Term Exact Sequence Research Articles

Partial generalized crossed products and a seven term exact sequence

Given a non-necessarily commutative unital ring R and a unital partial representation Θ of a group G into the Picard semigroup PicS(R) of the isomorphism classes of partially invertible R-bimodules, we construct an abelian group C(Θ/R) formed by the isomorphism classes of partial generalized crossed products related to Θ and identify an appropriate second partial cohomology group of G with a naturally defined subgroup C0(Θ/R) of C(Θ/R). Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings R⊆S with the same unity and a unital partial representation G→SR(S) of an arbitrary group G into the monoid SR(S) of the R-subbimodules of S. This generalizes the works by Kanzaki and Miyashita.

K-theoretic duality for extensions of Cuntz–Krieger algebras

We introduce the notion of K-theoretic duality for extensions of separable unital nuclear C⁎-algebras by using K-homology long exact sequence and cyclic six term exact sequence for K-theory groups of extensions. We then prove that the Toeplitz extension τA of a Cuntz-Krieger algebra OA is the K-theoretic dual of the Toeplitz extension τAt of the Cuntz-Krieger algebra OAt for the transposed matrix At of A. A pair of isomorphic Cuntz–Krieger algebras OA and OB does not necessarily yield the isomorphic pair of OAt and OBt. However, as an application, we may show that two Toeplitz algebras TA and TB are isomorphic as C⁎-algebras if and only if the Toeplitz algebras TAt and TBt of their transposed matrices are isomorphic.

A Wells type exact sequence for non-degenerate unitary solutions of the Yang–Baxter equation

Cycle sets are known to give non-degenerate unitary solutions of the Yang--Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions of linear cycle sets. This is an analogue of a similar exact sequence for group extensions known due to Wells. We also compare the exact sequence for linear cycle sets with that for their underlying abelian groups and discuss generalities on dynamical 2-cocycles.

Gysin sequences and SU(2)‐symmetries of C∗‐algebras

Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class.

Quandle cohomology, extensions and automorphisms

A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles.

Partial generalized crossed products and a seven-term exact sequence

For a partial Galois extension of commutative rings we give a seven term exact sequence which generalize the Chase-Harrison-Rosenberg sequence.

Abelian extensions and crossed modules of Hom-Lie algebras

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we study crossed modules of Hom-Lie algebras showing their equivalence with cat1-Hom-Lie algebras, and we introduce α-crossed modules to have a better understanding of the third cohomology group.

Exact sequences in homology of multiplicative Lie rings and a new version of Stallings' theorem

We show that the non-abelian tensor product of nilpotent, solvable and Engel multiplicative Lie rings is nilpotent, solvable and Engel, respectively. The six term exact sequence in homology of multiplicative Lie rings is obtained. We also prove a new version of Stallings' theorem.

A non-abelian exterior product and homology of Leibniz algebras

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.

Extensions and automorphisms of Lie algebras

Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].

An Explicit Seven-Term Exact Sequence for the Cohomology of a Lie Algebra Extension

We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra 𝔤, an ideal 𝔥 of 𝔤, and the quotient 𝔤/𝔥 with coefficients in a 𝔤-module. The existence of such a sequence follows from the Hochschild–Serre spectral sequence associated to the Lie algebra extension. However, some of the maps occurring in this induced sequence are not always explicitly known or easy to describe. In this article, we give alternative maps that yield an exact sequence of the same form, making use of the interpretations of the low-dimensional cohomology spaces in terms of derivations, extensions, etc. The maps are constructed using elementary methods. This alternative approach to the seven term exact sequence can certainly be useful, especially since we include straightforward cocycle descriptions of the constructed maps.

Crossed products by endomorphisms of C0(X)-algebras

In the first part of the paper, we develop a theory of crossed products of a $C^*$-algebra $A$ by an arbitrary (not necessarily extendible) endomorphism $\alpha:A\to A$. We consider relative crossed products $C^*(A,\alpha;J)$ where $J$ is an ideal in $A$, and describe up to Morita-Rieffel equivalence all gauge invariant ideals in $C^*(A,\alpha;J)$ and give six term exact sequences determining their $K$-theory. We also obtain certain criteria implying that all ideals in $C^*(A,\alpha;J)$ are gauge invariant, and that $C^*(A,\alpha;J)$ is purely infinite. In the second part, we consider a situation where $A$ is a $C_0(X)$-algebra and $\alpha$ is such that $\alpha(f a)=\Phi(f)\alpha(a)$, $a\in A$, $f\in C_0(X)$ where $\Phi$ is an endomorphism of $C_0(X)$. Pictorially speaking, $\alpha$ is a mixture of a topological dynamical system $(X,\varphi)$ dual to $(C_0(X),\Phi)$ and a continuous field of homomorphisms $\alpha_x$ between the fibers $A(x)$, $x\in X$, of the corresponding $C^*$-bundle. For systems described above, we establish efficient conditions for the uniqueness property, gauge-invariance of all ideals, and pure infiniteness of $C^*(A,\alpha;J)$. We apply these results to the case when $X=$Prim$(A)$ is a Hausdorff space. In particular, if the associated $C^*$-bundle is trivial, we obtain formulas for $K$-groups of all ideals in $C^*(A,\alpha;J)$. In this way, we constitute a large class of crossed products whose ideal structure and $K$-theory is completely described in terms of $(X,\varphi,\{\alpha_{x}\}_{x\in X};Y)$ where $Y$ is a closed subset of $X$.

Realizing the analytic surgery group of Higson and Roe geometrically, part I: the geometric model

© 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.

Exact sequences in the cohomology of a group extension

In [1] the authors constructed a seven term exact sequence in the cohomology of a group extension 1→N→G→Q→1 with coefficients in a G-module M. However, they were unable to establish the precise link between the maps in that sequence and the corresponding maps arising from the spectral sequence associated to the group extension and the G-module M. In this paper, we show that there is a close connection between [1] and the two earlier papers [6] and [7]. In particular, we show that the results in [6] and [7] entail that the maps of [1] other than the obvious inflation and restriction maps do correspond to the corresponding ones arising from the spectral sequence.

Groupoids and relations among Reidemeister and among Nielsen numbers

In this work we simplify, generalize and extend results and methods concerning the relationships between various Reidemeister numbers, and applications thereof to four different Nielsen theories (fixed point, ordinary coincidence, semi-index coincidence and root theory). We do this in two distinct contexts. The first context deals with the relationship between the Nielsen numbers of the maps involved, and those of representative lifts to regular covering spaces. We have a special interest in homogeneous spaces. The second context of our applications, namely to Nielsen theories of fibre-preserving maps, is rather curiously dual to the first. In both contexts, our results improve on those previously given.Our main tool is a collection of 8 term exact sequences (of groups and sets) whose inspiration and proof comes from the theory of fibrations of groupoids. We give a complete analysis of our sequences which yields previously unknown upper and lower bounds on the Reidemeister and Nielsen numbers we are wanting to compute (in one case it sharpens a previously known lower bound). When the upper and lower bounds coincide, generalizations of familiar formulas are forthcoming. The process also gives a uniform approach to proofs in both the underlying algebra (Reidemeister considerations) and to the two distinct contexts of our applications to the four Nielsen theories.New results include a new formula generalizing of the averaging formula in both the algebra and the geometry. We also generalize the original coincidence averaging formula for oriented infra-nilmanifolds to the smooth non-orientable category, and also to a pair of self-maps of smooth infra-solvmanifolds of type R. Other generalizations concern the finiteness of Reidemeister numbers. Finally we fill in proofs and details missing from previous work.

Cyclic cohomology after the excision theorem of Cuntz and Quillen

AbstractThe excision theorem of Cuntz and Quillen established the existence of a six term exact sequence in the bivariant periodic cyclic cohomology HP*(–,–) associated with an arbitrary algebra extension 0 →S→P→Q→ 0. This remarkable result enabled far reaching developments in the purely algebraic periodic cyclic cohomology. It also provided a new formalism that led to the creation of new versions of this theory for topological and bornological algebras. In this article we outline some of the developments that resulted from this breakthrough.

FREE CENTRE-BY-METABELIAN LIE RINGS

We study the free centre-by-metabelian Lie ring, that is, the free Lie ring with the property that the second derived ideal is contained in the centre. We exhibit explicit generating sets for the homogeneous and fine homogeneous components of the second derived ideal. Each of these components is a direct sum of a free abelian group and a (possibly trivial) elementary abelian $2$-group. Our generating sets are such that some of their elements generate the torsion subgroup while the remaining ones freely generate a free abelian group. A key ingredient of our approach is the determination of the dimensions of the corresponding homogeneous and fine homogeneous components of the free centre-by-metabelian Lie algebra over fields of characteristic other than $2$. For that we exploit a $6$-term exact sequence of modules over a polynomial ring that is originally defined over the integers, but turns into a sequence whose terms are projective modules after tensoring with a suitable field. Our results correct a partly erroneous theorem in the literature.

Baer Invariants and Cohomology of Precrossed Modules

In this paper we study Baer invariants of precrossed modules relative to the subcategory of crossed modules, following Frohlich and Furtado-Coelho’s general theory on Baer invariants in varieties of Ω-groups and Modi’s theory on higher dimensional Baer invariants. Several homological invariants of precrossed and crossed modules were defined in the last two decades. We show how to use Baer invariants in order to connect these various homology theories. First, we express the low-dimensional Baer invariants of precrossed modules in terms of a new non-abelian tensor product of a precrossed module. This expression is used to analyze the connection between the Baer invariants and the homological invariants of precrossed modules defined by Conduche and Ellis. Specifically we prove that the second homological invariant of Conduche and Ellis is in general a quotient of the first component of the Baer invariant we consider. The definition of classical Baer invariants is generalized using homological methods. These generalized Baer invariants of precrossed modules are applied to the construction of five term exact sequences connecting the generalized Baer invariants with the cohomology theory of crossed modules considered by Carrasco, Cegarra and R.-Grandjean and the cohomology theory of precrossed modules.

Chevalley-Eilenberg homology of crossed modules of Lie algebras in lower dimensions

In this paper we prove a five term exact sequence connecting in lower dimensions the Chevalley-Eilenberg homologies of the crossed module of Lie algebras (m,g,µ) and of the Lie algebra g/Im(µ). Moreover, a relationship between the ChevalleyEilenberg homology with coefficients and the homology of a crossed module of Lie algebras is established.

Unramified Brauer groups of finite and infinite groups

The Bogomolov multiplier is a group theoretical invariant isomorphic to the unramified Brauer group of a given quotient space. We derive a homological version of the Bogomolov multiplier, prove a Hopf-type formula, find a five term exact sequence corresponding to this invariant, and describe the role of the Bogomolov multiplier in the theory of central extensions. A new description of the Bogomolov multiplier of a nilpotent group of class two is obtained. We define the Bogomolov multiplier within $K$-theory and show that proving its triviality is equivalent to solving a long-standing problem posed by Bass. An algorithm for computing the Bogomolov multiplier is developed.