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We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and…
We give here the Hopf algebra structure describing the noncommutative renormalization of a recently introduced translation-invariant model on Moyal space. We define Hochschild one-cocyles $B_+^\gamma$ which allows us to write down the…
We study detailed classical-quantum correspondence for a cluster system of three spins with single-axis anisotropic exchange coupling. With autoregressive spectral estimation, we find oscillating terms in the quantum density of states…
An alternative "flipped" version of the quartification model is obtained by rearrangement of the particle assignments. The model has two standard (trinification) families and one flipped quartification family. An interesting…
A regular spectral triple is proposed for a two-dimensional $\kappa$-deformation. It is based on the naturally associated affine group $G$, a smooth subalgebra of $C^*(G)$, and an operator $\caD$ defined by two derivations on this…
We derive the non-relativistic $c\to\infty$ and ultra-relativistic $c\to 0$ limits of the $\kappa$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie…
We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role…
We discuss 2-cocycles of the Lie algebra $\Map(M^3;\g)$ of smooth, compactly supported maps on 3-dimensional manifolds $M^3$ with values in a compact, semi-simple Lie algebra $\g$. We show by explicit calculation that the…
We introduce the $C^*$-algebra $C^*(\kappa)$ generated by the Koopman representation $\kappa$ of an \'etale groupoid $G$ acting on a measure space $(X,\mu)$. We prove that for a level transitive self-similar action $(G,E)$ with $E$ finite…
We calculate the third homology of $\mathrm{SL}_2(\mathbb{Q})$ with half-integral coefficients. Corresponding to each prime $p$ there is an operator on this group with square the identity. The kernel of the (split surjective) homomorphism…
Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication. Models of Shimura curves are very important in number theory. Klein's icosahedral invariants $\mathfrak{A},\mathfrak{B}$ and $\mathfrak{C}$ give the…
We study a cross-ratio of four generic points of $S^3$ which comes from spherical CR geometry. We construct a homomorphism from a certain group generated by generic configurations of four points in $S^3$ to the pre-Bloch group $\mathcal…
We show that there are {\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\exp(X)\exp(Y)\exp(Z)=\exp({AX+BZ+CY+DI}) \ , $$ derived in arXiv:1502.06589, JHEP {\bf 1505}…
We construct by geometric methods a noncommutative model E of the algebra of regular functions on the universal (2-fold) cover M of certain nilpotent coadjoint orbits O for a complex simple Lie algebra g. Here O is the dense orbit in the…
Let X be a smooth variety over a field k, and l be a prime number invertible in k. We study the (\'etale) unramified H^3 of X with coefficients Q_l/Z_l(2) in the style of Colliot-Th\'el\`ene and Voisin. If k is separably closed, finite or…
The subgroup commutativity degree of a group G has been defined in [6] as the probability that two subgroups of G commute, or equivalently that the product of two subgroups is again a subgroup. Problem 4.3 of [6] asks whether there exist…
We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let $B$…
We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1, x_2, x_3) \mapsto (s x_1, tx_2, st x_3)$ by proving necessary and sufficient conditions for the boundedness and compactness of…
In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph $\Gamma$ and a vertex-transitive subgroup $G$ of $\mathrm{Aut}(\Gamma)$, a $2$-factor $\mathcal{C}$…
A new open spin chain hamiltonian is introduced. It is both integrable (Sklyanin`s type $K$ matrices are used to achieve this) and invariant under ${\cal U}_{\epsilon}(sl(2))$ transformations in nilpotent irreps for $\epsilon^3=1$. Some…