Nilpotent Subalgebra Research Articles

Solvable Lie A-algebras

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.

Generalized Lie triple derivations

Let be a complex semisimple Lie algebra, be a Borel subalgebra of and denote the maximal nilpotent subalgebra of . In this article, we prove that every generalized Lie triple derivation of can be written as the sum of a Lie triple derivation and a block diagonal map. We also show that every generalized Lie triple derivation for of each classical complex simple Lie algebra is the sum of a Lie triple derivation and a homothety.

Nil graded self-similar algebras

In [19], [24] we introduced a family of self-similar nil Lie algebras \bm L over fields of prime characteristic p > 0 whose properties resemble those of Grigorchuk and Gupta–Sidki groups. The Lie algebra \bm L is generated by two derivations \begin{aligned} v1 &= \partial_1 + t0p − 1(\partial_2 + t_1^{p − 1}(\partial_3 + t_2^{p − 1}(\partial_4 + t_3^{p − 1}(\partial_5 + t_4^{p − 1}(\partial_6 + \cdots ))))), \\ v2 &= \partial_2 + t_1^{p − 1}(\partial_3 + t_2^{p − 1}(\partial_4 + t_3^{p − 1}(\partial_5 + t_4^{p − 1}(\partial_6 + \cdots )))) \end{aligned} of the truncated polynomial ring K[t_i, i \in \mathbb N\ |\ t_i^p = 0,\ i \in \mathbb N] in countably many variables. The associative algebra \bm A generated by v_1,\ v_2 is equipped with a natural \mathbb Z \oplus \mathbb Z -gradation. In this paper we show that for p, which is not representable as p = m^2 + m + 1 , m \in \mathbb Z , the algebra \bm A is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi [3] and Ya. S. Krylyuk [15] proved that for p = m^2 + m + 1 the algebra \bm A is not graded nil. However, we show that the second family of self-similar Lie algebras introduced in [24] and their associative hulls are always \mathbb Z^p -graded, graded nil, and are sums of two locally nilpotent subalgebras.

Representations and cohomology for Frobenius–Lusztig kernels

Let U ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g , with parameter ζ specialized to an ℓ -th root of unity in a field of characteristic p > 0 . In this paper we study certain finite-dimensional normal Hopf subalgebras U ζ ( G r ) of U ζ , called Frobenius–Lusztig kernels, which generalize the Frobenius kernels G r of an algebraic group G . When r = 0 , the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible U ζ ( G r ) -modules and discuss their characters. We then study the cohomology rings for the Frobenius–Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G . We prove that the cohomology ring for the first Frobenius–Lusztig kernel is finitely-generated when g has type A or D , and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.

Laplacian spectrum for the nilpotent Kac–Moody Lie algebras

We prove that a maximal nilpotent subalgebra of a Kac‐Moody Lie algebra has an (essentially unique) Euclidean metric whose 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.

Filtering bases and cohomology of nilpotent subalgebras of the Witt algebra and the algebra of loops in sl 2

We study the cohomology with trivial coefficients of the Lie algebras Lk, k ≥ 1, of polynomial vector fields with zero k-jet on the circle and the cohomology of similar subalgebras ℒk of the algebra of polynomial loops with values in sl2 The main result is a construction of special bases in the exterior complexes of these algebras. Using this construction, we obtain the following results. We calculate the cohomology of Lk and ℒL. We obtain formulas in terms of Schur polynomials for cycles representing the homology of these algebras. We introduce “stable” filtrations of the exterior complexes of Lk and ℒk, thus generalizing Goncharova’s notion of stable cycles for Lk, and give a polynomial description of these filtrations. We find the spectral resolutions of the Laplace operators for L1 and ℒ1.

G2 dualities in D = 5 supergravity and black strings

Five-dimensional minimal supergravity dimensionally reduced on two commuting Killing directions gives rise to a G2 coset model. The symmetry group of the coset model can be used to generate new solutions by applying group transformations on a seed solution. We show that on a general solution the generators belonging to the Cartan and nilpotent subalgebras of G2 act as scaling and gauge transformations, respectively. The remaining generators of G2 form a subalgebra that can be used to generate non-trivial charges. We use these generators to generalize the five-dimensional Kerr string in a number of ways. In particular, we construct the spinning electric and spinning magnetic black strings of five-dimensional minimal supergravity. We analyze physical properties of these black strings and study their thermodynamics. We also explore their relation to black rings.

Automorphisms of a nilpotent Lie algebra over a commutative ring

Suppose that m ≥ 5 and that R is a commutative ring with identity in which 2 is invertible. This paper determines all automorphisms of a nilpotent subalgebra of the orthogonal Lie algebra o(2m, R).

Abelian ideals and cohomology of symplectic type

Let b be a Borel subalgebra of the symplectic Lie algebra sp (2n, C) and let n be the corresponding maximal nilpotent subalgebra. We find a connection between the abelian ideals of b and the cohomology of n with trivial coefficients. Using this connection, we are able to enumerate the number of abelian ideals of b with given dimension via the Poincare polynomials of Weyl groups of types A n- 1 and C n .

Identities for Lie superalgebras with a nilpotent commutator subalgebra

It is proved that the exponential growth rate of identities in any superalgebra with a nilpotent commutator subalgebra over a field of zero characteristic is an integer.

Tetrahedron Equations and Nilpotent Subalgebras of $${\fancyscript{U}}_q(sl_n)$$

A relation between q-oscillator R-matrix of the tetrahedron equation and decompositions of Poinkaré–Birkhoff–Witt type bases for nilpotent subalgebras $${\fancyscript{U}}_q({\mathfrak{n}}_+)\subset{\fancyscript{U}}_q(sl_n)$$ is observed.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

On central extensions of preprojective algebras

We show that the Hilbert polynomial P ( t ) of the trace space A / [ A , A ] of the centrally extended preprojective algebra A of an ADE quiver is equal to the Hilbert series of the maximal nilpotent subalgebra of the corresponding simple Lie algebra under the principal gradation. This implies that the Hilbert polynomial of the center of A is t 2 h − 4 P ( 1 / t ) , where h is the Coxeter number.

Higher Lamé Equations and Critical Points of Master Functions

Under certain conditions, we give an estimate from above on the number of differential equations of order r + 1 with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of solutions. The estimate is given in terms of a suitable weight subspace of the tensor power U(n−) ⊗(n−1) , where n is the number of singular points in C and U(n−) is the enveloping algebra of the nilpotent subalgebra of glr+1.

Elementary Lie algebras and Lie A-algebras

A finite-dimensional Lie algebra L over a field F is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. The present paper is primarily concerned with the classification of elementary Lie algebras. In particular, we provide a complete list in the case when F is algebraically closed and of characteristic different from 2, 3, reduce the classification over fields of characteristic 0 to the description of elementary semisimple Lie algebras, and identify the latter in the case when F is the real field. Additionally it is shown that over fields of characteristic 0 every elementary Lie algebra is almost algebraic; in fact, if L has no non-zero semisimple ideals, then it is elementary if and only if it is an almost algebraic A-algebra.

Lie algebras with almost constant-free derivations

Let L be a finite-dimensional Lie algebra of characteristic 0 admitting a nilpotent Lie algebra of derivations D. By a classical result of Jacobson, if D is constant-free (that is, without non-zero constants, xδ=0 for all δ∈D only if x=0), then L is nilpotent. We prove that if D is almost constant-free, then L is almost nilpotent in the following precise sense: if m is the dimension of the Fitting null-component with respect to D, then L contains a nilpotent subalgebra N of codimension bounded by a function of m and n, and the nilpotency class of N is bounded in terms of n only. This includes strengthening Jacobson's theorem by giving in the case m=0 a bound for the nilpotency class of L in terms of the number of weights of D. The main result can also be stated as a theorem about a locally finite Lie algebra over an algebraically closed field of characteristic 0 containing a nilpotent subalgebra with finitely many weights such that its Fitting null-component is of finite dimension—then the algebra contains a nilpotent subalgebra of finite codimension (with similar bounds for the codimension and the nilpotency class). The results on derivations are based on results on (Z/pZ)-graded Lie algebras, where p is a prime, with zero component of dimension m (these results in turn generalize the Higman–Kreknin–Kostrikin theorem on the nilpotency of Lie algebras with a regular automorphism of prime order).

On Lie Algebras with Many Nilpotent Subalgebras

ABSTRACT Let L be a torsion-free Lie algebra over a commutative domain. L is defined to be S-minimal non-nilpotent if L is not nilpotent but every proper subalgebra of L is nilpotent or its isolator is L. We give a description of the structure of L in the case that L is solvable.

Antinilpotent Lie Algebras

The class of antinilpotent Lie algebras closely related to the problem of constructing solutions with constant coefficients for the Yang-Mills equation is considered. A complete description of the antinilpotent Lie algebras is given. A Lie algebra is said to be antinilpotent if any of its nilpotent subalgebras is Abelian. The Yang-Mills equation with coefficients in a Lie algebra L has nontrivial solutions with constant coefficients if and only if the Lie algebra L is not antinilpotent. In Theorem 1, a description of all semisimple real antinilpotent Lie algebras is given. In Theorem 2, the problem of describing the antinilpotent Lie algebras is completely reduced to the case of semisimple Lie algebras (treated in Theorem 1) and solvable Lie algebras. The description of solvable antinilpotent Lie algebras is given in Theorem 3.

Cartan–Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems

In this paper, we prove Slodkowski version of the infinite-dimensional spectral mapping theorem and Cartan–Slodkowski version of the finite-dimensional spectral mapping theorem for nilpotent operator Lie subalgebras with respect to the various noncommutative functional calculi.

Gauged supergravity algebras from twisted tori compactifications with fluxes

Using the equivalence between Scherk–Schwarz reductions and twisted tori compactifications, we discuss the effective theories obtained by this procedure from M-theory and N = 4 type II orientifold constructions with Neveu–Schwarz and Ramond–Ramond form fluxes turned on. We derive the gauge algebras of the effective theories describing their general features, in particular the symplectic embedding in the duality symmetries of the theory. The generic gauge theory is non-Abelian and its gauge group is given by the semi-direct product of subgroups of SL ( 7 ) or SL ( p − 3 ) × SL ( 9 − p ) for p = 3 , … , 9 , with generators describing nilpotent subalgebras of e 7 ( 7 ) or so ( 6 , 6 ) (in M and type II theories, respectively).