Note on the cortex of two-step nilpotent Lie algebras

In this paper, we construct an example of a family of $4d$-dimensional two-step nilpotent Lie algebras $(\frak g_d)_{d\geq 2}$ so that the cortex of the dual of each $\frak g_d$ is a projective algebraic set. More precisely, we show that the cortex of each dual $\frak g_d^*$ of $\frak g_d$ is the zero set of a homogeneous polynomial of degree $d$. This example is a generalization of one given in "Irreducible representations of locally compact groups that cannot be Hausdorff separated from the identity representation" by "{\sc M.E.B. Bekka, and E. Kaniuth}"

Linear family of Lie brackets on the space of matrices Mat(n\times m,\K) and Ado's Theorem

In this paper we classify a linear family of Lie brackets on the space of rectangular matrices Mat(n\times m,\K) and we give an analogue of the Ado's Theorem. We give also a similar classification on the algebra of the square matrices Mat(n, \K) and as a consequence, we prove that we can't built a faithful representation of the $(2n+1)$-dimensional Heisenberg Lie algebra $\mathfrak{H}_n$ in a vector space $V$ with $\dim V\leq n+1$. Finally, we prove that in the case of the algebra of square matrices Mat(n,\K), the corresponding Lie algebras structures are a contraction of the canonical Lie algebra \mathfrak{gl}(n,\K)

Frobenius algebras and root systems: the trigonometric case

We construct Frobenius structures on the $\mathbb{C}^{\times}$-bundle of the complement of a toric arrangement associated with a root system, by making use of a one-parameter family of torsion free and flat connections on it. This gives rise to a trigonometric version of Frobenius algebras in terms of root systems and a new class of Frobenius manifolds. We also determine their potential functions.Comment: 18 pages. Comments welcom

Effective codescent morphisms in some varieties of universal algebras

The paper gives the sufficient condition formulated in the syntactical form for all codescent morphisms of a variety of universal algebras satisfying the amalgamation property to be effective. This result is further used in proving that all codescent morphisms of quasigroups are effective.Comment: 15 page

On the implication T0⇒T312T_{0} \Rightarrow T_{3 \frac{1}{2}} for some topological protomodular algebras

The notion of a right-cancellable protomodular algebra is introduced. It is proved that a right-cancellable topological protomodular algebra that satisfies the separation axiom $T_{0}$ is completely regular

Associative Protomodular Algebras

The notion of associativity (which differs from the straightforward generalization of the usual associativity given by the move of parentheses in the relevant expression) for operations of high arity is introduced. It is proved that the algebraic theory of a variety of universal algebras contains a group operation if and only if it contains a semi-abelian operation which is associative in the sense introduced

A sharp bound on the Hausdorff dimension of the singular set of an n-uniform measure

The study of the geometry of $n$-uniform measures in $\mathbb{R}^{d}$ has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely $3$-Hausdorff measure restricted to the Kowalski-Preiss cone. Using this cone one can construct an $n$-uniform measure whose singular set has Hausdorff dimension $n-3$. In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any $n$-uniform measure is at most $n-3$

Uniformly Distributed Measures have Big Pieces of Lipschitz Graphs locally

The study of uniformly distributed measures was crucial in Preiss' proof of his theorem on rectifiability of measures with positive density. It is known that the support of a uniformly distributed measure is an analytic variety. In this paper, we provide quantitative information on the rectifiability of this variety. Tolsa had already shown that $n$-uniform measures have Big Pieces of Lipschitz Graphs(BPLG) . Here, we prove that a uniformly distributed measure has BPLG locally

Singular Sets of Uniformly Asymptotically Doubling Measures

In the following paper, we prove a dimension bound on the singular set of a Radon measure assuming its doubling ratio converges uniformly on compact sets. More precisely, we prove that if a Radon measure is $n$-Uniformly Asymptotically Doubling, then $\dim(\mathcal{S}_{\mu}) \leq n-3$, where $\mathcal{S}_{\mu}$ is the singular set of the measure.Comment: arXiv admin note: text overlap with arXiv:1510.0373

Multi-anisotropic gevrey regularity of hypoelliptic operators

We show a multi-anisotropic Gevrey regularity of solutions of hypoelliptic equations. This result is a precision of a classical result of H\"ormanderComment: This is the preprint version of our paper in the journal Operator Theory : Advances and Applications, Vol. 189, 265-27