Dmitri I. Panyushev
We study Abelian ideals of a Borel subalgebra consisting of long roots. It is shown that methods of Cellini and Papi can be extended to this situation. A uniform expression for the number of long Abelian ideals is given. We also show that there is a one-to-one correspondence between the long Abelian ideals and B-stable commutative subalgebras in the little adjoint representation of the Langlands dual Lie algebra.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let g be a complex simple Lie algebra and b a fixed Borel subalgebra of g. We shall describe the abelian ideals of b in a uniform way, that is, independent of the classification of complex simple Lie algebras.
Let $g$ be a simple Lie algebra and $Ab$ the poset of non-trivial abelian ideals of a fixed Borel subalgebra of $g$. In 2003 (IMRN, no.35, 1889--1913), we constructed a partition of $Ab$ into the subposets $Ab_μ$, parameterised by the long positive roots of $g$, and established some properties of these subposets. In this note, we show that this partition is compatible with intersections, relate it to the Kostant-Peterson parameterisation of abelian ideals and to the centralisers of abelian ideals. We also prove that the poset of positive roots of $g$ is a join-semilattice.
Elaborating on a paper by Suter, we provide a detailed description of the automorphism group of the poset of abelian ideals in a Borel subalgebra of a finite dimensional complex simple Lie algebra.
Let $\b$ be a Borel subalgebra of a simple Lie algebra $\g$ and let $\Ab$ denote the set of all Abelian ideals of $\b$. We consider $\Ab$ as poset with respect to inclusion, the zero ideal being the unique minimal element of $\Ab$. It was shown in my paper with G.Roehrle (Adv. Math. v.159 (2001)) that there is a one-to-one correspondence between the maximal Abelian ideals and the long simple roots of $\g$. But the very existence of it was demonstrated in a case-by-case fashion. Here a conceptual explanation for that empirical observation is given. The main results are: 1) there is a natural ma…
Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak b$ a fixed Borel subalgebra, and $W$ the Weyl group of $\mathfrak g$. In this note, we study a relationship between the maximal abelian ideals of $\mathfrak b$ and the minimal inversion complete sets of $W$. The latter have been recently defined by Malvenuto et al. (J. Algebra, 424 (2015), 330-356.)
Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$ and $\mathfrak{Ab}$ the set of abelian ideals of $\mathfrak b$. Let $Δ^+$ be the corresponding set of positive roots. We continue our study of combinatorial properties of the partition of $\mathfrak{Ab}$ parameterised by the long positive roots. In particular, the union of an arbitrary set of maximal abelian ideals is described, if $\mathfrak g\ne\mathfrak{sl}_n$. We also characterise the greatest lower bound of two positive roots, when it exists, and point out interesting subposets of $Δ^+$ that are modular lattic…
Let $g$ be a simple Lie algebra and $Ab(g)$ the set of Abelian ideals of a Borel subalgebra of $g$. In this note, an interesting connection between $Ab(g)$ and the subsets of the Dynkin diagram of $g$ is discussed. We notice that the number of abelian ideals with $k$ generators equals the number of subsets of the Dynkin diagram with $k$ connected components. For $g$ of type $A_n$ or $C_n$, we provide a combinatorial explanation of this coincidence by constructing a suitable bijection. We also construct another general bijection between $Ab(g)$ and the subsets of the Dynkin diagram, which is ba…
Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$. Let $Δ^+$ be the corresponding (po)set of positive roots and $θ$ the highest root. A pair $\{η,η'\}\subset Δ^+$ is said to be glorious, if $η,η'$ are incomparable and $η+η'=θ$. Using the theory of abelian ideals of $\mathfrak b$, we (1) establish a relationship of $η,η'$ to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform conn…
Let $\mathfrak g$ be a simple Lie algebra and $\mathfrak{Ab}$ the poset of all abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. If $\mathfrak a\in\mathfrak{Ab}$, then the normaliser of $\mathfrak a$ is a standard parabolic subalgebra of $\mathfrak g$. We give an explicit description of the normaliser for a class of abelian ideals that includes all maximal abelian ideals. We also elaborate on a relationship between abelian ideals and $\mathbb Z$-gradings of $\mathfrak g$ associated with their normalisers.
Recollements of abelian categories are used as a basis of a homological and recursive approach to quasi-hereditary algebras. This yields a homological proof of Dlab and Ringel's characterisation of idempotent ideals occuring in heredity chains, which in turn characterises quasi-hereditary algebras recursively. Further applications are given to hereditary algebras and to Morita context rings.
We consider a nonlinear representation of a Lie algebra which is regular on an abelian ideal, we define a normal form which generalizes that defined in [D. Arnal, M. Ben Ammar, M. Selmi, {\rm Normalisation d'une représentation non linéaire d'une algèbre de Lie}. {\it Annales de la Faculté des Sciences de Toulouse}, 5e série, tome 9, No 3, 1988, p 355--379.]
In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not two. Throughout the paper, we also give several examples to clarify some results.
Let $\frb$ be a fixed Borel subalgebra of a finite-dimensional complex simple Lie algebra $\frg$. The Shi bijection associates to every ad-nilpotent ideal $\fri$ of $\frb$ a region $V_{\fri}$. In this paper, we show that $\fri$ is abelian if and only if $V_{\fri}\cap 2A$ is empty, if and only if the volume of $V_{\fri}\cap 2A$ equals to that of $A$, where $A$ is the fundamental alcove of the affine Weyl group. For certain flag of abelian ideals, we record an ascending property of their associated regions. We also determine the maximal eigenvalue $m_{r-1}$ of the Casimir operator on $\wedge^{r-…
For a semisimple algebraic group $G$ of adjoint type with Lie algebra $\mathfrak g$ over the complex numbers, we establish a bijection between the set of closed orbits of the group $G \ltimes \mathfrak g^{\ast}$ acting on the variety of Lagrangian subalgebras of $\mathfrak g \ltimes \mathfrak g^{\ast}$ and the set of abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. In particular, the number of such orbits equals $2^{\text{rk} \mathfrak g}$ by Peterson's theorem on abelian ideals.
In a previous paper we constructed rank and support variety theories for "quantum elementary abelian groups," that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a tensor product property for the support varieties.
Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$. To any long positive root $γ$, one associates two ideals of $\mathfrak b$: the abelian ideal $I(γ)_{max}$ and not necessarily abelian ideal $I\langle{\succcurlyeq}γ\rangle$. It is known that $I(γ)_{max} \subset I\langle{\succcurlyeq}γ\rangle$, and $γ$ is said to be amazing if the equality holds. The set of amazing roots, $\mathcal A$, is closed under the operation `$\vee$' in $Δ^+$, and $γ\in\mathcal A$ is said to be primitive, if it cannot be written as $γ_1\veeγ_2$ with incomparable amazing roots $γ_1,γ_2$. We c…
Tensor ideals of abelian type and quantum groups
研究线性刚性幺半范畴中作为到阿贝尔幺半范畴的线性幺半函子核的张量理想，发展一般方法并应用于量子群上的倾斜模范畴和有限群表示范畴。
Let $B$ be a Borel subgroup of a semisimple algebraic group $G$, and let $\mathfrak a$ be an abelian ideal of $\mathfrak b=Lie(B)$. The ideal $\mathfrak a$ is determined by certain subset $Δ_{\mathfrak a}$ of positive roots, and using $Δ_{\mathfrak a}$ we give an explicit classification of the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$. Our description visibly demonstrates that there are finitely many $B$-orbits in both cases. We also describe the Pyasetskii correspondence between the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$ and the invariant algebras $\Bbbk[\mathfrak a]^U$ and $\B…
The Bruhat order on abelian ideals of Borel subalgebras
本文在特征不是非常坏的代数闭域上，对拟单代数群Borel子代数幂零根中B-稳定阿贝尔子代数内的B-轨道进行参数化，并刻画其闭包关系。
The lowest discriminant ideal of central extensions of abelian groups
本文研究任意有限群中心扩张的群代数，证明其最低判别理想的零点集为整个极大谱，并针对阿贝尔群中心扩张研究纤维代数与自同构轨道，给出同构不可提升的例子。
In this paper, we consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and full, in another word, the ideal quotient becomes abelian. Moreover, a new equivalent characterization of cluster-tilting subcategories is given by applying homological methods according to this functor. As an application, we show that in a connected 2-Calabi-Yau triangulated category B, a functorially finite, extension closed subcategory T of B is cluster til…
We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer's prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked…
Let $Δ$ be a root system with a subset of positive roots, $Δ^+$. We consider edges of the Hasse diagrams of some posets associated with $Δ^+$. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For $Δ^+$, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple roots. We give several descriptions of the set of edges of given type and uniform expressions for the number of edges. By a result of Peterson, the number of Abelian ideals is $2^n$, where $n$ is the rank of $Δ$. We prove that the …
Using GAP4, we determine the number of ad-nilpotent and abelian ideals of a parabolic subalgebra of a simple Lie algebra of exceptional types E, F or G.
L. Salce introduced the notion of a cotorsion pair (F,C) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proven useful in a variety of settings. A significant result of cotorsion theory proven by Eklof and Trlifaj is that if a pair (F,C) of classes of R-modules is cogenerated by a set, then it is complete. Recently Herzog, Fu, Asensio and Torrecillas developed the ideal approximation theory. In this article we look at a result motivated by the Eklof-Trlifaj argument for an ideal I when it is generated by a set of h…
Tensor Abelian Geometry of VI-modules
本文研究了VI-modules的素Serre理想谱，证明其与N的单点紧化同胚，并探讨了该方法在FI-modules和循环p群全局表示中的应用。
Extended Module Categories in Higher Cluster Tilting Theory
本文通过研究三角范畴关于高丛倾斜子范畴的理想商，证明其自然DG增强是阿贝尔d-截断DG范畴，并利用Morita型定理将其等价于d-截断DG范畴上的d-扩展模范畴。
We extend the results of Cellini-Papi on the characterizations of nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra of a classical Lie algebra, we give formulas for the number of these ideals.
This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these posets. We also obtain results on ad-nilpotent ideals which complete the analysis started in \cite{CP2}, \cite{CP3}.
In this note, we show that the long $n$-exact sequence theorem holds in any $n$-abelian category. As an application, the $n$-abelian version of Wakamatsu's Lemma is given.
Let $R$ be a commutative local uniserial ring of length $n$, $p$ a generator of the maximal ideal, and $k$ the radical factor field. The pairs $(B,A)$ where $B$ is a finitely generated $R$-module and $A\subset B$ a submodule of $B$ such that $p^mA=0$ form the objects in the category $S_m(R)$. We show that in case $m=2$ the categories $S_m(R)$ are in fact quite similar to each other: If also $R'$ is a commutative local uniserial ring of length $n$ and with radical factor field $k$, then the categories $S_2(R)/\mathcal N_R$ and $S_2(R')/\mathcal N_{R'}$ are equivalent for certain nilpotent categ…
In this paper a characterisation is given of solvable complemented Lie algebras. They decompose as a direct sum of abelian subalgebras and their ideals relate nicely to this decomposition. The class of such algebras is shown to be a formation whose residual is the ideal closure of the prefrattini subalgebras.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。