associated right-angled Artin groups. Consequently, special cube-complexes have linear fundamental groups. In the word-hyperbolic case, we prove the separability of quasiconvex subgroups of fundamental groups of special cube-complexes. Finally, we give a linear variant of Rips’s short exact sequence.

The Quarterly Journal of Mathematics, 58 (4), ISSN:0033-5606

Let k be a field, H a Hopf k-algebra with bijective antipode, A a right H-comodule algebra and C a Hopf algebra with bijective antipode which is also a right H-module coalgebra. Under some appropriate assumptions, and assuming that the set of grouplike elements G(A ⊗ C) of the coring A ⊗ C is a group, we show how to calculate, via an exact sequence, the Picard group of the subring of coinvariants in terms of the Picard group of A and various subgroups of G(A ⊗ C).

We show that there is an Avramov–Martsinkovsky type exact sequence with , Gtor, and Tor. We prove that if R is a Gorenstein ring, then the modules , n ≥ 1 can be computed using either a complete resolution of MR or using a complete resolution of RN. We show that over a Gorenstein ring a left R-module N is Gorenstein flat if and only if . We also show that over commutative Gorenstein rings the modules can be computed by the combined use of a flat resolution and a Gorenstein flat resolution of M.

Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the setting of rational homotopy theory via differential graded Lie algebras, we identify the long exact sequence induced on rational homotopy groups by w in terms of (generalized) derivation spaces and adjoint maps. As a consequence, we obtain a unified description of the rational homotopy theory of function spaces, at the level of rational homotopy groups, in terms of derivations of Quillen models and adjoints. In particular, as a natural extension of a result of Tanre, we identify the rationalization of the evaluation subgroups of a map f: X -> Y in this setting. As applications, we consider a generalization of a question of Gottlieb, within the context of rational homotopy theory. We also identify the rationalization of the G-sequence of f and make explicit computations of the homology of this sequence. In a separate result of independent interest, we give an explicit Quillen minimal model of a product AxX, in the case in which A is a rational co-H-space., 33 pages

We prove that if E and F are large ideals of B(G) for which the associated coaction functors are exact, then the same is true for the intersection of E and F. We also give an example of a coaction functor whose restriction to the maximal coactions does not come from any large ideal., minor revision

In this chapter and the following one, we shall describe contrapuntal rules in terms of singular homology of gestures. Chapter 9 describes the mathematical theory of gestures, whereas Chapter 10 develops singular homology for gestures and applies it to restate contrapuntal rules in terms of homological dimension.

In this article we introduce and study the first cohomology categorical group ℋ1(,) of a categorical group  with coefficients in a braided categorical group  provided of a coherent -action. The fundamental exact sequence connecting ℋ0 and ℋ1 in this context is then established.

We define noncommutative Leibniz–Poisson algebras (NLP-algebras) and their dual algebras, construct free objects in the corresponding categories, and relate free objects on one element set with the set of planar binary rooted trees. We define actions, crossed modules, representations, extensions, and cohomology of NLP-algebras and study their properties; in particular we derive a long exact sequence, relating this cohomology with Hochschild and Leibniz cohomologies, and with the cohomology of algebras with bracket defined in Casas and Pirashvili (2006).