Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups

The complete classification of classical r-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincar ́e groups such that their Lorentz sector is a quantum sub-group is presented. It is found that there exists three classes of such r-matrices, one of them being a novel two-parametric one. The (A)dS and Minkowskian Poisson homogeneous spaces corresponding to these three deformations are explicitly constructed in both local and ambient coordinates. Their quantization is performed, thus giving rise to the associated noncommutative spacetimes, that in the Minkowski case are naturally expressed in terms of quantum null-plane coordinates, and they are always defined by homogeneous quadratic algebras. Finally, non-relativistic and ultra-relativistic limits giving rise to novel Newtonian and Carrollian noncommutative spacetimes are also presented.

Manin Matrices for Quadratic Algebras

We give a general definition of Manin matrices for arbitrary quadratic algebras in terms of idempotents. We establish their main properties and give their interpretation in terms of the category theory. The notion of minors is generalised for a general Manin matrix. We give some examples of Manin matrices, their relations with Lax operators and obtain the formulae for some minors. In particular, we consider Manin matrices of the types B, C and D introduced by A. Molev and their relation with Brauer algebras. Infinite-dimensional Manin matrices and their connection with Lax operators are also considered.

Certain partitions on a set and their applications to different classes of graded algebras

Let (𝔘, ɛu ) and (𝔅, ɛb ) be two pointed sets. Given a family of three maps ℱ = {f 1 : 𝔘 → 𝔘; f 2 : 𝔘 × 𝔘 → 𝔘; f 3 : 𝔘 × 𝔘 → 𝔅}, this family provides an adequate decomposition of 𝔘 \ {ɛu } as the orthogonal disjoint union of well-described ℱ-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak H *-algebras.

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.