Lie Algebras Of Maximal Class Research Articles

Biderivations, commuting mappings and 2-local derivations of -graded Lie algebras of maximal class

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with one-dimensional homogeneous components: m 0 , L 1 and m 2 . In this paper, we study their biderivations by considering the embedded mapping to derivation algebras. Then we determine commuting mappings on these algebras as an application of biderivations. Finally, local and 2-local derivations for these three algebras are characterized as the given gradings.

Thin subalgebras of Lie algebras of maximal class

For every field $$\mathbb{F}$$ which has a quadratic extension $$\mathbb{E}$$ we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension 2. We construct such Lie algebras as $$\mathbb{F}$$ -subalgebras of Lie algebras M of maximal class over $$\mathbb{E}$$ . We characterise the thin Lie $$\mathbb{F}$$ -subalgebras of M generated in degree 1. Moreover, we show that every thin Lie algebra L whose ring of graded endomorphisms of degree zero of L3 is a quadratic extension of $$\mathbb{F}$$ can be obtained in this way. We also characterise the 2-generator $$\mathbb{F}$$ -subalgebras of a Lie algebra of maximal class over $$\mathbb{E}$$ which are ideally r-constrained for a positive integer r.

Graded Lie algebras of maximal class of type n

Let n>1 be an integer. The algebras of the title, which we abbreviate as algebras of type n, are infinite-dimensional graded Lie algebras L=⨁i=1∞Li, which are generated by an element of degree 1 and an element of degree n, and satisfy [Li,L1]=Li+1 for i≥n. Algebras of type 2 were classified by Caranti and Vaughan-Lee in 2000 over any field of odd characteristic. In this paper we lay the foundations for a classification of algebras of arbitrary type n, over fields of sufficiently large characteristic relative to n. Our main result describes precisely all possibilities for the first constituent length of an algebra of type n, which is a numerical invariant closely related to the dimension of its largest metabelian quotient.

Graded Lie algebras of maximal class of type p

The algebras of the title are infinite-dimensional graded Lie algebras L=⨁i=1∞Li, over a field of positive characteristic p, which are generated by an element of degree 1 and an element of degree p, and satisfy [Li,L1]=Li+1 for i≥p. In case p=2 such algebras were classified by Caranti and Vaughan-Lee in 2003. We announce an extension of that classification to arbitrary prime characteristic, and prove several major steps in its proof.

Constituents of graded Lie algebras of maximal class and chains of thin Lie algebras

A thin Lie algebra is a Lie algebra L, graded over the positive integers, with its first homogeneous component L 1 of dimension two and generating L, and such that each nonzero ideal of L lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. If L 1 is the only diamond, then L is a graded Lie algebra of maximal class. We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method, with emphasis on a polynomial interpretation. Among else, we determine the possible values for the most fundamental parameter of such algebras, which is one less than the dimension of their largest metabelian quotient.

The schur multiplier of a nilpotent Lie algebra with derived subalgebra of maximum dimension

We give a new bound for the dimension of the Schur multiplier of an n-dimensional nilpotent Lie algebra of class c with the derived subalgebra of dimension n − 2. This new bound sharpens the earlier upper bounds on the dimension of Schur multiplier of such Lie algebras, especially for Lie algebras of maximal class.

The Schur multipliers of Lie algebras of maximal class

Let [Formula: see text] be a non-abelian nilpotent Lie algebra of dimension [Formula: see text] and [Formula: see text] be its Schur multiplier. It was proved by the second author the dimension of the Schur multiplier is equal to [Formula: see text] for some [Formula: see text]. In this paper, we classify all nilpotent Lie algebras of maximal class for [Formula: see text]. The dimension of Schur multiplier of such Lie algebras is also bounded by [Formula: see text]. Here, we give the structure of all nilpotent Lie algebras of maximal class [Formula: see text] when [Formula: see text] and then we show that all of them are capable.

Enumerating submodules invariant under an endomorphism

We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of (S-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta functions and establish meromorphic continuation of these zeta functions to the complex plane. As an application, we show that ideal zeta functions associated with nilpotent Lie algebras of maximal class have abscissa of convergence 2.

On Varieties of Lie Algebras of Maximal Class

AbstractWe study complex projective varieties that parametrize (finite-dimensional) filiform Lie algebras over ℂ, using equations derived by Millionshchikov. In the infinite-dimensional case we concentrate our attention on ℕ-graded Lie algebras of maximal class. As shown by A. Fialowski there are only three isomorphism types of ℕ-graded Lie algebras of maximal class generated by L1 and L2, L = 〈L1; L2〉. Vergne described the structure of these algebras with the property L = 〈L1〉. In this paper we study those generated by the first and q-th components where q > 2, L = 〈L1; Lq〉. Under some technical condition, there can only be one isomorphism type of such algebras. For q = 3 we fully classify them. This gives a partial answer to a question posed by Millionshchikov.

ON SCHUR MULTIPLIERS OF LIE ALGEBRAS AND GROUPS OF MAXIMAL CLASS

The Lie algebra analogue to the Schur multiplier has been investigated in a number of recent articles. We consider the multipliers of Lie algebras of maximal class, classifying these algebras with a certain additional property. The classification leads to a conjecture about a bound on the dimension of the multiplier for each of these algebras and also for p-groups of maximal class. The conjectures are then shown to hold.

THE STRUCTURE OF THIN LIE ALGEBRAS WITH CHARACTERISTIC TWO

Thin Lie algebras are graded Lie algebras [Formula: see text] with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L1, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if Lk is the second diamond of L, then the quotient L/Lk is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/Lk is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/Lk need not be metabelian in characteristic two. We describe here all the possibilities for L/Lk up to isomorphism. In particular, we prove that k + 1 equals a power of two.

The variety of Lie algebras of maximal class

We present an explicit description of the affine variety M Fil of Lie algebras of maximal class (filiform Lie algebras): the formulas of polynomial equations that determine this variety are written down. The affine variety M Fil can be considered as the base of the nilpotent versal deformation of the ℕ-graded Lie algebra m0.

Cohomology of graded lie algebras of maximal class with coefficients in the adjoint representation

We compute explicitly the adjoint cohomology of two ℕ-graded Lie algebras of maximal class (infinite-dimensional filiform Lie algebras) m0 and m2. It is known that up to an isomorphism there are only three ℕ-graded Lie algebras of maximal class. The third algebra from this list is the “positive” part L1 of the Witt (or Virasoro) algebra, and its adjoint cohomology was computed earlier by Feigin and Fuchs.

Cohomology of graded Lie algebras of maximal class

It was shown by A. Fialowski that an arbitrary infinite-dimensional N -graded “filiform type” Lie algebra g = ⊕ i = 1 ∞ g i with one-dimensional homogeneous components g i such that [ g 1 , g i ] = g i + 1 , ∀ i ⩾ 2 over a field of zero characteristic is isomorphic to one (and only one) Lie algebra from three given ones: m 0 , m 2 , L 1 , where the Lie algebras m 0 and m 2 are defined by their structure relations: m 0 : [ e 1 , e i ] = e i + 1 , ∀ i ⩾ 2 and m 2 : [ e 1 , e i ] = e i + 1 , ∀ i ⩾ 2 , [ e 2 , e j ] = e j + 2 , ∀ j ⩾ 3 and L 1 is the “positive” part of the Witt algebra. In the present article we compute the cohomology H ∗ ( m 0 ) and H ∗ ( m 2 ) with trivial coefficients, give explicit formulas for their representative cocycles and describe the multiplicative structure in the cohomology. Also we discuss the relations with combinatorics and representation theory. The cohomology H ∗ ( L 1 ) was calculated by L. Goncharova in 1973.

Graded Lie algebras of maximal class, III

We describe the isomorphism classes of infinite-dimensional N-graded Lie algebras of maximal class generated by their first homogeneous component over fields of characteristic two. This complements the analogous work by Caranti and Newman in the odd characteristic case.

Graded Lie algebras of maximal class. V

We describe the isomorphism classes of certain infinite-dimensional graded Lie algebras of maximal class, generated by an element of weight one and an element of weight two, over fields of even characteristic.

NON EXISTENCE OF COMPLEX STRUCTURES ON FILIFORM LIE ALGEBRAS

ABSTRACT The aim of this work is to prove the nonexistence of complex structures over nilpotent Lie algebras of maximal class (also called filiform).

Noncomplete affine structures on Lie algebras of maximal class

Every affine structure on Lie algebra 𝔤 defines a representation of 𝔤 in aff(ℝn). If 𝔤 is a nilpotent Lie algebra provided with a complete affine structure then the corresponding representation is nilpotent. We describe noncomplete affine structures on the filiform Lie algebra Ln. As a consequence we give a nonnilpotent faithful linear representation of the 3‐dimensional Heisenberg algebra.

Graded Lie Algebras of Maximal Class, II

We describe the isomorphism classes of infinite-dimensional N-graded Lie algebras of maximal class over fields of odd characteristic generated by their first homogeneous component.

Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class

AbstractWe investigate a class of infinite-dimensional, modular, graded Lie algebra in which the homogeneous components have dimension at most two. A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type.Another subclass of these algebras is strictly related to certain graded Lie algebras of maximal class, and exhibits a wide range of behaviours.