We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. As all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.
For an ad-nilpotent ideal $i$ of a Borel subalgebra of $sl_{l+1}(C)$, we denote by $I_i$ the maximal subset $I$ of the set of simple roots such that $i$ is an ad-nilpotent ideal of the standard parabolic subalgebra $p_I$. We use the bijection given by G.E. Andrews, C. Krattenthaler, L. Orsina and P. Papi between the set of ad-nilpotent ideals of a Borel subalgebra in $sl_{l+1}(C)$ and the set of Dyck paths of length $2l+2$, to explicit a bijection between ad-nilpotent ideals $i$ of the Borel subalgebra such that the cardinality of $I_i$ is equal to $r$ and the Dyck paths of length $2l+2$ havin…
We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of $sl(n+1,\Bbb C)$. We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q,t)-analogue of the Catalan number $C_n$. These (q,t)-Catalan numbers count on the one hand ad-nilpotent ideals with respect to dimension and class of nilpote…
Nilpotent Leibniz algebras with isomorphic maximal subalgebras are considered. The algebras are classified for coclass zero, one, and two. The results are field dependent.
In this paper, the structure of all finite-dimensional nilpotent Lie algebras of class two with derived subalgebra of dimension two over an arbitrary field $ \mathbb{F} $ is determined. Furthermore, we give the structure of the Schur multiplier of such Lie algebras.
By considering the nilpotent Lie algebra with the derived subalgebra of dimension $ 2$, we compute some functors include the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such Lie algebras.
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$, and of nilpotent length $\leq k$, and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-${\mathcal N}$ Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index $\leq 3$.
Let A be an associative algebra of arbitrary dimension over a field F and G a finite soluble group of automorphisms of A oforder n, prime to the characteristic of F. We prove that if the fixed-point subalgebra of A under the action of G contains a two-sided nilpotent ideal of nilpotency index d and of finite codimension m in I, then A contains a nilpotent two-sided ideal of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of m, n and d.
We study certain types of ideals in the standard Borel subalgebra of an untwisted affine Lie algebra. We classify these ideals in terms of the root combinatorics and give an explicit formula for the number of such ideals in type $A$. The formula involves various aspects of combinatorics of Dyck paths and leads to a new interesting integral sequence.
We obtain the functions that bound the dimensions of finite dimensional nilpotent associative or Lie algebras of class 2 over an algebraically closed field in terms of the dimensions of their commutative subalgebras. As a result, we also compute a similar function for complex nilpotent Lie groups of class 2.
We show that semi-simple lie algebras can be characterized by their maximal nilpotent subalgebra, which is the same as the nilpotent radical of a Borel subalgebra.
Let $k$ be a field and let $A=\bigoplus_{n\ge 1}A_n$ be a positively graded $k$-algebra. We recall that $A$ is graded nilpotent if for every $d\ge 1$, the subalgebra of $A$ generated by elements of degree $d$ is nilpotent. We give a method of producing grading nilpotent algebras and use this to prove that over any base field $k$ there exists a finitely generated graded nilpotent algebra that contains a free $k$-subalgebra on two generators.
The paper is devoted to give a full classification of all finite dimensional nilpotent Lie algebras $ L $ of class $4$ such that $ \dim L^2=3. $ Moreover, we classify the capable ones.
This is an old paper put here for archeological purposes. It is proved that a finite-dimensional Lie algebra over a field of characteristic p>5, that can be written as a vector space (not necessarily direct) sum of two nilpotent subalgebras, is solvable. The same result (but covering also the cases of low characteristics) was established independently by V. Panyukov (Russ. Math. Surv. 45 (1990), N4, 181-182), and the homological methods utilized in the proof were developed later in arXiv:math/0204004. Many inaccuracies in the English translation are corrected, otherwise the text is identical t…
Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exis…
In this paper we study subalgebras of complex finite dimensional evolution algebras. We obtain the classification of nilpotent evolution algebras whose any subalgebra is an evolution subalgebra with a basis which can be extended to a natural basis of algebra. Moreover, we formulate three conjectures related to description of such non-nilpotent algebras.
We provide explicit formulas for the number of ad-nilpotent ideals of a Borel subalgebra of a complex simple Lie algebra having fixed class of nilpotence.
It has been conjectured by Gene Freudenburg that for a polynomial ring, the triangular Lie algebra is the maximal Lie algebra which lies in the set of locally nilpotent derivations of the ring. Also it was conjectured that each other maximal Lie algebra which lies in the set of locally nilpotent derivations of the polynomial ring is conjugated to the triangular Lie algebra. In the present work we prove the first part of this conjecture and provide the counterexample to the second part. Also we show that the second part of the conjecture holds for the maximal Lie algebras among locally nilpoten…
We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n\geq 1, let L_{n+2} be a Lie algebra with generator set x_1,..., x_{n+2} and the following relations: for k\leq n, any commutator of length $k$ which consists of fewer than k different symbols from {x_1,...,x_{n+2}} is zero. As an application of this result about automata algebras, we prove that for every n\geq 1, L_{n+2} contains a free subalgebra. We also prove the similar result about groups defined by commutator relations.
Let $K$ be an algebraically closed field of characteristic zero and $A$ an integral $K$-domain. The Lie algebra $Der_{K}(A)$ of all $K$-derivations of $A$ contains the set $LND(A)$ of all locally nilpotent derivations. The structure of $LND(A)$ is of great interest, and the question about properties of Lie algebras contained in $LND(A)$ is still open. An answer to it in the finite dimensional case is given. It is proved that any finite dimensional (over $K$) subalgebra of $Der_{K}(A)$ consisting of locally nilpotent derivations is nilpotent. In the case $A=K[x, y],$ it is also proved that any …
We investigate (twisted) rings of differential operators on the resolution of singularities of a particular irreducible component of the (Zarisky) closure of the minimal orbit $\bar O_{\mathrm{min}}$ of $\mathfrak{sp}_{2n}$, intersected with the Borel subalgebra $\mathfrak n_+$ of $\mathfrak{sp}_{2n}$, using toric geometry and show that they are homomorphic images of a subalgebra of the Universal Enveloping Algebra (UEA) of $\mathfrak{sp}_{2n}$, which contains the maximal parabolic subalgebra $\mathfrak p$ determining the minimal nilpotent orbit. Further, using Fourier transforms on Weyl algeb…
The main result of this paper is the following: if F is any field and R is any F-subalgebra of the algebra of nxn matrices over F with Lie nilpotence index m, then the F-dimension of R is less or equal than M(m+1,n), where M(m+1,n) is the maximum of a certain expression of m+1 nonnegative integers and n. The case m=1 reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if F is an algebraically closed field of characteristic zero, and R is any commutative F-subalgebra of the full nxn matrix algebra over F, then the F-dimension of…
Let $K$ be an arbitrary field of characteristic zero and $A$ a commutative associative $ K$-algebra which is an integral domain. Denote by $R$ the fraction field of $A$ and by $W(A)=RDer_{\mathbb K}A,$ the Lie algebra of $\mathbb K$-derivations of $R$ obtained from $Der_{\mathbb K}A$ via multiplication by elements of $R.$ If $L\subseteq W(A)$ is a subalgebra of $W(A)$ denote by $rk_{R}L$ the dimension of the vector space $RL$ over the field $R$ and by $F=R^{L}$ the field of constants of $L$ in $R.$ Let $L$ be a nilpotent subalgebra $L\subseteq W(A)$ with $rk_{R}L\leq 3$. It is proven that the …
A finite-dimensional Lie algebra $L$ over a field $F$ is called an $A$-algebra if all of its nilpotent subalgebras are abelian. This is analogous to the concept of an $A$-group: a finite group with the property that all of its Sylow subgroups are abelian. These groups were first studied in the 1940s by Philip Hall, and are still studied today. Rather less is known about $A$-algebras, though they have been studied and used by a number of authors. The purpose of this paper is to obtain more detailed results on the structure of solvable Lie $A$-algebras. It is shown that they split over each term…
We prove that the maximal nilpotent subalgebra of a Kac-Moody Lie algebra has an (essentially unique) Euclidean metric with respect to which the Laplace operator in the chain complex is scalar on each component of a given degree. Moreover, both the Lie algebra structure and the metric are uniquely determined by this property.
Let $\mathbb K$ be a field of characteristic zero and $A$ an integral domain over $\mathbb K.$ The Lie algebra $\Der_{\mathbb K} A$ of all $\mathbb K$-derivations of $A$ carries very important information about the algebra $A.$ This Lie algebra is embedded into the Lie algebra $R\Der_{\mathbb K} A\subseteq \Der_{\mathbb K}R$, where $R={\rm Frac}(A)$ is the fraction field of $A.$ The rank $rk_{R}L$ of a subalgebra $L$ of $R\Der_{\mathbb K} A$ is defined as dimension $\dim_R RL.$ We prove that every locally nilpotent subalgebra $L$ of $R\Der_{\mathbb K} A$ with $rk_{R}L=n$ has a series of ideals…
Zassenhaus has proved that if U is a subnormal subalgebra of a finite-dimensional Lie algebra L and V is a finite-dimensional irreducible L-module, then all U-module composition factors of V are isomorphic. Schenkman has proved that if U is a subnormal subalgebra of a finite-dimensional Lie algebra L, then the nilpotent residual of U is an ideal of L. These useful results generalise to Leibniz algebras.
Let $\mathbb K$ be a field of characteristic zero, $A$ an integral domain over $\mathbb K$ with the field of fractions $R = \text{Frac}(A),$ and $\text{Der}_{\mathbb{K}}A$ the Lie algebra of all $\mathbb K$-derivations on $A$. Let $W(A):=R\text{Der}_{\mathbb{K}} A$ and $L$ a nilpotent subalgebra of rank $n$ over $R$ of the Lie algebra $W(A).$ We prove that if the center $Z=Z(L)$ is of rank $\geq n-2$ over $R$ and $F=F(L)$ is the field of constants for $L$ in $R,$ then the Lie algebra $FL$ is contained in a locally nilpotent subalgebra of $ W(A)$ of rank $n$ over $R$ with a natural basis over t…
Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that a left Leibniz algebra, all of whose maximal subalgebras are right ideals, is nilpotent. A primitive Leibniz algebra is shown to split over its minimal ideal and that all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.
In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Leibniz algebras. We study Leibniz algebras containing abelian subalgebras of codimension 1, solvable and supersolvable Leibniz algebras for codimensions 1 and 2, nilpotent Leibniz algebras in case of codimension 2 and we also analyze the case of k-abelian p-filiform Leibniz algebras. Throughout the paper, we also give examples to clarify some results and the need for restrictions on the underlying field.
In this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given.
Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive 2-soluble n-Lie algebra is shown to split over its minimal ideal and that all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.
We compare the maximal dimension of abelian subalgebras and the maximal dimension of abelian ideals for finite-dimensional Lie algebras. We show that these dimensions coincide for solvable Lie algebras over an algebraically closed field of characteristic zero. We compute this invariant for all complex nilpotent Lie algebras of dimension n less than 8. Furthermore we study the case where there exists an abelian subalgebra of codimension 2. Here we explicitly construct an abelian ideal of codimension 2 in case of nilpotent Lie algebras.
Let $\mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $\mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$. We show that, except for some explicit exceptional cases, every derivation of the nilpotent subalgebra $\mathfrak{n}$ that preserves its restricted root space decomposition is of the form $\text{ad}( W)$, where $W \in \mathfrak{m}\oplus .
In this note we establish several basic properties of nilpotent Sabinin algebras. Namely, we show that nilpotent Sabinin algebras (1) can be integrated to produce nilpotent loops, (2) satisfy an analogue of the Ado theorem, (3) have nilpotent Lie envelopes. We also give a new set of axioms for Sabinin algebras. These axioms reflect the fact that a complementary subspace to a Lie subalgebra in a Lie algebra is a Sabinin algebra. Finally, we note that the non-associative version of the Jennings theorem produces a version of Ado theorem for loops whose commutator-associator filtration is of finit…
Let $\mathbb F$ be a field $\mathbb F $ of characteristic zero. Let $W_{n}(\mathbb F)$ be the Lie algebra of all $\mathbb F$-derivations with the Lie bracket $[D_1, D_2]:=D_1D_2-D_2D_1$ on the polynomial ring $\mathbb F [x_1, \ldots , x_n]$. The problem of classifying finite dimensional subalgebras of $W_{n}(\mathbb F)$ was solved if $ n\leq 2$ and $\mathbb F=\mathbb C$ or $\mathbb F=\mathbb R.$ We prove that this problem is wild if $n\geq 4$, which means that it contains the classical unsolved problem of classifying matrix pairs up to similarity. The structure of finite dimensional subalgebra…
Let $\mathbb{F}$ be an algebraically closed field of characteristic $p>3$, and $\ggg$ the $p$-dimensional Witt algebra over $\mathbb{F}$. Let $\N$ be the nilpotent cone of $\ggg$. Explicit description of $\N$ is given, so that the conjugacy classes of Borel subalgebras of $\ggg$ under the automorphism group are determined. In contrast with only one conjugacy class of Borel subalgebras in a classical simple Lie algebra, there are two conjugacy classes of Borel subalgebras in $\ggg$. The representatives of conjugacy classes of Borel subalgebras, i.e., the so-called standard Borel subalgebras, ar…
Let B be an algebra over a field k and let Der(B) be the set of k-derivations from B to B. We define what it means for a subset of Der(B) to be a locally nilpotent set. We prove some basic results about that notion and explore the following questions. Let L be a Lie subalgebra of Der(B); if every element of L is a locally nilpotent derivation then does it follow that L is a locally nilpotent set? Does it follow that L is a nilpotent Lie algebra?
Let g = Lie(G) be the Lie algebra of a simple algebraic group G over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g_e = Lie(G_e) where G_e stands for the stabiliser of e in G. For g classical, we give an explicit combinatorial formula for the codimension of [g_e, g_e] in g_e and use it to determine those e in g for which the largest commutative quotient U(g,e)^{ab} of the finite W-algebra U(g,e) is isomorphic to a polynomial algebra. It turns out that this happens if and only if e lies in a unique sheet of g. The nilpotent elements with this prop…
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.
Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ such that the characteristic of the ground field does not divide $|H|$. It is proved that if the subalgebra $C_L(F)$ of fixed points of the kernel has finite dimension $m$ and the subalgebra $C_L(H)$ of fixed points of the complement is nilpotent of class $c$, then $L$ has a nilpotent subalgebra of finite codimension bounded in terms of $m$, $c$, $|H|$, and $|F|$ whose nilpotency class is bounded in terms of only $|H|$ and $c$. Examples show that the condition of the k…
For a Lie algebra $L$ and a subalgebra $M$ of $L$ we say that a subalgebra $U$ of $L$ is a {\em supplement} to $M$ in $L$ if $L = M + U$. We investigate those Lie algebras all of whose maximal subalgebras have abelian supplements, those that have nilpotent supplements, those that have nil supplements, and those that have supplements with the property that their derived algebra is inside the maximal subalgebra being supplemented. For the algebras over an algebraically closed field of characteristic zero in the last three of these classes we find complete descriptions; for those in the first cla…
All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to thes…
In this paper we determine the precise extent to which the classical sl_2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson--Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer--Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example i…
For a non-abelian Lie algebra $L$ of dimension $n$ with the derived subalgebra of dimension $m$ , the first author earlier proved that the dimension of its Schur multiplier is bounded by $\frac{1}{2}(n+m-2)(n-m-1)+1$. In the current work, we give some new inequalities on the exterior square and the Schur multiplier of Lie algebras and then we obtain the class of all nilpotent Lie algebras which attains the above bound. Moreover, we also improve this bound as much as possible.
We approach celebrated theorems of Burnside and Wedderburn via simultaneous triangularization. First, for a general field $F$, we prove that $M_n(F)$ is the only irrreducible subalgebra of triangularizable matrices in $M_n(F)$ provided such a subalgebra exists. This provides a slight generalization of a well-known theorem of Burnside. Next, for a given $n > 1$, we characterize all fields $F$ such that Burnside's Theorem holds in $M_n(F)$, i.e., $M_n(F)$ is the only irreducible subalgebra of itself. In fact, for a subfield $F$ of the center of a division ring $D$, our simple proof of the aforem…
For every finite $p$-group $G$ of order $p^n$ with derived subgroup of order $p^m$, Rocco in \cite{roc} proved that the order of tensor square of $G$ is at most $p^{n(n-m)}$. This upper bound has been improved recently by author in \cite{ni}. The aim of the present paper is to obtain a similar result for a non-abelian nilpotent Lie algebra of finite dimension. More precisely, for any given $n$-dimensional non-abelian nilpotent Lie algebra $L$ with derived subalgebra of dimension $m$ we have $\mathrm{dim} (L\otimes L)\leq (n-m)(n-1)+2$. Furthermore for $m=1$, the explicit structure of $L$ is gi…
Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and $\mathscr{C}(\mathscr{N}):=\{(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathscr{C}(\mathscr{N})$ is reducible and equidimens…
To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained in a proper Zariski-closed subset. We compute the maximal dimension of an abelian subalgebra of a generic Lie algebra and give a sufficient condition for a generic Lie algebra to admit no surjective homomorphism onto a non-abelian Lie algebra of a given dimension. Also we consider analogous questions for finitely generated torsion free nilpotent groups of cl…
We adapt methods coming from additive combinatorics in groups to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Diderrich-Kneser's and Hamidoune's theorems on sumsets and Tao's theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Diderrich-Kneser's theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group …
Finite-dimensional Lie subalgebras of the universal enveloping algebra of $\mathfrak{sl}(2,\mathbb{C})$
Institute for Quantum Computing Analytics (PGI-12), Forschungszentrum Jülich（于利希研究中心量子计算分析研究所）；Theoretical Physics, Universität des Saarlandes（萨尔兰大学理论物理系）
