One-Relator Maximal Pro-p Galois Groups and the Koszulity Conjectures

Let p be a prime number and let ${\mathbb{K}}$ be a field containing a root of 1 of order p. If the absolute Galois group $G_{\mathbb{K}}$ satisfies $\dim\, H^1(G_{\mathbb{K}},\mathbb{F}_p)\lt\infty$ and $\dim\, H^{\,2}(G_{\mathbb{K}},\mathbb{F}_p)=1$, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for ${\mathbb{K}}$. Also, under the above hypothesis, we show that the $\mathbb{F}_p$-cohomology algebra of $G_{\mathbb{K}}$ is the quadratic dual of the graded algebra ${\rm gr}_\bullet\mathbb{F}_p[G_{\mathbb{K}}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{\mathbb{K}}]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s elementary type conjecture.

The quotient of a Kauffman bracket skein algebra by the square of an augmentation ideal

We give an explicit basis [Formula: see text] of the quotient of the Kauffman bracket skein algebra [Formula: see text] on a surface [Formula: see text] by the square of an augmentation ideal. Moreover, we construct an embedding of the mapping class group of a compact connected oriented surface of genus [Formula: see text] into the Kauffman bracket skein algebra on the surface completed with respect to a filtration coming from the augmentation ideal.

Dimension quotients of metabelian Lie rings

For a Lie ring [Formula: see text] over the ring of integers, we compare its lower central series [Formula: see text] and its dimension series [Formula: see text] defined by setting [Formula: see text], where [Formula: see text] is the augmentation ideal of the universal enveloping algebra of [Formula: see text]. While [Formula: see text] for all [Formula: see text], the two series can differ. In this paper, it is proved that if [Formula: see text] is a metabelian Lie ring, then [Formula: see text], and [Formula: see text], for all [Formula: see text].

Units of group rings, the Bogomolov multiplier and the fake degree conjecture

Let π be a finite p-group and ${\mathbb{F}_{q}}$ a finite field with q = pn elements. Denote by $\I_{\mathbb{F}_{q}}$ the augmentation ideal of the group ring ${\mathbb{F}_{q}}$[π]. We have found a surprising relation between the abelianization of 1 + $\I_{\mathbb{F}_{q}}$, the Bogomolov multiplier B0(π) of π and the number of conjugacy classes k(π) of π: $$ \left | (1+\I_{\Fq})_{\ab} \right |=q^{\kk(\pi)-1}|\!\B_0(\pi)|. In particular, if π is a finite p-group with a non-trivial Bogomolov multiplier, then 1 + $\I_{\mathbb{F}_{q}}$ is a counterexample to the fake degree conjecture proposed by M. Isaacs.

Bott periodicity in the Hit Problem

In this short paper, we use Robert Bruner's $\cal{A}$(1)-resolution of $P = {\mathbb{F}_2[t]$ to shed light on the Hit Problem. In particular, the reduced syzygies Pn of P occur as direct summands of $\widetilde{P}^{\otimes n}$, where $\widetilde{P}$ is the augmentation ideal of the map $P \to \mathbb{F}_2$. The complement of Pn in $\widetilde{P}^{\otimes n}$ is free, and the modules Pn exhibit a type of “Bott periodicity” of period 4: Pn+4 = Σ8Pn. These facts taken together allow one to analyse the module of indecomposables in $\widetilde{P}^{\otimes n}$, that is, to say something about the “$\cal{A}$(1)-hit Problem”. Our study is essentially in two parts: first, we expound on the approach to the Hit Problem begun by William Singer, in which we compare images of Steenrod squares to certain kernels of squares. Using this approach, the author discovered a nontrivial element in bidegree (5, 9) that is neither $\cal{A}$(1)-hit nor in kerSq1 + kerSq3. Such an element is extremely rare, but Bruner's result shows clearly why these elements exist and detects them in full generality; second, we describe the graded ${\mathbb{F}_2$-space of $\cal{A}$(1)-hit elements of $\widetilde{P}^{\otimes n}$ by determining its Hilbert series.

COMMUTATOR IDENTITIES ON SYMMETRIC ELEMENTS OF GROUP ALGEBRAS

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈ log 2 t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

Milnor-Witt K-Theory and Tensor Powers of the Augmentation Ideal of ℤ[F×]

Let F be a field of characteristic not equal to 2. We describe the relation between the non-negative dimensional Milnor-Witt K-theory of F and the tensor algebra over the group ring ℤ[F×] of the augmentation ideal [Formula: see text]. In the process, we clarify the structure of the additive group [Formula: see text], giving a simple presentation in particular.

Notes on the Drinfeld Double of Finite-dimensional Group Algebras

Let k be an algebraically closed field of characteristic p > 0. We characterize the finite groups G for which the Drinfeld double D(kG) of the group algebra kG has the Chevalley property. We also show that this is the case if and only if the tensor product of every simple D(kG)-module with its dual is semisimple. The analogous result for the group algebra kG is also true, but its proof requires the classification of the finite simple groups. A further result concerns the largest Hopf ideal contained in the Jacobson radical of D(kG). We prove that this is generated by the augmentation ideal of kOp(Z(G)), where Z(G) is the center of G and Op(Z(G)) the largest p-subgroup of this center.