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Rupert W. T. Yu

Publications and source records attributed to Rupert W. T. Yu.

11 recordsLinked to original sources
arXiv2025

On adjoint orbits in nilpotent ideals of a Borel subalgebra

Let $\mathfrak{m}$ be a nilpotent ideal in the Borel subalgebra $\mathfrak{b}$ of a complex finite-dimensional semisimple Lie algebra, and $\mathfrak{m}^{\bullet}$ the subset of (ad-)nilpotent elements in $\mathfrak{b}$ such that $\mathfrak{m}$ is the minimal ideal containing them. This set is stable under the adjoint action of the corresponding Borel subgroup $B$. We prove that $\mathfrak{m}^{\bullet}$ contains a unique closed $B$-orbit which is the orbit of a nilpotent element whose support is the set of minimal roots associated to the root space decomposition of $\mathfrak{m}$.

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arXiv2015

Jet schemes of the closure of nilpotent orbits

We study in this paper the jet schemes of the closure of nilpotent orbits in a finite-dimensional complex reductive Lie algebra. For the nilpotent cone, which is the closure of the regular nilpotent orbit, all the jet schemes are irreducible. This was first observed by Eisenbud and Frenkel, and follows from a strong result of Mustautca (2001). Using induction and restriction of "little" nilpotent orbits in reductive Lie algebras, we show that for a large number of nilpotent orbits, the jet schemes of their closure are reducible. As a consequence, we obtain certain geometrical properties of these nilpotent orbit closures.

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arXiv2014

On compositions associated to Frobenius parabolic and seaweed subalgebras of $\mathrm{sl}_{n}(\Bbbk )$

By using a free monoid of operators on the set of compositions (resp. pairs of compositions), we establish in this paper a bijective correspondence between Frobenius standard parabolic (resp. seaweed) subalgebras and certain elements of this monoid. We prove via this correspondence a conjecture of one of the authors on the number of Frobenius standard parabolic (resp. seaweed) subalgebras of $\mathrm{sl}_{n}(\Bbbk )$ associated to compositions (resp. pairs of compositions) with $n-t$ parts (resp. parts in total).

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arXiv2009

On the index of the quotient of a Borel subalgebra by an ad-nilpotent ideal

In this paper, we give upper bounds for the index of the quotient of the Borel subalgebra of a simple Lie algebra or its nilpotent radical by an ad-nilpotent ideal. For the nilpotent radical quotient, our bound is a generalization of the formula for the index given by Panov in the type A case. In general, this bound is not exact. Using results from Panov, we show that the upper bound for the Borel quotient is exact in the type $A$ case, and we conjecture that it is exact in general.

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arXiv2007

Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we study the set of $Δ$-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type $A$. Our main result is that for any fixed $Δ$-dimension vector, there is a unique (up to isomorphism) exceptional $Δ$-filtered module. We then apply this result to show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in $\mathrm{gl}_{n}(\field)$, thus answering positively in this $\mathrm{gl}_{n}(\field)$ case to a question raised independently by Michel Duflo and Dmitri Panyushev. An example of a seaweed Lie algebra in a simple Lie algebra of type $E_{8}$ not admitting an open orbit in its nilpotent radical is given.

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arXiv2006

On the sum of the index of a parabolic subalgebra and of its nilpotent radical

In this short note, we investigate the following question of Panyushev : ``Is the sum of the index of a parabolic subalgebra of a semisimple Lie algebra $\mathfrak{g}$ and the index of its nilpotent radical always greater than or equal to the rank of $\mathfrak{g}$?''. Using the formula for the index of parabolic subalgebras conjectured by Tauvel and the author, and proved by Millet-Fauquant and Joseph, we give a positive answer to this question. Moreover, we also obtain a necessary and sufficient condition for this sum to be equal to the rank of $\mathfrak{g}$. This provides new examples of direct sum decomposition of a semisimple Lie algebra verifying the ``index additivity condition'' as stated by Ra{\"ı}s.

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arXiv2004

On the irreducibility of the commuting variety of a symmetric pair associated to a parabolic subalgebra with abelian unipotent radical

In this paper, we study the commuting variety of symmetric pairs associated to parabolic subalgebras with abelian unipotent radical in a simple complex Lie algebra. By using the ``cascade'' construction of Kostant, we construct a Cartan subspace which in turn provides, in certain cases, useful information on the centralizers of non $\mathfrak{p}$-regular semisimple elements. In the case of the rank 2 symmetric pair $(\mathrm{so}_{p+2},\mathrm{so}_{p}\times \mathrm{so}_{2})$, $p\geq 2$, this allows us to apply induction, in view of previous results of the authors, and reduce the problem of the irreducibility of the commuting variety to the consideration of evenness of $\mathfrak{p}$-distinguished elements. Finally, via the correspondence of Kostant-Sekiguchi, we check that in this case, $\mathfrak{p}$-distinguished elements are indeed even, and consequently, the commuting variety is irreducible.

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arXiv2004

On the index of certain Lie algebras

We give an upper bound for the index of certain Lie algebras, called of seaweed type, introduced by V. Dergachev, A. Kirillov and D. Panyushev. We deduce from this a conjecture of D. Panyushev stated in "Inductive formulas for the index of seaweed Lie algebra, Moscow Math. Journal (2001) 221-241". We conjecture that our upper bound gives in fact a formula for the index of such Lie algebras, and we prove it in certain cases. Finally, we obtain a result concerning the index of parabolic subalgebras of a semisimple Lie algebra.

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arXiv2003

Index and stable linear forms of Lie algebras

We characterize, in a purely algebraic manner, certain linear forms, called stable, on a Lie algebra. As an application, we determine the index of a Borel subalgebra of a semi-simple Lie algebra. Finally, we give an example of a parabolic subalgebra of a semi-simple Lie algebra which does not admit any stable linear form.

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arXiv2001

Centralizers of distinguished nilpotent pairs and related problems

In this paper, by establishing an explicit and combinatorial description of the centralizer of a distinguished nilpotent pair in a classical simple Lie algebra, we solve in the classical case Panyushev's Conjecture which says that distinguished nilpotent pairs are wonderful, and the classification problem on almost principal nilpotent pairs. More precisely, we show that disinguished nilpotent pairs are wonderful in types A, B and C, but they are not always wonderful in type D. Also, as the corollary of the classification of almost principal nilpotent pairs, we have that almost principal nilpotent pairs do not exist in the simply-laced case and that the centralizer of an almost principal nilpotent pair in a classical simple Lie algebra is always abelian.

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