Nilpotent Subalgebra Research Articles

Simple modules for untwisted affine Lie algebras induced from nilpotent loop subalgebras

We construct large families of simple modules for untwisted affine Lie algebras using induction from one-dimensional modules over nilpotent loop subalgebras. We also show that the vector space of the first self-extensions for these module has uncountable dimension and that generic tensor products of these modules are simple.

The Universe Inside Hall Algebras of Coherent Sheaves on Toric Resolutions

Abstract Let $\mathfrak {g}\neq \mathfrak {s}\mathfrak {o}_{8}$ be a simple Lie algebra of type $A,D,E$ with $\widehat {\mathfrak {g}}$ the corresponding affine Kac–Moody algebra and $\mathfrak {n}_{+}\subset \widehat {\mathfrak {g}}$ is the standard positive nilpotent subalgebra. Given $\mathfrak {n}_{+}$ as above, we provide an infinite collection of cyclic finite abelian subgroups of $SL_{3}(\mathbb {C})$ with the following properties. Let $G$ be any group in the collection, $Y=G\operatorname {-}\mbox {Hilb}(\mathbb {C}^{3})$, and $\Psi : D^{b}_{G}(Coh(\mathbb {C}^{3}))\rightarrow D^{b}(Coh(Y))$ the derived equivalence of Bridgeland, King, and Reid. We present an (explicitly described) subset of objects in $Coh_{G}(\mathbb {C}^{3})$, s.t. the Hall algebra generated by their images under $\Psi $ is isomorphic to $U(\mathfrak {n}_{+})$. In case the field $\Bbbk $ (in place of $\mathbb {C}$) is finite and $\mbox {char}(\Bbbk )$ is coprime with the order of $G$, we establish isomorphisms of the corresponding “counting” Ringel–Hall algebras and the specializations of quantized universal enveloping algebras $U_{v}(\mathfrak {n}_{+})$ at $v=\sqrt {|\Bbbk |}$.

On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $W_n(K)$

Let $K$ be an algebraically closed field of characteristic zero, $P_n=K[x_1,\ldots ,x_n]$ the polynomial ring, and $W_n(K)$ the Lie algebra of all $K$-derivations on $P_n$. One of the most important subalgebras of $W_n(K)$ is the triangular subalgebra $u_n(K) = P_0\partial_1+\cdots+P_{n-1}\partial_n$, where $\partial_i:=\partial/\partial x_i$ are partial derivatives on $P_n$ and $P_0=K.$ This subalgebra consists of locally nilpotent derivations on $P_n.$ Such derivations define automorphisms of the ring $P_n$ and were studied by many authors. The subalgebra $u_n(K) $ is contained in another interesting subalgebra $s_n(K)=(P_0+x_1P_0)\partial_1+\cdots +(P_{n-1}+x_nP_{n-1})\partial_n,$ which is solvable of the derived length $ 2n$ that is the maximum derived length of solvable subalgebras of $W_n(K).$ It is proved that $u_n(K)$ is a maximal locally nilpotent subalgebra and $s_n(K)$ is a maximal solvable subalgebra of the Lie algebra $W_n(K)$.

Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive scalar products. To each rook placement $D$ and each map $\xi$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one can naturally assign the coadjoint orbit $\Omega_{D,\xi}$ in the dual space $\mathfrak{n}^*$. By definition, $\Omega_{D,\xi}$ is the orbit of $f_{D,\xi}$, where $f_{D,\xi}$ is the sum of root covectors $e_{\alpha}^*$ multiplied by $\xi(\alpha)$, $\alpha\in D$. (In fact, almost all coadjoint orbits studied at the moment have such a form for certain $D$ and $\xi$.) It follows from the results of Andr\`e that if $\xi_1$ and $\xi_2$ are distinct maps from $D$ to $\mathbb{C}^{\times}$ then $\Omega_{D,\xi_1}$ and $\Omega_{D,\xi_2}$ do not coincide for classical root systems $\Phi$. We prove that this is true if $\Phi$ is of type $G_2$, or if $\Phi$ is of type $F_4$ and $D$ is orthogonal.

A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations

The main purpose of this paper is to give a combinatorial realization of Kirillov-Reshetikhin (KR simply) crystals Br,s for type En(1) with a minuscule node r and s≥1. To do this, we describe explicitly the crystal of the quantum nilpotent subalgebra associated with the translation by the negative of the r-th fundamental weight. Then the crystal can be extended as an affine crystal, in which a certain subcrystal characterized by the εr⁎-statistic is isomorphic to Br,s as an affine crystal, where εr⁎ is also realized precisely in terms of triple and quadruple paths.

On the solvability of graded Novikov algebras

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a [Formula: see text]-graded Novikov algebra [Formula: see text] over a field [Formula: see text] with solvable [Formula: see text]-component [Formula: see text] is solvable, where [Formula: see text] is a finite additive abelean group and the characteristic of [Formula: see text] does not divide the order of the group [Formula: see text]. We also show that any Novikov algebra [Formula: see text] with a finite solvable group of automorphisms [Formula: see text] is solvable if the algebra of invariants [Formula: see text] is solvable.

Tetrahedron and 3D Reflection Equation from PBW Bases of the Nilpotent Subalgebra of Quantum Superalgebras

In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov–Sergeev solution to the Zamolodchikov tetrahedron equation is characterized as the transition matrix for a particular case of type A, which clarifies an algebraic origin of it. Our work is inspired by the recent developments connecting transition matrices for quantum non-super algebras with intertwiners of irreducible representations of quantum coordinate rings. We also discuss the crystal limit of transition matrices, which gives a super analog of transition maps of Lusztig’s parametrizations of the canonical basis.

Fractal nil graded Lie, associative, Poisson, and Jordan superalgebras

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The first author constructed their analogue in case of restricted Lie algebras of characteristic 2 [50], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [68]. Thus, we have examples of finitely generated restricted Lie algebras with a nil p-mapping. In characteristic zero, similar examples of Lie and Jordan algebras do not exist by results of Martinez and Zelmanov [43] and [78]. The first author constructed analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of arbitrary characteristic, the virtue of that construction is that Lie superalgebras have clear monomial bases [51], they have slow polynomial growth. As an analogue of periodicity, Z2-homogeneous elements are ad-nilpotent. A recent example of a Lie superalgebra is of linear growth, of finite width 4, just infinite but not hereditary just infinite [13]. By that examples, an extension of the result of Martinez and Zelmanov [43] for Lie superalgebras of characteristic zero is not valid.Now, we construct a just infinite fractal 3-generated Lie superalgebra Q over arbitrary field, which gives rise to an associative hull A, a Poisson superalgebra P, and two Jordan superalgebras J and K, the latter being a factor algebra of J. In case charK≠2, A has a natural filtration, which associated graded algebra has a structure of a Poisson superalgebra such that grA≅P, also P admits an algebraic quantization using a deformed superalgebra A(t). The Lie superalgebra Q is finely Z3-graded by multidegree in the generators, A, P are also Z3-graded, while J and K are Z4-graded by multidegree in four generators. By virtue of our construction, these five superalgebras have clear monomial bases and slow polynomial growth. We describe multihomogeneous coordinates of bases of Q, A, P in space as bounded by “almost cubic paraboloids”. We determine a similar hypersurface in R4 that bounds monomials of J and K. Constructions of the paper can be applied to Lie (super)algebras studied before to obtain Poisson and Jordan superalgebras as well.The algebras Q, A, and the algebras without unit Po, Jo, Ko are direct sums of two locally nilpotent subalgebras and there are continuum such decompositions. Also, Q=Q0¯⊕Q1¯ is a nil graded Lie superalgebra, so, Q again shows that an extension of the result of Martinez and Zelmanov for Lie superalgebras of characteristic zero is not valid. In case charK=2, Q has a structure of a restricted Lie algebra with a nil p-mapping. The Jordan superalgebra K is nil finely Z4-graded, in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk group) of Jordan algebras in characteristic distinct from 2 [78]. Also, K is of slow polynomial growth, just infinite, but not hereditary just infinite.We call the superalgebras Q, A, P, J, K fractal because they contain infinitely many copies of themselves.

Leibniz A-algebras

Abstract A finite-dimensional Lie algebra is called an A-algebra if all of its nilpotent subalgebras are abelian. These arise in the study of constant Yang-Mills potentials and have also been particularly important in relation to the problem of describing residually finite varieties. They have been studied by several authors, including Bakhturin, Dallmer, Drensky, Sheina, Premet, Semenov, Towers and Varea. In this paper we establish generalisations of many of these results to Leibniz algebras.

Nilpotent Lie algebras of derivations with the center of small corank

Let $\mathbb K$ be a field of characteristic zero, $A$ be an integral domain over $\mathbb K$ with the field of fractions $R=Frac(A),$ and $Der_{\mathbb K}A$ be the Lie algebra of all $\mathbb K$-derivations on $A$. Let $W(A):=RDer_{\mathbb K} A$ and $L$ be 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 the field $R.$ It is also proved that the Lie algebra $FL$ can be isomorphically embedded (as an abstract Lie algebra) into the triangular Lie algebra $u_n(F)$, which was studied early by other authors.

On the structure of Kac–Moody algebras

AbstractLet A be a symmetrisable generalised Cartan matrix, and let $\mathfrak {g}(A)$ be the corresponding Kac–Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak {g}(A)$ : given two homogeneous elements $x,y\in \mathfrak {g}(A)$ , when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak {g}(A)$ .

Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and geometric Satake equivalence

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Quantum nilpotent subalgebras of classical quantum groups and affine crystals

We study the crystal of quantum nilpotent subalgebra of Uq(Dn) associated to a maximal Levi subalgebra of type An−1. We show that it has an affine crystal structure of type Dn(1) isomorphic to a limit of perfect Kirillov-Reshetikhin crystal Bn,s for s⩾1, and give a new polytope realization of Bn,s. We show that an analogue of RSK correspondence for type D due to Burge is an isomorphism of affine crystals and give a generalization of Greene's formula for type D.

Analysis of the symmetry group and exact solutions of the dispersionless KP equation in n + 1 dimensions

The Lie algebra of the symmetry group of the (n + 1)-dimensional generalization of the dispersionless Kadomtsev–Petviashvili equation is obtained and identified as a semi-direct sum of a finite dimensional simple Lie algebra and an infinite dimensional nilpotent subalgebra. Group transformation properties of solutions under the subalgebra sl(2,R) are presented. Known explicit analytic solutions in the literature are shown to be actually group-invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group.

Almost nilpotency of an associative algebra with an almost nilpotent fixed-point subalgebra

Let A be an associative algebra of arbitrary dimension over a field F and G a finite group of automorphisms of A of order n, prime to the characteristic of F. Denote by AG={a∈A|ag=afor allg∈G} the fixed-point subalgebra. By the classical Bergman–Isaacs theorem, if AG is nilpotent of index d, i.e. (AG)d=0, then A is also nilpotent and its nilpotency index is bounded by a function depending only on n and d. We prove, under the additional assumption of solubility of G, that if AG contains a two-sided nilpotent ideal I◁AG of nilpotency index d and of finite codimension m in AG, then A contains a nilpotent two-sided ideal H◁A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of m, n and d. An even stronger result is provided for graded associative algebras: if G is a finite (not necessarily soluble) group of order n and A=⨁g∈GAg is a G-graded associative algebra over a field F, i.e. AgAh⊂Agh, such that the identity component Ae has a two-sided nilpotent ideal Ie◁AG of nilpotency index d and of finite codimension m in Ae, then A has a homogeneous nilpotent two-sided ideal H◁A of nilpotency index bounded by a function of n and d and of finite codimension bounded by a function of n, d and m.

Cyclotomic Discriminantal Arrangements and Diagram Automorphisms of Lie Algebras

Abstract We identify a class of affine hyperplane arrangements that we call cyclotomic discriminantal arrangements. We establish correspondences between the flag and Aomoto complexes of such arrangements and chain complexes for nilpotent subalgebras of Kac–Moody type Lie algebras with diagram automorphisms. As part of this construction, we find that flag complexes naturally give rise to a certain cocycle on the fixed-point subalgebras of such diagram automorphisms. As a byproduct, we show that the Bethe vectors of cyclotomic Gaudin models associated to diagram automorphisms are nonzero. We also obtain the Poincare polynomial for the cyclotomic discriminantal arrangements.

Quantum Twist Maps and Dual Canonical Bases

In this paper, we show that quantum twist maps, introduced by Lenagan-Yakimov, induce bijections between dual canonical bases of quantum nilpotent subalgebras. As a corollary, we show the unitriangular property between dual canonical bases and Poincar\'e-Birkhoff-Witt type bases under the "reverse" lexicographic order. We also show that quantum twist maps induce bijections between certain unipotent quantum minors.

О СВОЙСТВАХ ПЕРВИЧНОГО РАДИКАЛА СЛАБОАРТИНОВОЙ АЛГЕБРЫ ЛИ

This article deals with the issues of the structural theory of Lie algebras. The construction of the structural theory of algebraic systems implies the existence of certain structures of a special form, which are simpler than the base system. The important tool to study algebraic systems is the radical. The development of the structural theory of Lie algebras led to the emergence of various radicals. There are many radicals of Lie algebras in numerous publications. For example, the Killing radical, the Parfenov radical, the Jacobson radical and the prime radical are considered in various articles. The important area of research is the study of radicals of innite-dimensional Lie algebras. The article is devoted to proving properties of prime radical of a weakly artinian Lie algebra. A Lie algebra is said to be a weakly artinian if the Lie algebra satises the descending chain condition on ideals. In the rst section of the paper we introduced the concept of the prime radical in the following way. A Lie algebra L is said to be prime if [U; V ] = 0 implies U = 0 or V = 0 for any ideals U and V of L. We say that the ideal P of a Lie algebra L is prime if the factor algebra L=P is prime. The intersection of all prime ideals is called the prime radical P(L) of a Lie algebra L. In the second section it is shown that any nite set of elements of the prime radical of a weakly artinian Lie algebra generates the nilpotent subalgebra. This means that the prime radical is locally nilpotent. The third section is devoted to the solvability of the prime radical of a weakly artinian Lie algebra. There is a history of solving Mikhalev' s problem about the prime radical of a weakly artinian Lie algebra in this section also.

On derivations of subalgebras of real semisimple 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 $$W \in \mathfrak {m}\oplus \mathfrak {a}$$ .

Commutative Nilpotent Subalgebras with Nilpotency Index n-1 in the Algebra of Matrices of Order n

The paper establishes the existence of an element with nilpotency index n − 1 in an arbitrary nilpotent commutative subalgebra with nilpotency index n−1 in the algebra of upper niltriangular matrices N n (𝔽) over a field 𝔽 with at least n elements for all n ≥ 5, and also, as a corollary, in the full matrix algebra M n (𝔽). The result implies an improvement with respect to the base field of known classification theorems due to D. A. Suprunenko, R. I. Tyshkevich, and I. A. Pavlov for algebras of the class considered.