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Leandro Cagliero

Publications and source records attributed to Leandro Cagliero.

At least 19 recordsLinked to original sources
arXiv2026

Explicit Laplace Spectra of Homogeneous Principal Bundles

We present a unified representation-theoretic method to compute the Laplace-Beltrami spectrum on homogeneous principal bundles. For this setting, we introduce a multi-parameter family of metric deformations called generalized canonical variations. Building upon the geometric realization of such fibrations as naturally reductive spaces, we establish a simplified spectral branching criterion. We apply this method to derive the full spectra (yielding all eigenvalues and multiplicities) for several prominent geometric families. Specifically, we compute the full spectra for the entire classical series of homogeneous 3-$(α,δ)$-Sasaki manifolds (Types A, B, C, and D) and for real and complex Stiefel manifolds over Grassmannians. These explicit formulas provide the analytical data to investigate related problems in geometric analysis. As an application, we classify the scalar stability of these spaces under Perelman's $ν$-entropy and, for the 3-$(α,δ)$-Sasaki manifolds, determine the exact thresholds for Yamabe bifurcations.

math.DG↗
arXiv2026

A family of groups extending McLain's

Given a strict partial order $Δ$ on a set $Λ$ and an arbitrary ring $R$ with $1\neq 0$, the corresponding McLain group $M(Δ)$ has been studied in depth. We construct a larger family of McLain groups $G(Δ)$, where $Δ$ is neither asymmetric nor transitive, while satisfying two weaker axioms. Structural properties common to all members~$G(Δ)$ of this new family are investigated, including a group presentation, a description of the factors of its descending central series, a canonical form for its elements relative to any total order on~$Δ$, and a recursive determination of its upper central series. In addition, we prove the natural isomorphism $G(Δ)/G(Γ)\cong G(Δ\setminusΓ)$, where $Γ$ is a normal subset $Γ$ of $Δ$, and $G(Γ)$ and $G(Δ\setminusΓ)$ are extended McLain groups on their own right. This result has no parallel in the classical context.

math.GR↗
arXiv2025

Artin-Schreier towers of finite fields

Given a prime number $p$, we consider the tower of finite fields $F_p=L_{-1}\subset L_0\subset L_1\subset\cdots$, where each step corresponds to an Artin-Schreier extension of degree $p$, so that for $i\geq 0$, $L_{i}=L_{i-1}[c_{i}]$, where $c_i$ is a root of $X^p-X-a_{i-1}$ and $a_{i-1}=(c_{-1}\cdots c_{i-1})^{p-1}$, with $c_{-1}=1$. We extend and strengthen to arbitrary primes prior work of Popovych for $p=2$ on the multiplicative order of the given generator $c_i$ for $L_i$ over $L_{i-1}$. In particular, for $i\geq 0$, we show that $O(c_i)=O(a_i)$, except only when $p=2$ and $i=1$, and that $O(c_i)$ is equal to the product of the orders of $c_j$ modulo $L_{j-1}^\times$, where $0\leq j\leq i$ if $p$ is odd, and $i\geq 2$ and $1\leq j\leq i$ if $p=2$. We also show that for $i\geq 0$, the $\mathrm{Gal}(L_i/L_{i-1})$-conjugates of $a_i$ form a normal basis of $L_i$ over $L_{i-1}$. In addition, we obtain the minimal polynomial of $c_1$ over $F_p$ in explicit form.

math.NT↗
arXiv2024

$G$-tables and the Poisson structure of the even cohomology of cotangent bundle of the Heisenberg Lie group

In the first part of the paper, we define the concept of a $G$-table of a $G$-(co)algebra and we compute the $G$-table of some $G$-(co)algebras (here a $G$-algebra is an algebra on which $G$ acts, semisimply, by algebra automorphisms). The $G$-table of a $G$-(co)algebra $A$ is a set of scalars that provides very precise and concise information about both the algebra structure and the $G$-module structure of $A$. In particular, the ordinary multiplication table of $A$ can be derived from the $G$-table of $A$. From the $G$-table of a $G$-algebra $A$ we define a plain algebra $P(A)$ associated to it and we present some basic functoriality results about $P$. Obtaining the $G$-table of a given $G$-algebra $A$ requires a considerable amount of work but, the result, is a very powerful tool as shown in the second part of the paper. Here we compute the $SL(2)$-tables of the Poisson algebra structure of the even-degree part of the cohomology associated to the cotangent bundle of the 3-dimensional Heisenberg Lie group with Lie algebra $h$, that is $H_E(h)=H_E^{\bullet}(h,\bigwedge^{\bullet}h)$. This Poisson $SL(2)$-algebra has dimension 18. From these $SL(2)$-tables we deduce that the underlying Lie algebra of $H_E(h)$ is isomorphic to $gl(3)\ltimes gl(3)_{ab}$ with the first factor acting on the second (abelian) one by the adjoint representation. We find it remarkable that the Lie algebra structure on $H_{E}(h)$ contains a semisimple Lie subalgebra (in this case $sl(3)$) strictly larger than the Levi factor of $\text{Der}(h)$, which in this case is $sl(2)\subset H^{1}(h,h)$. This means that the Levi factor of the Lie algebra $H_{E}(h)$ has nontrivial elements outside $H^{1}(h,h)$. Finally, this leads us to find a family of commutative Poisson algebras whose underlying Lie structure is $gl(n)\ltimes gl(n)_{ab}$ (arbitrary $n$) such that, for $n=3$, is isomorphic to $H_E(h)$.

math.RT↗
arXiv2024

Tensor products and intertwining operators between two uniserial representations of the Galilean Lie algebra $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$

Let $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$, $n\ge 1$, be the Galilean Lie algebra over a field of characteristic zero, here $\mathfrak{h}_{n}$ is the Heisenberg Lie algebra of dimension $2n+1$, and $\mathfrak{sl}(2)$ acts on $\mathfrak{h}_{n}$ so that $\mathfrak{h}_n\simeq V(2n-1)\oplus V(0)$ as $\mathfrak{sl}(2)$-modules (here $V(k)$ denotes the irreducible $\mathfrak{sl}(2)$-module of highest weight $k$). In this paper, we study the tensor product of two uniserial representations of $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$. We obtain the $\mathfrak{sl}(2)$-module structure of the socle of $V\otimes W$ and we describe the space of intertwining operators $\text{Hom}_{\mathfrak{sl}(2)\ltimes \mathfrak{h}_n}(V,W)$, where $V$ and $W$ are uniserial representations of $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$. The structure of the radical of $V\otimes W$ follows from that of the socle of $V^*\otimes W^*$. The result is subtle and shows how difficult is to obtain the whole socle series of arbitrary tensor products of uniserials. In contrast to the serial associative case, our results for $\mathfrak{sl}(2)\ltimes \mathfrak{h}_n$ reveal that these tensor products are far from being a direct sum of uniserials; in particular, there are cases in which the tensor product of two uniserial $\big(\mathfrak{sl}(2)\ltimes \mathfrak{h}_n\big)$-modules is indecomposable but not uniserial. Recall that a foundational result of T. Nakayama states that every finitely generated module over a serial associative algebra is a direct sum of uniserial modules. This article extends a previous work in which we obtained the corresponding results for the Lie algebra $\mathfrak{sl}(2)\ltimes \mathfrak{a}_m$ where $\mathfrak{a}_m$ is the abelian Lie algebra of dimension $m+1$ and $\mathfrak{sl}(2)$ acts so that $\mathfrak{a}_m\simeq V(m)$ as $\mathfrak{sl}(2)$-modules.

math.RT↗
arXiv2022

Tensor products and intertwining operators for uniserial representations of the Lie algebra $\mathfrak{sl}(2)\ltimes V(m)$

Let $\mathfrak{g}_m=\mathfrak{sl}(2)\ltimes V(m)$, $m\ge 1$, where $V(m)$ is the irreducible $\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial $\mathfrak{g}_m$-modules consist of a family, say of type $Z$, containing modules of arbitrary composition length, and some exceptional modules with composition length $\le 4$. Let $V$ and $W$ be two uniserial $\mathfrak{g}_m$-modules of type $Z$. In this paper we obtain the $\mathfrak{sl}(2)$-module decomposition of $\text{soc}(V\otimes W)$ by giving explicitly the highest weight vectors. It turns out that $\text{soc}(V\otimes W)$ is multiplicity free. Roughly speaking, $\text{soc}(V\otimes W)=\text{soc}(V)\otimes \text{soc}(W)$ in half of the cases, and in these cases we obtain the full socle series of $V\otimes W$ by proving that $ \text{soc}^{t+1}(V\otimes W)=\sum_{i=0}^{t} \text{soc}^{i+1}(V)\otimes \text{soc}^{t+1-i}(W)$ for all $t\ge0$. As applications of these results, we obtain for which $V$ and $W$, the space of $\mathfrak{g}_m$-module homomorphisms $\text{Hom}_{\mathfrak{g}_m}(V,W)$ is not zero, in which case is 1-dimensional. Finally we prove, for $m\ne 2$, that if $U$ is the tensor product of two uniserial $\mathfrak{g}_m$-modules of type $Z$, then the factors are determined by $U$. We provide a procedure to identify the factors from $U$.

math.RT↗
arXiv2020

Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank $r$

Given a finite dimensional Lie algebra $\mathfrak{g}$, let $\mathfrak{z}(\mathfrak{g})$ denote the center of $\mathfrak{g}$ and let $μ(\mathfrak{g})$ be the minimal possible dimension for a faithful representation of $\mathfrak{g}$. In this paper we obtain $μ(\mathcal{L}_{r,2})$, where $\mathcal{L}_{r,k}$ is the free $k$-step nilpotent Lie algebra of rank $r$. In particular we prove that $μ(\mathcal{L}_{r,2})= \left\lceil \sqrt{2r(r-1)} \right\rceil + 2$ for $r \geq 4$. It turns out that $μ(\mathcal{L}_{r,2}) \simμ\big(\mathfrak{z}(\mathcal{L}_{r,2})\big) \sim 2\sqrt{\dim\mathcal{L}_{r,2}} $ (as $r\to\infty$) and we present some evidence that this could be true for $\mathcal{L}_{r,k}$ for any $k$, this is considerably lower than the known bounds for $μ(\mathcal{L}_{r,k})$, which are (for fixed $k$) polynomial in $\dim\mathcal{L}_{r,k}$.

math.RT↗
arXiv2020

Nilpotency degree of the nilradical of a solvable Lie algebra on two generators

Given a sequence $\vec d=(d_1,\dots,d_k)$ of natural numbers, we consider the Lie subalgebra $\mathfrak{h}$ of $\mathfrak{gl}(d,\mathbb{F})$, where $d=d_1+\cdots +d_k$ and $\mathbb{F}$ is a field of characteristic 0, generated by two block upper triangular matrices $D$ and $E$ partitioned according to $\vec d$, and study the problem of computing the nilpotency degree $m$ of the nilradical $\mathfrak{n}$ of $\mathfrak{h}$. We obtain a complete answer when $D$ and $E$ belong to a certain family of matrices that arises naturally when attempting to classify the indecomposable modules of certain solvable Lie algebras. Our determination of $m$ depends in an essential manner on the symmetry of $E$ with respect to an outer automorphism of $\mathfrak{sl}(d)$. The proof that $m$ depends solely on this symmetry is long and delicate. As a direct application of our investigations on $\mathfrak{h}$ and $\mathfrak{n}$ we give a full classification of all uniserial modules of an extension of the free $\ell$-step nilpotent Lie algebra on $n$ generators when $\mathbb{F}$ is algebraically closed.

math.RT↗
arXiv2018

Jordan-Chevalley decomposition in Lie algebras

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.

math.RT↗
arXiv2017

Free 2-step nilpotent Lie algebras and indecomposable modules

Given an algebraically closed field $F$ of characteristic 0 and an $F$-vector space $V$, let $L(V)=V\oplusΛ^2(V)$ denote the free 2-step nilpotent Lie algebra associated to $V$. In this paper, we classify all uniserial representations of the solvable Lie algebra $\mathfrak g=\langle x\rangle\ltimes L(V)$, where $x$ acts on $V$ via an arbitrary invertible Jordan block.

math.RT↗
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↗
arXiv2016

Nilradicals of parabolic subalgebras admitting symplectic structures

In this paper we describe all the nilradicals of parabolic subalgebras of split real simple Lie algebras admitting symplectic structures. The main tools used to obtain this list are Kostant's description of the highest weight vectors (hwv) of the cohomology of these nilradicals and some necessary conditions obtained for the $\mathfrak g$-hwv's of $H^2(\mathfrak n)$ for a finite dimensional real symplectic nilpotent Lie algebra $\mathfrak n$ with a reductive Lie subalgebra of derivations $\mathfrak g$ acting on it.

math.DG↗
arXiv2015

A new generalization of Hermite's reciprocity law

Given a partition $λ$ of $n$, the {\it Schur functor} $\mathbb{S}_λ$ associates to any complex vector space $V$, a subspace $\mathbb{S}_λ(V)$ of $V^{\otimes n}$. Hermite's reciprocity law, in terms of the Schur functor, states that $ \mathbb{S}_{(p)}\left(\mathbb{S}_{(q)}(\mathbb{C}^2)\right)\simeq \mathbb{S}_{(q)}\left(\mathbb{S}_{(p)}(\mathbb{C}^2)\right). $ We extend this identity to many other identities of the type $\mathbb{S}_λ\left(\mathbb{S}_δ(\mathbb{C}^2)\right)\simeq \mathbb{S}_μ\left(\mathbb{S}_ε(\mathbb{C}^2)\right)$.

math.CO↗
arXiv2014

Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

Let ${\mathfrak g}$ be a finite dimensional Lie algebra over a field of characteristic 0, with solvable radical ${\mathfrak r}$ and nilpotent radical ${\mathfrak n}=[{\mathfrak g},{\mathfrak r}]$. Given a finite dimensional ${\mathfrak g}$-module $U$, its nilpotency series $ 0\subset U({\mathfrak n}^1)\subset\cdots\subset U({\mathfrak n}^m)=U$ is defined so that $U({\mathfrak n}^1)$ is the 0-weight space of ${\mathfrak n}$ in $U$, $U({\mathfrak n}^2)/U({\mathfrak n}^1)$ is the 0-weight space of ${\mathfrak n}$ in $U/U({\mathfrak n}^1)$, and so on. We say that $U$ is linked if each factor of its nilpotency series is a uniserial ${\mathfrak g}/{\mathfrak n}$-module, i.e., its ${\mathfrak g}/{\mathfrak n}$-submodules form a chain. Every uniserial ${\mathfrak g}$-module is linked, every linked ${\mathfrak g}$-module is indecomposable with irreducible socle, and both converse fail. In this paper we classify all linked ${\mathfrak g}$-modules when ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$ and $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Moreover, we identify and classify all uniserial ${\mathfrak g}$-module amongst them.

math.RT↗
arXiv2014

Indecomposable modules of 2-step solvable Lie algebras in arbitrary characteristic

Let $F$ be an algebraically closed field and consider the Lie algebra ${\mathfrak g}=\langle x\rangle\ltimes {\mathfrak a}$, where $\mathrm{ad}\, x$ acts diagonalizably on the abelian Lie algebra ${\mathfrak a}$. Refer to a ${\mathfrak g}$-module as admissible if $[{\mathfrak g},{\mathfrak g}]$ acts via nilpotent operators on it, which is automatic if $\mathrm{char}(F)=0$. In this paper we classify all indecomposable ${\mathfrak g}$-modules $U$ which are admissible as well as uniserial, in the sense that $U$ has a unique composition series.

math.RT↗
arXiv2014

A lower bound for faithful representations of nilpotent Lie algebras

In this paper we present a lower bound for the minimal dimension $μ(\mathfrak{n})$ of a faithful representation of a finite dimensional $p$-step nilpotent Lie algebra $\mathfrak{n}$ over a field of characteristic zero. Our bound is given as the minimum of a quadratically constrained linear optimization problem, it works for arbitrary $p$ and takes into account a given filtration of $\mathfrak{n}$. We present some estimates of this minimum which leads to a very explicit lower bound for $μ(\mathfrak{n})$ that involves the dimensions of $\mathfrak{n}$ and its center. This bound allows us to obtain $μ(\mathfrak{n})$ for some families of nilpotent Lie algebras.

math.RT↗
arXiv2014

Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials

For a two-parameter family of lower triangular matrices with entries involving Jacobi polynomials an explicit inverse is given, with entries involving a sum of two Jacobi polynomials. The formula simplifies in the Gegenbauer case and then one choice of the parameter solves an open problem in a recent paper by Koelink, van Pruijssen & Roman. The two-parameter family is closely related to two two-parameter groups of lower triangular matrices, of which we also give the explicit generators. Another family of pairs of mutually inverse lower triangular matrices with entries involving Jacobi polynomials, unrelated to the family just mentioned, was given by J. Koekoek & R. Koekoek (1999). We show that this last family is a limit case of a pair of connection relations between Askey-Wilson polynomials having one of their four parameter in common.

math.CA↗