Split Extension Research Articles

COQUASITRIANGULAR STRUCTURES FOR EXTENSIONS OF HOPF ALGEBRAS. APPLICATIONS

AbstractLet A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A ⋉ H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A ⋉ H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite-dimensional quantum double Dλ(A, H) = A ⋈τH to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.

Actions in the Category of Precrossed Modules in Lie Algebras

For any object L in the category of precrossed modules in Lie algebras PXLie, we construct the object Act(L), which we call the actor of this object. From this construction, we derive the notions of action, center, semidirect product, derivation, commutator, and abelian precrossed module in PXLie. We show that the notion of action is equivalent to the one given in semi-abelian categories, and Act(L) is the split extension classifier for L. In the case of a crossed module in Lie algebras we show how to recover its actor in the category of crossed modules from its actor in the category of precrossed modules.

On cohomology theory for topological groups

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of the identity. We show that if G and A are locally compact and second countable, then the second cohomology group based on locally continuous measurable cochains as above parametrizes the collection of locally split extensions of G by A.

An equivariant generalization of the Miller splitting theorem

Let G be a compact Lie group. We build a tower of G-spectra over the suspension spectrum of the space of linear isometries from one G-representation to another. The stable cofibres of the maps running down the tower are certain interesting Thom spaces. We conjecture that this tower provides an equivariant extension of Miller's stable splitting of Stiefel manifolds. We provide a cohomological obstruction to the tower producing a splitting in most cases; however, this obstruction does not rule out a split tower in the case where the Miller splitting is possible. We claim that in this case we have a split tower which would then produce an equivariant version of the Miller splitting, we prove this claim in certain special cases though the general case remains a conjecture. To achieve these results we construct a variation of the functional calculus with useful homotopy-theoretic properties and explore the geometric links between certain equivariant Gysin maps and residue theory.

Intrinsic Schreier-Mac Lane extension theorem II: The case of action accessible categories

In (Bourn, 2008 [11]), the Schreier-Mac Lane extension theorem in the category Gp of groups, which describes a simply transitive group action on a certain indexed class of non-abelian extensions, was extended to any exact pointed protomodular category with split extension classifiers. We show here that the same scheme of proofs allows us to extend it to any exact action accessible category in the sense of Bourn and Janelidze (2009) [13], which includes the case of any category of interest in the sense of Orzech (1972) [21].

Finite quantum groups and quantum permutation groups

We give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups. We also give examples of finite quantum groups which are not quantum permutation groups: one such example occurs as a split abelian extension associated to the exact factorization S4=Z4S3 and has dimension 24. We show that, in fact, this is the smallest possible dimension that a non-quantum permutation group can have.

Remarks Concerning Lubotzky's Filtration

A discrete group which admits a faithful, finite dimensional, linear representation over a field $\mathbb F$ of characteristic zero is called linear. This note combines the natural structure of semi-direct products with work of A. Lubotzky on the existence of linear representations to develop a technique to give sufficient conditions to show that a semi-direct product is linear. Let $G$ denote a discrete group which is a semi-direct product given by a split extension $1 \to \pi \to G \to \Gamma \to 1$. This note defines an additional type of structure for this semi-direct product called a stable extension below. The main results are as follows: 1. If and $\Gamma$ are linear, and the extension is stable, then $G$ is also linear. Restrictions concerning this extension are necessary to guarantee that $G$ is linear as seen from properties of the Formanek-Procesi poison group. 2. If the action of $\Gamma$ on has a Galois-like property that it factors through the automorphisms of certain natural of groups over $\pi$ (to be defined below), then the associated extension is stable and thus $G$ is linear. 3. The condition of a stable extension also implies that $G$ admits filtration quotients which themselves give a natural structure of Lie algebra and which also imply earlier results of Kohno, and Falk-Randell on the Lie algebra attached to the descending central series associated to the fundamental groups of complex hyperplane complements. The methods here suggest that a possible technique for obtaining new linearity results may be to analyze automorphisms of towers of groups.

UNITS OF THE GROUP ALGEBRA 𝔽2κ(C2× D8)

The structure of the unit group of the group algebra of the group C2× D8over any field of characteristic 2 is established in terms of split extensions of cyclic groups.

A quasivariety lattice of torsion-free soluble groups

Let L q (qG) be a lattice of quasivarieties contained in a quasivariety generated by a group G. It is proved that if G is a torsion-free finitely generated group in $\mathcal{AB}$ pk , where p is a prime, p ≠ 2, and k ∈ N, which is a split extension of an Abelian group by a cyclic group, then the lattice L q (qG) is a finite chain.

Modular centralizer algebras corresponding to p-groups

We study the Loewy structure of the centralizer algebra k P Q for P a p-group with subgroup Q and k a field of characteristic p. Here k P Q is a special type of Hecke algebra. The main tool we employ is the decomposition k P Q = k C P ( Q ) ⋉ I of k P Q as a split extension of a nilpotent ideal I by the group algebra k C P ( Q ) . We compute the Loewy structure for several classes of groups, investigate the symmetry of the Loewy series, and give upper and lower bounds on the Loewy length of k P Q . Several of these results were discovered through the use of MAGMA, especially the general pattern for most of our computations. As a final application of the decomposition, we determine the representation type of k P Q .

The Structure of the Unit Group of the Group Algebra

AbstractLetRGdenote the group ring of the groupGover the ringR. Using an isomorphism betweenRGand a certain ring ofn×nmatrices in conjunction with other techniques, the structure of the unit group of the group algebra of the dihedral group of order 8 over any finite field of chracteristic 2 is determined in terms of split extensions of cyclic groups.

The structure of the unit group of the group algebra $\mathbb{F}_{2^k } A_4 $

The structure of the unit group of the group algebra of the group A4 over any finite field of characteristic 2 is established in terms of split extensions of cyclic groups.

Characteristic classes for cohomology of split Hopf algebra extensions

We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. These classes can be seen as obstructions for the vanishing of differentials in the spectral sequence. We give a decomposition theorem and interpret our results in the settings of group and Lie algebra extensions. As applications, we derive several results concerning the collapse of the (Lyndon–)Hochschild–Serre spectral sequence and the order of characteristic classes.

FINITE SELF-SIMILAR p-GROUPS WITH ABELIAN FIRST LEVEL STABILIZERS

We determine all finite p-groups that admit a faithful, self-similar action on the p-ary rooted tree such that the first level stabilizer is abelian. A group is in this class if and only if it is a split extension of an elementary abelian p-group by a cyclic group of order p.The proof is based on use of virtual endomorphisms. In this context the result says that if G is a finite p-group with abelian subgroup H of index p, then there exists a virtual endomorphism of G with trivial core and domain H if and only if G is a split extension of H and H is an elementary abelian p-group.

Stable ranks of split extensions of Banach algebras

In this paper, we estimate the stable ranks of a Banach algebra in terms of the stable ranks of its quotient algebra and ideal under the assumption that the quotient map splits. As an application, several results about the Bass and the connected stable ranks of nest algebras are obtained.

Lie Algebras with Abelian Centralizers

We classify all finite-dimensional Lie algebras over an algebraically closed field of characteristic 0, whose nonzero elements have abelian centralizers. These algebras are either simple or solvable, where the only simple such Lie algebra is [Formula: see text]. In the solvable case, they are either abelian or a one-dimensional split extension of an abelian Lie algebra.

A NEW EXAMPLE FOR MINIMALITY OF MONOIDS

By considering the split extension of a free abelian monoid having finite rank by a finite monogenic monoid, the main purposes of this paper are to present examples of efficient monoids and, also, minimal but inefficient monoids. Although results presented in this paper seem as in the branch of pure mathematics, they are actually related to applications of Combinatorial and Geometric Group-Semigroup Theory, especially computer science, network systems, cryptography and physics etc., which will not be handled here.

A Stochastic Model for Chemotaxis Based on the Ordered Extension of Pseudopods

Many amoeboid cells move by extending pseudopods. Here I present a new stochastic model for chemotaxis that is based on pseudopod extensions by Dictyostelium cells. In the absence of external cues, pseudopod extension is highly ordered with two types of pseudopods: de novo formation of a pseudopod at the cell body in random directions, and alternating right/left splitting of an existing pseudopod that leads to a persistent zig-zag trajectory. We measured the directional probabilities of the extension of splitting and de novo pseudopods in chemoattractant gradients with different steepness. Very shallow cAMP gradients can bias the direction of splitting pseudopods, but the bias is not perfect. Orientation of de novo pseudopods require much steeper cAMP gradients and can be more precise. These measured probabilities of pseudopod directions were used to obtain an analytical model for chemotaxis of cell populations. Measured chemotaxis of wild-type cells and mutants with specific defects in these stochastic pseudopod properties are similar to predictions of the model. These results show that combining splitting and de novo pseudopods is a very effective way for cells to obtain very high sensitivity to stable gradient and still be responsive to changes in the direction of the gradient.

The Structure of the Unit Group of the Group Algebra 𝔽3 k (C 3 × D 6)

The Structure of the Unit Group of the Group Algebra of the group C 3 × D 6 over a finite field of characteristic 3 is established in terms of split extensions of cyclic groups.

THE STRUCTURE OF THE UNIT GROUP OF THE GROUP ALGEBRA OF PAULI'S GROUP OVER ANY FIELD OF CHARACTERISTIC 2

The structure of the unit group of the group algebra of Pauli's group over any field of characteristic 2 is established in terms of split extensions of cyclic groups.