arXiv subjects

Alfredo Brega

Publications and source records attributed to Alfredo Brega.

3 recordsLinked to original sources

arXiv2016

The Nash-Moser Theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras

We apply the Nash-Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space $H_{k-nil}^2(\mathfrak{g},\mathfrak{g})$, and certain subspaces of it, that provide fine information about the deformations of $\mathfrak{g}$ in the variety of $k$-step nilpotent Lie algebras. Then we focus on degenerations and rigidity in the variety of $k$-step nilpotent Lie algebras of dimension $n$ with $n\le7$ and, in particular, we obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent Lie algebras of dimension 7. We also recover some known results and point out a possible error in a published article related to this subject.

math.RA↗

arXiv2011

The image of the Lepowsky homomorphism for the group $F_4$

Let $G_o$ be a semisimple Lie group, let $K_o$ be a maximal compact subgroup of $G_o$ and let $\mathfrak{k}\subset\mathfrak{g}$ denote the complexification of their Lie algebras. Let $G$ be the adjoint group of $\mathfrak{g}$ and let $K$ be the connected Lie subgroup of $G$ with Lie algebra $ad(\mathfrak{k})$. If $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$ then $U(\mathfrak{g})^K$ will denote the centralizer of $K$ in $U(\mathfrak{g})$. Also let $P:U(\mathfrak{g})\longrightarrow U(\mathfrak{k})\otimes U(\mathfrak{a})$ be the projection map corresponding to the direct sum $U(\mathfrak{g})=\bigl(U(\mathfrak{k})\otimes U(\mathfrak{a})\bigr)\oplus U(\mathfrak{g})\mathfrak{n}$ associated to an Iwasawa decomposition of $G_o$ adapted to $K_o$. In this paper we give a characterization of the image of $U(\mathfrak{g})^K$ under the injective antihomorphism $P:U(\mathfrak{g})^K\longrightarrow U(\mathfrak{k})^M\otimes U(\mathfrak{a})$, considered by Lepowsky, when $G_o$ is locally isomorphic to F$_4$.

math.RT↗

arXiv2008

LU-decomposition of a noncommutative linear system and Jacobi polynomials

In this paper we obtain the LU-decomposition of a noncommutative linear system of equations that, in the rank one case, characterizes the image of the Lepowsky homomorphism $U(\lieg)^{K}\to U(\liek)^{M}\otimes U(\liea)$. This LU-decomposition can be transformed into very simple matrix identities, where the entries of the matrices involved belong to a special class of Jacobi polynomials. In particular, each entry of the L part of the original system is expressed in terms of a single ultraspherical Jacobi polynomial. In turns, these matrix identities yield a biorthogonality relation between the ultraspherical Jacobi polynomials.

math.RT↗