Nilpotent Subalgebra Research Articles

Finding the radical of matrix algebras using Fitting decompositions

We present a new approach to calculating the radical of a matrix algebra, based on the Fitting decomposition with respect to the simultaneous adjoint action of certain commutative subalgebras. This idea results in a reduction to finding the radical of a Lie nilpotent subalgebra or a commutative factor thereof. We also describe a probabilistic version for computing elements which generate the radical as an ideal.

Exponents of varieties of lie algebras with a nilpotent commutator subalgebra

We prove that the growth exponent of any variety of Lie algebras with a nilpotent commutator subalgebra is integral.

Associated variety, Kostant-Sekiguchi correspondence, and locally free U(n)-action on Harish-Chandra modules

Let g be acomplex semisimple Lie algebra with symmetric decomposition g=k+p. For each irreducible Harish-Chandra (g,k)-module X, we construct a family of nilpotent Lie subalgebras n(O) of g whose universal enveloping algebras U(n(O)) act on X locally freely. The Lie subalgebras n(O) are parametrized by the nilpotent orbits O in the associated variety of X, and they are obtained by making use of the Cayley tranformation of sI2-triples(Kostant-Sekiguchi correspondence). As aconsequence, it is shown that an irreducible Harish-Chandra module has the possible maximal Gelfand-Kirillov dimension if and only if it admits locally free U(nm)-action for nm=n(Omax) attached to aprincipal nilpotent orbit Omax in p.

A proof of Feigin’s conjecture

The paper is devoted to the proof of the following conjecture due to B. Feigin. Let $\frak u_\ell$ be the small quantum group a the primitive $\ell$-th root of unity. Then it is known that the usual $Ext$ algebra of the trivial $\frak u_\ell$-module is isomorphic to the algebra of regular functions on the nilpotent variety $\cal N$ in the corresponding simple Lie algebra $\frak g$ (see [GK]). Consider semiinfinite cohomology of the trivial $\frak u_\ell$-module introduced in [Ar1], [Ar2]. It was shown in [Ar2] that the $Ext$ algebra of the trivial $\frak u_\ell$-module acts naturally on semiinfinite cohomology. Moreover semiinfinite cohomology space of the trivial $\frak_\ell$-module is equipped with a natural $\frak g$-module structure. B. Feigin conjectured that the described $\frak g$-module and $F(\cal N)$-module structures coincide with the ones on the space of local cohomology of the structure sheaf on $\cal N$ with support in the standard positive nilpotent subalgebra $\frak n^+\subset\cal N\subset \frak g$. We give a detailed proof of the conjecture. Moreover we generalize the statement and describe semiinfinite cohomology of contragradient Weyl modules in terms of local cohomology of certain coherent sheaves on the nilpotent cone and on its desingularization $T^*(G/B)$. To do this we construct a certain specialization of the quantum BGG resolution of the simple module $L(\lambda)$ defined for generic values of the quantizing parameter into the root of unity called the quasi-BGG complex. The complex conists of direct sums of quasi-Verma modules and provides conjecturally a resolution of the Weyl module $W(\lambda)$.

A Proof of Hozo's Conjecture on the Homology of a Class of Nilpotent Lie Algebras

LetPbe a rooted tree with rootrand vertex set {1,2,…,n}. ConsiderPto be a poset by sayingu≤vifuis on the path fromrtov. LetZ[P] be the span of all matriceszuvsuch thatu<Pv, wherezuvis then×nmatrix with a 1 in theu,ventry and 0's elsewhere. Note thatZ[P] is a nilpotent Lie subalgebra of sℓn(C). I. Hozo (Electron. J. Combin., to appear) studied the Laplacian of the Koszul complex for computing the Lie algebra homology ofZ[P]. He showed that the tree structure ofPforces significant simplifications in the Laplacian. He went on to conjecture that the eigenvalues of the Laplacian are indexed by certain labellings ofPby partitions. He gave a simple method for determining the eigenvalue indexed by a labelling. In this paper we prove Hozo's conjecture. As a consequence, we deduce that all eigenvalues of the Laplacian are integers.

Global Intersection Cohomology of Quasimaps' Spaces

Let $C$ be a smooth projective curve of genus 0. Let $\CB$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha\in\BN[I]$ of positive integers one can consider the space $\CQ_\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\CB$. This space admits some remarkable compactifications $\CQ^D_\alpha$ (Quasimaps), $\CQ^L_\alpha$ (Quasiflags), $\CQ^K_\alpha$ (Stable Maps) of $\CQ_\alpha$ constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map $\pi: \CQ^L_\alpha\to \CQ^D_\alpha$ is a small resolution of singularities. The aim of the present note is to study the cohomology $H^\bullet(\CQ^L_\alpha,\BQ)$ of Laumon's spaces or, equivalently, the Intersection Cohomology $H^\bullet(\CQ^L_\alpha,IC)$ of Drinfeld's Quasimaps' spaces. We calculate the generating function $P_G(t)$ (``Poincar\'e polynomial'') of the direct sum $\oplus_{\alpha\in\BN[I]}H^\bullet(\CQ^D_\alpha,IC)$ and construct a natural action of the Lie algebra ${\frak{sl}}_n$ on this direct sum by some middle-dimensional correspondences between Quasiflags' spaces. We conjecture that this module is isomorphic to distributions on nilpotent cone supported at nilpotent subalgebra.

Rings that are sums of two locally nilpotent subrings, II

A family of simple examples of algebras which are sums of two locally nilpotent subalgebras and are not nil, and which have the min-imal possible Gelfand-Kirillov dimension, is given. Semigroup algebras of subsemigroups of the semigroup of all partial translations on the real line are used in the construction.

Strings fromN=2 gauged Wess-Zumino-Witten models

We present an algebraic approach to string theory. An embedding of $sl(2|1)$ in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of the $N=2$ superconformal algebra. The extension is completely determined by the $sl(2|1)$ embedding. The realization of the superconformal algebra is determined by the grading. For a particular choice of grading, one obtains in this way, after twisting, the BRST structure of a string theory. We classify all embeddings of $sl(2|1)$ into Lie super algebras and give a detailed account of the branching of the adjoint representation. This provides an exhaustive classification and characterization of both all extended $N=2$ superconformal algebras and all string theories which can be obtained in this way.

Cohomology of nilpotent subalgebras of the algebra $$\widehat{sl_2 }$$

Cohomology of nilpotent subalgebras of the algebra $$\widehat{sl_2 }$$

Catenarity in quantum algebras

Various quantum algebras are shown to be catenary, i.e., all saturated chains of prime ideals between any two fixed primes have the same length. Further, Tauvel's formula relating the height of a prime ideal to the Gelfand-Kirillov dimension of the corresponding factor ring is established. These results are obtained for coordinate rings of quantum affine spaces, for quantized Weyl algebras, and for coordinate rings of complex quantum general linear groups, as well as for quantized enveloping algebras of maximal nilpotent subalgebras of semisimple complex Lie algebras.

NONPOLYNOMIAL REALIZATIONS OF ${\mathcal W}$ ALGEBRAS

Relaxing first-class constraint conditions in the usual Drinfeld—Sokolov Hamiltonian reduction leads, after symmetry fixing, to realizations of [Formula: see text] algebras expressed in terms of all the J current components. General results are given for [Formula: see text] a nonexceptional simple (finite and affine) algebra. Such calculations directly provide the commutant, in the (closure of) [Formula: see text] enveloping algebra, of the nilpotent subalgebra [Formula: see text], where the subscript refers to the chosen gradation in [Formula: see text]. In the affine case, explicit expressions are presented for the Virasoro, [Formula: see text], and Bershadsky algebras at the quantum level.

Rings that are sums of two locally nilpotent subrings

A family of simple examples of algebras which are sums of two locally nilpotent subalgebras and are not nil, and which have the min-imal possible Gelfand-Kirillov dimension, is given. Semigroup algebras of subsemigroups of the semigroup of all partial translations on the real line are used in the construction.

Study of q-commutations in Uq(n+) Algebra

Let g be a semi-simple Lie algebra with maximal nilpotent subalgebra n+. Let Ǔq(g) and Uq(n+) be the quantized enveloping algebras. We describe the set of normal elements of Uq(n+). We prove that each ideal of Uq(n+) can be generated by a normal sequence.

GENERALIZED HIROTA EQUATIONS AND REPRESENTATION THEORY I: THE CASE OF SL(2) AND SLq(2)

This paper begins investigation of the concept of a “generalized τ function,” defined as a generating function of all the matrix elements of a group element g∈G in a given highest weight representation of a universal enveloping algebra [Formula: see text]. In the generic situation, the time variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such τ “functions” are not c numbers but take their values in noncommutative algebras (of functions on the quantum group G). Despite all these differences from the particular case of conventional τ functions of integrable (KP and Toda lattice) hierarchies [which arise when G is a Kac-Moody (one-loop) algebra of level k=1], these generic τ functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to k&gt;1 Kac-Moody and multiloop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (zero-loop) algebra SL (2) and its quantum counterpart SL q(2), as well as for the system of fundamental representations of SL (n).

GEOMETRICAL DESCRIPTION OF THE LOCAL INTEGRALS OF MOTION OF MAXWELL-BLOCH EQUATION

We represent a classical Maxwell-Bloch equation and relate it to positive part of the AKNS hierarchy in geometrical terms. The Maxwell-Bloch evolution is given by an infinitesimal action of a nilpotent subalgebra n+ of affine Lie algebra [Formula: see text] on a Maxwell–Bloch phase space treated as a homogeneous space of n+. A space of local integrals of motion is described using cohomology methods. We show that Hamiltonian flows associated with the Maxwell–Bloch local integrals of motion (i.e. positive AKNS flows) are identified with an infinitesimal action of an Abelian subalgebra of the nilpotent subalgebra n− on a Maxwell–Bloch phase space. Possibilities of quantization and lattice setting of Maxwell–Bloch equation are discussed.

ON SOLVABLE LIE ALGEBRAS DECOMPOSABLE INTO A SUM OF TWO NILPOTENT SUBALGEBRAS

Finite-dimensional Lie algebras over an algebraically closed field of positive characteristic representable as a sum (not necessarily direct) of two nilpotent subalgebras are studied. Necessary and sufficient conditions are obtained for a Lie algebra to be decomposable into a sum of two Cartan subalgebras. Bibliography: 4 titles.

Cohomology of Nilpotent Subalgebras of Affine Lie Algebras

Cohomology of Nilpotent Subalgebras of Affine Lie Algebras

Fusion algebra at a rational level and cohomology of nilpotent subalgebras of $$\widehat{\mathfrak{s}\mathfrak{l}}$$ (2)

We define and calculate the fusion algebra of a WZW model at a rational level using cohomological methods. As a byproduct, we obtain a cohomological characterization of admissible representations of $$\widehat{\mathfrak{s}\mathfrak{l}}$$ 2.

Maximal Abelian subalgebras of complex Euclidean Lie algebras

Maximal Abelian subalgebras (MASAs) of the complex Euclidean Lie algebra [Formula: see text] are classified into conjugacy classes under the action of the Lie group [Formula: see text] Use is made of an earlier classification of MASAs of the orthogonal Lie algebra [Formula: see text] These are then extended to nonsplitting MASAs of [Formula: see text] in which maximal Abelian nilpotent subalgebras of [Formula: see text] are coupled with translations in a nontrivial manner. The methods presented are applicable to the classification of MASAs of any affine Lie algebra.

Residual finiteness of color Lie superalgebras

A (color) Lie superalgebra L over a field K of characteristic ¬= 2, 3 is called residually finite if any of its nonzero elements remains nonzero in a finite-dimensional homomorphic image of L. In what follows we are looking for necessary and sufficient conditions under which all finitely generated Lie superalgebras satisfying a fixed system of identical relations are residually finite. In the case char K = 0 we show that a variety V satisfies this property if and only if V does not contain all center-by-metabelian algebras and every finitely generated algebra of V has nilpotent commutator subalgebra