arXiv subjects

Leandro Cagliero

Publications and source records attributed to Leandro Cagliero.

25 records · Page 2Linked to original sources

arXiv2014

Total cohomology of solvable Lie algebras and linear deformations

Given a finite dimensional Lie algebra $\mathfrak{g}$, let $Γ_\circ(\mathfrak{g})$ be the set of irreducible $\mathfrak{g}$-modules with non-vanishing cohomology. We prove that a $\mathfrak{g}$-module $V$ belongs to $Γ_\circ(\mathfrak{g})$ only if $V$ is contained in the exterior algebra of the solvable radical $\mathfrak{s}$ of $\mathfrak{g}$, showing in particular that $Γ_\circ(\mathfrak{g})$ is a finite set and we deduce that $H^*(\mathfrak{g},V)$ is an $L$-module, where $L$ is a fixed subgroup of the connected component of $\operatorname{Aut}(\mathfrak{g})$ which contains a Levi factor. We describe $Γ_\circ$ in some basic examples, including the Borel subalgebras, and we also determine $Γ_\circ(\mathfrak{s}_n)$ for an extension $\mathfrak{s}_n$ of the 2-dimensional abelian Lie algebra by the standard filiform Lie algebra $\mathfrak{f}_n$. To this end, we described the cohomology of $\mathfrak{f}_n$. We introduce the \emph{total cohomology} of a Lie algebra $\mathfrak{g}$, as $TH^*(\mathfrak{g})=\bigoplus_{V\in Γ_\circ(\mathfrak{g})} H^*(\mathfrak{g},V)$ and we develop further the theory of linear deformations in order to prove that the total cohomology of a solvable Lie algebra is the cohomology of its nilpotent shadow. Actually we prove that $\mathfrak{s}$ lies, in the variety of Lie algebras, in a linear subspace of dimension at least $\dim (\mathfrak{s}/\mathfrak{n})^2$, $\mathfrak{n}$ being the nilradical of $\mathfrak{s}$, that contains the nilshadow of $\mathfrak{s}$ and such that all its points have the same total cohomology.

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arXiv2013

Some remarks on graded nilpotent Lie algebras and the Toral Rank Conjecture

If $\mathfrak{n}$ is a $\mathbb{Z}^d_+$-graded nilpotent finite dimensional Lie algebra over a field of characteristic zero, it is well known that $\dim H^{\ast }(\mathfrak{n})\geq L(p) $ where $p$ is the polynomial associated to the grading and $L(p)$ is the sum of the absolute values of the coefficients of $p$. From this result Deninger and Singhof derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that $\dim H^{\ast }(\mathfrak{n})\geq 2^{\dim (\mathfrak{z)}} $ for any finite dimensional Lie algebra $\mathfrak{n}$ with center $\mathfrak{z}$. The TRC is more that 25 years old and remains open even for $\mathbb{Z}^d_+$-graded 3-step nilpotent Lie algebras. Investigating to what extent the above bound for $\dim H^{\ast }(\mathfrak{n})$ could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra $\mathfrak{n}$ with center $\mathfrak{z}$: (A) If $\mathfrak{n}$ admits a $\mathbb{Z}_+^d$-grading $\mathfrak{n}=\bigoplus_{α\in\mathbb{Z}_+^d} \mathfrak{n}_α$, such that its associated polynomial $p$ satisfies $L(p)>2^{\dim\mathfrak{z}}$, does it admit a grading $\mathfrak{n}=\mathfrak{n}'_{1}\oplus \mathfrak{n}'_{2}\oplus \dots\oplus \mathfrak{n}'_{k}$ such that its associated polynomial $p'$ satisfies $L(p')>2^{\dim\mathfrak{z}}$? (B) If $\mathfrak{n}$ is $r$-step nilpotent admitting a grading $\mathfrak{n}=\mathfrak{n}_{1}\oplus \mathfrak{n}_{2}\oplus \dots\oplus \mathfrak{n}_{k}$ such that its associated polynomial $p$ satisfies $L(p)>2^{\dim\mathfrak{z}}$, does it admit a grading $\mathfrak{n}=\mathfrak{n}'_{1}\oplus \mathfrak{n}'_{2}\oplus \dots\oplus \mathfrak{n}'_{r}$ such that its associated polynomial $p'$ satisfies $L(p')>2^{\dim\mathfrak{z}}$? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.

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arXiv2013

On the theorem of the primitive element with applications to the representation theory of associative and Lie algebras

We describe of all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,y\in K$. When is $F[x,y]=F[αx+βy]$ for some non-zero elements $α,β\in F$?

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arXiv2012

A new interpretation of the Racah-Wigner $6j$-symbol and the classification of uniserial $sl(2)\ltimes V(m)$-modules

All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let $V(m)$ be the irreducible $\sl(2)$-module with highest weight $m\geq 1$ and consider the perfect Lie algebra $\g=\sl(2)\ltimes V(m)$. Recall that a $\g$-module is uniserial when its submodules form a chain. In this paper we classify all uniserial $\g$-modules. The main family of uniserial $\g$-modules is actually constructed in greater generality for the perfect Lie algebra $\g=\s\ltimes V(μ)$, where $\s$ is a semisimple Lie algebra and $V(μ)$ is the irreducible $\s$-module with highest weight $μ\neq 0$. The fact that the members of this family are, but for a few exceptions of lengths 2, 3 and~4, the only uniserial $\sl(2)\ltimes V(m)$-modules depends in an essential manner on the determination of certain non-trivial zeros of Racah-Wigner $6j$-symbol.

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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$_4$.

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arXiv2010

Jordan-Chevalley decomposition in finite dimesional Lie algebras

Let $\g$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\g$ is said to have an \emph{abstract Jordan-Chevalley decomposition} if there exist unique $s,n\in\g$ such that $x=s+n$, $[s,n]=0$ and given any finite dimensional representation $π:\g\to\gl(V)$ the Jordan-Chevalley decomposition of $π(x)$ in $\gl(V)$ is $π(x)=π(s)+π(n)$. In this paper we prove that $x\in\g$ has an abstract Jordan-Chevalley decomposition if and only if $x\in [\g,\g]$, in which case its semisimple and nilpotent parts are also in $[\g,\g]$ and are explicitly determined. We derive two immediate consequences: (1) every element of $\g$ has an abstract Jordan-Chevalley decomposition if and only if $\g$ is perfect; (2) if $\g$ is a Lie subalgebra of $\gl(n,k)$ then $[\g,\g]$ contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods. Our proof only uses elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lie's Theorem, in addition to the fundamental theorems of Ado and Levi.

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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.

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