arXiv subjects

Manfred Hartl

Publications and source records attributed to Manfred Hartl.

15 recordsLinked to original sources

arXiv2025

Intrinsic tensor products and a Ganea-type extension of the five-term exact sequence

We define an intrinsic symmetric bi-right-exact (and for varieties, bi-cocontinuous) bilinear product on objects of a semi-abelian category, constructed as the cosmash product in the two-nilpotent reflection. When applied to abelian objects, this recovers classical tensor products in many cases. A recognition theorem states that any symmetric bi-cocontinuous bifunctor on an abelian variety of algebras is realised as the bilinear product in the variety of algebras over a suitable 2-nilpotent symmetric operad in the monoidal category of abelian groups. For abelian groups replaced with any commutative ring, the bilinear product of algebras over such an operad is associative as long as the only unary operations are given by multiplication with scalars, but not in general. This relies on a right-exactness theorem for cross-effects of bifunctors, and consequently for cosmash products. We develop basic properties, compare the bilinear product to the Brown-Loday non-abelian tensor product, and prove a categorical version of Ganea's six-term exact homology sequence. We further characterise abelian extensions via internal action cores, obtaining explicit descriptions of bilinear products in categories of representations; in particular, the bilinear product of the associated Beck modules generalises the classical tensor product of representations for groups and Lie algebras.

math.CT↗

arXiv2014

A formula for the second cohomology of two-step nilpotent groups

In [5], the notion of polynomial cocycles is used to give an expression for the second cohomology of T-groups with coefficients in a torsion-free nilpotent module. We make this expression concrete in the case of a T-group G of nilpotency class <=2 and coefficients in a trivial G-module, using a Lie algebra associated to the group. This approach allows us to construct explicit cocycles representing the elements of the second cohomology group.

math.GR↗

arXiv2012

Polynomial functors from Algebras over a set-operad and non-linear Mackey functors

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (fi nitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial functors from (finitely generated free) P-algebras to abelian groups and from (finitely generated free) groups to abelian groups are described explicitely by their cross-e ffects and maps relating them which satisfy a list of relations.

math.AT↗

arXiv2012

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 $\Lg$, an ideal $\Lh$ of $\Lg$ and the quotient $\Lg / \Lh$ with coefficients in a $\Lg$-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. Although we don't know whether the new maps coincide with the ones induced by the spectral sequence, the alternative sequence can certainly be useful, especially since we include straight-forward cocycle descriptions of the constructed maps.

math.RA↗

arXiv2012

The ternary commutator obstruction for internal crossed modules

In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their normalisations is measured by a ternary commutator, so that the Smith is Huq property itself can be characterised by the relation between the latter two commutators. This allows to show that the category of loops does not have the Smith is Huq property, which also implies that ternary commutators are generally not decomposable into nested binary ones. Thus, in contexts where Smith is Huq need not hold, we obtain a new description of internal categories, Beck modules and double central extensions, as well as a decomposition formula for the Smith commutator. The ternary commutator now also appears in the Hopf formula for the third homology with coefficients in the abelianisation functor.

math.CT↗

arXiv2012

A seven-term exact sequence for the cohomology of a group extension

In this paper, we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occuring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions).

math.GR↗

arXiv2010

On Fox and augmentation quotients of semidirect products

Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and some subgroup $T$. Let $I^n(G)$, $n\ge 1$, denote the powers of the augmentation ideal $I(G)$ of the group ring $\Z(G)$. Using homological methods the groups $Q_n(G,H) = I^{n-1}(G)I(H)/I^{n}(G)I(H)$, $H=G,N,T$, are functorially expressed in terms of enveloping algebras of certain Lie rings associated with $N$ and $T$, in the following cases: for $n\le 4$ and arbitrary $G,N,T$ (except from one direct summand of $Q_4(G,N)$), and for all $n\ge 2$ if certain filtration quotients of $N$ and $T$ are torsionfree.

math.GR↗

arXiv2010

On Fox quotients of arbitrary group algebras

For a group $G$, N-series $\cal G$ of $G$ and commutative ring $R$ let $I^n_{R,\cal G}(G)$, $n\ge 0$, denote the filtration of the group algebra $R(G)$ induced by $\cal G$, and $I_R(G)$ its augmentation ideal. For subgroups $H$ of $G$, left ideals $J$ of $R(H)$ and right $H$-submodules $M$ of $I_Z(G)$ the quotients $I_R(G)J/MJ$ are studied by homological methods, notably for $M= I_Z(G)I_Z(H)$, $I_Z(H)I_Z(G) + I_Z([H,G])Z(G)$ and $Z(G)I_Z(N) +I^n_{Z,\cal G}(G)$ with $N \lhd G$ where the group $I_R(G)J/MJ$ is completely determined for $n=2$. The groups $I^{n-1}_{Z,\cal G}(G)I_Z(H)/I^n_{Z,\cal G}(G)I_Z(H)$ are studied and explicitly computed for $n\le 3$ in terms of enveloping rings of certain graded Lie rings and of torsion products of abelian groups.

math.GR↗

arXiv2010

Internal object actions in homological categories

Let $G$ and $A$ be objects of a finitely cocomplete homological category $\mathbb C$. We define a notion of an (internal) action of $G$ of $A$ which is functorially equivalent with a point in $\mathbb C$ over $G$, i.e. a split extension in $\mathbb C$ with kernel $A$ and cokernel $G$. This notion and its study are based on a preliminary investigation of cross-effects of functors in a general categorical context. These also allow us to define higher categorical commutators. We show that any proper subobject of an object $E$ (i.e., a kernel of some map on $E$ in $\mathbb C$) admits a "conjugation" action of $E$, generalizing the conjugation action of $E$ on itself defined by Bourn and Janelidze. If $\mathbb C$ is semi-abelian, we show that for subobjects $X$, $Y$ of some object $A$, $X$ is proper in the supremum of $X$ and $Y$ if and only if $X$ is stable under the restriction to $Y$ of the conjugation action of $A$ on itself. This amounts to an elementary proof of Bourn and Janelidze's functorial equivalence between points over $G$ in $\mathbb C$ and algebras over a certain monad $\mathbb T_G$ on $\mathbb C$. The two axioms of such an algebra can be replaced by three others, in terms of cross-effects, two of which generalize the usual properties of an action of one group on another.

math.CT↗

arXiv2010

Morphismes quadratiques entre modules sur un anneau carré

We introduce the notions of a commutative square ring $R$ and of a quadratic map between modules over $R$, called $R$-quadratic map. This notion generalizes various notions of quadratic maps between algebraic objects in the literature. We construct a category of quadratic maps between $R$-modules and show that it is a right-quadratic category and has an internal Hom-functor. Along our way, we recall the notions of a general square ring $R$ and of a module over $R$, and discuss their elementary properties in some detail, adopting an operadic point of view. In particular, it turns out that the associated graded object of a square ring $R$ is a nilpotent operad of class 2, and the associated graded object of an $R$-module is an algebra over this operad, in a functorial way. This generalizes the well-known relation between groups and graded Lie algebras (in the case of nilpotency class 2). We also generalize some elementary notions from group theory to modules over square rings.

math.RA↗

arXiv2009

Quadratic functors on pointed categories

We study polynomial functors of degree 2, called quadratic, with values in the category of abelian groups $Ab$, and whose source category is an arbitrary category $\C$ with null object such that all objects are colimits of copies of a generating object $E$ which is small and regular projective; this includes all pointed algebraic varieties. More specifically, we are interested in such quadratic functors $F$ from $\C$ to $Ab$ which preserve filtered colimits and suitable coequalizers; one may take reflexive ones if $\C$ is Mal'cev and Barr exact. A functorial equivalence is established between such functors $F:\C\to Ab$ and certain minimal algebraic data which we call quadratic $\C$-modules: these involve the values on $E$ of the cross-effects of $F$ and certain structure maps generalizing the second Hopf invariant and the Whitehead product. Applying this general result to the case where $E$ is a cogroup these data take a particularly simple form. This application extends results of Baues and Pirashvili obtained for $\C$ being the category of groups or of modules over some ring; here quadratic $\C$-modules are equivalent with abelian square groups or quadratic $R$-modules, respectively.

math.AT↗

arXiv2009

A "working mathematician's" definition of semi-abelian categories

Semi-abelian and finitely cocomplete homological categories are characterized in terms of four resp. three simple axioms, in terms of the basic categorical notions introduced in the first few chapters of MacLane's classical book. As an immediate application we show that categories of diagrams in semi-abelian and similar categories are of the same type; in particular, the category of simplicial or Γ-objects in a semi-abelian category is semi-abelian.

math.CT↗

arXiv2008

Quadratic maps between groups

The notion of quadratic maps between arbitrary groups appeared at several places in the literature on quadratic algebra. Here a unified extensive treatment of their properties is given; the relation with a relative version of Passi's polynomial maps and groups of degree 2 is established and used to study the structure of the latter.

math.GR↗

arXiv2007

The relative second Fox and third dimension subgroup of arbitrary groups

Let $I_R(G)$ denote the augmentation ideal of the group algebra $R(G)$ of a group $G$ with coefficients in a commutative ring $R$. We give a complete description of the third relative dimension subgroup $G\cap(1+I_R(K)I_R(G)+I^3_R(G))$ and the second relative Fox subgroup $G\cap(1+I_R(K)I_R(H)+I^2_R(G)I_R(H))$ for any subgroups $K$ and $H$ of $G$.

math.GR↗

arXiv2007

On the second cohomology of semidirect products

Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and a subgroup $T$, and let $M$ be a $G$-module with not necessarily trivial $G$-action. Then we embed the simultaneous restriction map $res=(res^G_N,res^G_T)^t : H^2(G,M) \to H^2(N,M)^T \times H^2(T,M)$ into a natural five term exact sequence consisting of one and two-dimensional cohomology groups of the factors $N$ and $T$. The elements of $H^2(G,M)$ are represented in terms of group extensions of $G$ by $M$ constructed from extensions of $N$ and $T$.

math.GR↗