Oz Ben-Shimol | Searcharxiv

arXiv2014

Centers associated with the Borel subalgebra of certain simple Lie algebras

We continue the study in Ben-Shimol [1],[2] and consider a Borel subalgebra $\mathfrak{b}$ and its nil radical $\mathfrak{n}$ of the simple Lie algebras of types $G_2$, $F_4$, $C_n$ over arbitrary field. Let $\mathcal{L}\in\{\mathfrak{n}, \mathfrak{b}\}$. We establish here explicit realizations of the center $Z(\mathcal{L})$ and semi-center $Sz(\mathcal{L})$ of the enveloping algebra, the Poisson center $S(\mathcal{L})^{\mathcal{L}}$ and Poisson semi-center $S(\mathcal{L})^{\mathcal{L}}_{\operatorname{si}}$ of the symmetric algebra. We describe their structure as commutative rings and establish isomorphisms $Z(\mathcal{L})\cong~S(\mathcal{L})^{\mathcal{L}}$, $Sz(\mathcal{L})\cong S(\mathcal{L})^{\mathcal{L}}_{\operatorname{si}}$.

math.RT↗

arXiv2014

Centers associated with the Borel subalgebra of the general linear Lie algebra

We consider a Borel subalgebra $\fg$ of the general linear algebra and its subalgebra $\BB$ which is a Borel subalgebra of the special linear algebra, over arbitrary field. Let $\cL\in\{\fg, \BB\}$. We establish here explicit realizations of the center $Z(\cL)$ and semi-center $Sz(\cL)$ of the enveloping algebra, the Poisson center $S(\cL)^{\cL}$ and Poisson semi-center $S(\cL)^{\cL}_{\si}$ of the symmetric algebra. We describe their structure as commutative rings and establish isomorphisms $Z(\cL)\cong S(\cL)^{\cL}$, $Sz(\cL)\cong S(\cL)^{\cL}_{\si}$

math.RT↗

arXiv2008

Explicit Constructions of the non-Abelian $\mathbf{p^3}$-Extensions Over $\mathbf{\QQ}$

Let $p$ be an odd prime. Let $F/k$ be a cyclic extension of degree $p$ and of characteristic different from $p$. The explicit constructions of the non-abelian $p^{3}$-extensions over $k$, are induced by certain elements in ${F(μ_{p})}^{*}$. In this paper we let $k=\QQ$ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over $\QQ$ are constructed. We describe explicit realizations of those groups with exactly two ramified primes, without consider Scholz conditions.

math.NT↗

arXiv2008

Explicit Constructions of the non-Abelian $p^3$-Extensions Over $\QQ$

Let $p$ be an odd prime. Let $F/k$ be a cyclic extension of degree $p$ and of characteristic different from $p$. The explicit constructions of the non-abelian $p^{3}$-extensions over $k$, are induced by certain elements in ${F(μ_{p})}^{*}$. In this paper we let $k=\QQ$ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over $\QQ$ are constructed.

math.NT↗

arXiv2007

On Galois Groups of Prime Degree Polynomials with Complex Roots

Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens Höchsmann (1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T.Shaska. If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree $p$ over $\QQ$ having complex roots.

math.NT↗