Jonathan Brown, Jonathan Brundan
We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the Lie algebra gl_N(C). In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova (math.RT/0610049).
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
In this paper we characterize all nilpotent orbits under the action by conjugation that intersect the nilpotent centralizer of a nilpotent matrix $B$ consisting of two Jordan blocks of the same size. We list all the possible Jordan canonical forms of the nilpotent matrices that commute with $B$ by characterizing the corresponding partitions.
Nilpotent Lie algebras with two centralizer dimensions over a finite field
本文在有限域上分类了恰好具有两个中心化子维数且幂零类为 3 的有限维幂零李代数，完善了 Barnea 和 Isaacs 相关结果。
Let $K$ be an algebraically closed field of characteristic zero, $A = K[x_1,\dots,x_n]$ the polynomial ring, $R = K(x_1,\dots,x_n)$ the field of rational functions, and let $W_n(K) = \Der_{K}A$ be the Lie algebra of all $K$-derivations on $A$. If $D \in W_n(K),$ $D\not =0$ is linear (i.e. of the form $D = \sum_{i,j=1}^n a_{ij}x_j \frac{\partial}{\partial x_i}$) we give a description of the centralizer of $D$ in $W_n(K)$ and point out an algorithm for finding generators of $C_{W_n(K)}(D)$ as a module over the ring of constants in case when $D$ is the basic Weitzenboeck derivation. In more gener…
We study the Loewy structure of the centralizer algebra kP^Q for P a p-group with subgroup Q and k a field of characteristic p. Here kP^Q is a special type of Hecke algebra. The main tool we employ is the decomposition of kP^Q as a split extension of a nilpotent ideal I by the group algebra kC_P(Q). We compute the Loewy structure for several classes of groups, investigate the symmetry of the Loewy series, and give upper and lower bounds on the Loewy length of $P^Q. Several of these results were discovered through the use of MAGMA, especially the general pattern for most of our computations. As…
Computing component groups of stabilizers of nilpotent orbit representatives
本文提出计算方法，为简单李代数幂零元素中心化子的分量群构造显式生成元，经典类型用直接转化法，例外类型用双重中心化法，并在GAP中实现。
We prove that the centralizer Cen(f) in Hom_R(M,M) of a nilpotent endomorphism f of a finitely generated semisimple left R-module M (over an arbitrary ring R) is the homomorphic image of the opposite of a certain Z(R)-subalgebra of the full m x m matrix algebra M_m(R[z]), where m is the dimension (composition length) of ker(f). If R is a local ring, then we provide an explicit description of the above Cen(f). If in addition Z(R) is a field and R/J(R) is finite dimensional over Z(R), then we give a formula for the Z(R)-dimension of Cen(f). If R is a local ring, f is as above and g is an arbitra…
This paper is primarily concerned with studying finite-dimensional anti-commutative nonassociative algebras in which every centralizer is an ideal. These are shown to be anti-associative and are classified over a general field $F$; in particular, they are nilpotent of class at most $3$ and metabelian. These results are then applied to show that a Leibniz algebra over a field of charactersitic zero in which all centralizers are ideals is solvable.
Derived and stable equivalences of centralizer matrix algebras
该研究通过矩阵初等因子定义方阵新等价关系，将主中心化子矩阵代数的各类范畴等价归约为线性代数问题，证明了置换矩阵对应代数的Morita等价与导出等价等价等结论。
Singular equivalences and homological conjectures
通过中心化子矩阵代数的奇异范畴和奇异等价，验证了Auslander-Reiten和Cartan行列式猜想，进而证明所有经典同调猜想对域上中心化子矩阵代数成立。
We report on some computations with nilpotent orbits in simple Lie algebras of exceptional type within the SLA package of GAP4. Concerning reachable nilpotent orbits our computations firstly confirm the classification of such orbits in Lie algebras of exceptional type by Elashvili and Grelaud, secondly they answer a question by Panyushev, and thirdly they show in what way a recent result of Yakimova for the Lie algebras of classical type extends to the exceptional types. The second topic of this note concerns abelianizations of centralizers of nilpotent elements. We give tables with their dime…
A 9-dimensional nilpotent Lie algebra isomorphic to its complex conjugate admitting no real form
该研究构造了一个9维5步幂零复李代数，其与自身复共轭同构却无实形式，且证明过程较早期10维例子更初等简短。
Let $A$ be a unital dense algebra of linear mappings on a complex vector space $X$. Let $ϕ=\sum_{i=1}^n M_{a_i,b_i}$ be a locally quasi-nilpotent elementary operator of length $n$ on $A$. We show that, if $\{a_1,\ldots,a_n\}$ is locally linearly independent, then the local dimension of $V(ϕ)=\spa\{b_ia_j: 1 \leq i,j \leq n\}$ is at most $\frac{n(n-1)}{2}$. If $\lDim V(ϕ)=\frac{n(n-1)}{2} $, then there exists a representation of $ϕ$ as $ϕ=\sum_{i=1}^n M_{u_i,v_i}$ with $v_iu_j=0$ for $i\geq j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of leng…
Minimal nilpotent finite $W$-algebra and cuspidal module category of $\mathfrak{sp}_{2n}$
本文证明最小幂零有限W-代数同构于局部化代数的中心化子，并作为张量积因子，进而建立尖点模范畴与有限维模范畴的等价，解释其半单性。
Nilpotent Lie Algebras of breadth type $(0,3)$
本文分类了奇数特征有限域上宽度类型为 (0,3) 的有限维幂零李代数，并给出偶数特征域及实复数域上的部分分类，同时讨论了 2-步幂零 Camina 李代数。
8-dimensional 2-step nilpotent Lie algebras over algebraically closed fields of char $\ne 2, 3$
本文在特征不等于2,3的代数闭域上，利用代数几何方法和基础线性代数，对8维2步 nil 拉格朗日代数进行了自包含的分类。
In this work large families of naturally graded nilpotent Lie algebras in arbitrary dimension and characteristic sequence (n,q,1), with n odd, satisfying the centralizer property, are given. This condtion constitutes a generalization, for a nilpotent Lie agebra, of the structural properties charactrizing the Lie algebra $Q_{n}$. By considering certain cohomological classes of the space $H^{2}(\frak{g},\mathbb{C})$, it is shown that, with few exceptions, the isomorphism classses of these algebras are given by central extensions of $Q_{n}$ by $\mathbb{C}^{p}$ which preserve the nilindex and the …
Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e \in \mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\mathfrak g,e)$ associated to $e$ and the problem of determining the variety $\mathcal E(\mathfrak g,e)$ of 1-dimensional representations of $U(\mathfrak g,e)$. For $\mathfrak g$ of low rank, we report on computer calculations that have been used to determine the structure of $\mathcal E(\mathfrak g,e)$, and the action of the component group $Γ_e$ of the centralizer of $e$ on $\mathcal E(\mathfrak g,e)$. As a consequence, we provide two examples where the n…
Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques.
Irreducible cuspidal $\mathfrak{sl}_{n+1}$-modules from finite-dimensional modules over the minimal nilpotent finite $W$-algebra
本文通过 W(e) 的中心化实现和显式嵌入，证明了有限维不可约 W(e)-模块与有限维不可约 $\mathfrak{gl}_n$-模块的关系，并给出了不可约 cuspidal $\mathfrak{sl}_{n+1}$-模块的明确实现方法。
We study almost inner derivations of $2$-step nilpotent Lie algebras of genus $2$, i.e., having a $2$-dimensional commutator ideal, using matrix pencils. In particular we determine all almost inner derivations of such algebras in terms of minimal indices and elementary divisors over an arbitrary algebraically closed field of characteristic not $2$ and over the real numbers.
A complete description of the coadjoint orbits for A_{n-1}^{+}, the nilpotent Lie algebra of n-by-n strictly upper triangular matrices, has not yet been obtained, though there has been steady progress on it ever since the orbit method was devised. We apply methods developed by Andre' to find defining equations for the elementary coadjoint orbits for the maximal nilpotent Lie subalgebras of the orthogonal Lie algebras, and we also determine all the possible dimensions of coadjoint orbits in the case of A_{n-1}^{+}.
A finite-dimensional Lie algebra $L$ over a field $F$ of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an $A$-algebra if every nilpotent subalgebra is abelian. This paper is a continuation of the study of these algebras initiated by the authors in `Elementary Lie Algebras and Lie A-algebras', J. Algebra 312 (2007), 891--901. If we denote by $\mathcal{A}$, $\mathcal{G}$, $\mathcal{E}$, $\mathcal{L}$, $Φ$ the classes of $A$-algebras, almost algebraic algebras, $E$-algebras, elementary algebras and $ϕ$-free algebras respectively, then it is …
Let K be a field, then we exhibit two matrices in the full nxn matrix algebra M_{n}(K) which generate M_{n}(K) as a Lie K-algebra with the commutator Lie product. We also study Lie centralizers of a not necessarily commutative unitary algebra and obtain results which we hope will eventually be a step in the direction of, firstly, proving that a Lie-nilpotent K-subspace (or a sub Lie K-algebra) of a finite-dimensional associative algebra over K of index k (say) generates a Lie-nilpotent associative subalgebra of much higher nilpotency index, and secondly, in the light of the sharp upper bound f…
Let $A$ be a finite commutative nilpotent $\mathbb{F}_p$-algebra structure on $G$, an elementary abelian group of order $p^n$. If $K/k$ is a Galois extension of fields with Galois group $G$ and $A^p = 0$, then corresponding to $A$ is an $H$-Hopf Galois structure on $K/k$ of type $G$. For that Hopf Galois structure we may study the image of the Galois correspondence from $k$-subHopf algebras of $H$ to subfields of $K$ containing $k$ by utilizing the fact that the intermediate subfields correspond to the $\mathbb{F}_p$-subspaces of $A$, while the subHopf algebras of $H$ correspond to the ideals …
Independence Polynomials of 2-step Nilpotent Lie Algebras
本文将图的独立多项式概念扩展到任意2步 nilpotent 李代数，推导了独立数的上下界，并基于量子力学解释提出度量依赖的广义概念，应用于推导阿贝尔子代数维度的初等界。
Let $L/K$ be a Galois extension of fields with Galois group $G$, an elementary abelian $p$-group of rank $n$ for $p$ an odd prime. It is known that nilpotent $\mathbb{F}_p$-algebra structures $A$ on $G$ yield regular subgroups of the holomorph of $G$, hence Hopf Galois structures on $L/K$. In this paper we illustrate the richness of Hopf Galois structures on $L/K$ by examining the case where $A$ is abelian of dimension $n$ where the dimension of $A^2 = 1$. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent …
In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of [relative] $K_1$; (3) bounded word length of commutators. To this end we enlarge the elementary group, construct a generic element for the extended elementary group, and use localization in the universal ring. The key step is a construction of a generic element for a principle congruence subgroup, corresponding to a principle ideal.
On the Galois correspondence ratio for Hopf-Galois extensions arising from nilpotent $\mathbb{F}_p$-algebras
本文研究由幂零 $\mathbb{F}_p$-代数产生的 Hopf-Galois 扩张的 Galois 对应比率，通过初等线性代数计算左理想个数，确定了所有 4 维幂零代数的 GCR，并证明在参数很大时 GCR 趋近于 0。
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。