Finite-dimensional Lie Research Articles

Irreducible Representations for the Baby Tits–Kantor–Koecher Algebra

The baby Tits–Kantor–Koecher (TKK) algebra constructed from the smallest (nonlattice) semilattice is related to the “smallest” extended affine Lie algebras other than the finite dimensional simple Lie algebras and the affine Kac–Moody algebras. In this article, we classify the finite dimensional irreducible representations for the baby TKK algebra. It turns out that such representations can be lifted from modules of direct sums of finitely many copies of the simple Lie algebra sp 4(ℂ).

On the non-abelian tensor product of Lie algebras

Ellis (G. Ellis, A non-abelian tensor product of Lie algebras, Glasgow Math. J. 39 (1991), pp. 101–120.) introduced the notion of the non-abelian tensor product L ⊗ K for a pair of Lie algebras L, K and investigated some of its fundamental properties. In this article, we study some common properties between Lie algebras and their tensor products, and present some bounds on the nilpotency class and solvability length of L ⊗ K, provided such information is given on L or K. Also, we give some upper and lower bounds for the dimension of L ⊗ K if L and K are finite-dimensional nilpotent Lie algebras and ideals of a single Lie algebra.

Nilpotent-by-abelian Lie algebras of type FPm

AbstractLet L be a finitely generated Lie algebra which is a split extension of a free nilpotent Lie algebra N by a finite dimensional abelian Lie algebra. Let V denote the quotient of N by its commutator subalgebra; we can regard V as a module for L/N. We discuss the relationship between the homological finiteness properties of V and those of L. In particular, we show that if L has type FPm and N has class c then V is 1 + c(m − 1)-tame (equivalently, the (1 + c(m − 1))th tensor power of V is finitely generated under the diagonal action of L/N).

Nilpotency of n-tuple Lie algebras and associative n-tuple algebras

We obtain conditions for the nilpotency of finite-dimensional n-tuple Lie algebras and finite-dimensional associative n-tuple algebras. The established conditions are analogous to theorems of Engel and Wedderburn for Lie algebras and associative algebras.

FINITE-DIMENSIONAL SPECIAL ODD HAMILTONIAN SUPERALGEBRAS IN PRIME CHARACTERISTIC

In this paper, we study a new family of finite-dimensional simple Lie superalgebras of Cartan type over a field of characteristic p > 3, the so-called special odd Hamiltonian superalgebras. The spanning sets are first given and then the grading structures are described explicitly. Finally, the simplicity and the dimension formulas are determined. As application, using the dimension formulas, we make a comparison between the special odd Hamiltonian superalgebras and the other known families of finite-dimensional simple modular Lie superalgebras of Cartan type.

Cohomology of Oriented Tree Diagram Lie Algebras

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this article, we use Hodge Laplacian to study the cohomology of these Lie algebras. The “total rank conjecture” and “b 2-conjecture” for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler–Poincaré principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.

On varieties parametrizing graded complex Lie algebras

We study spaces parametrizing graded complex Lie algebras from geometric as well as algebraic point of view. If R is a finite-dimensional complex Lie algebra, which is graded by a finite abelian group of order n, then a graded contraction of R, denoted by \({R^{(\varepsilon)}}\), is defined by a complex n × n-matrix \({\varepsilon=(\varepsilon_{ij})}\), i, j = 1, . . . , n. In order for \({R^{(\varepsilon)}}\) to be a Lie algebra, \({\varepsilon_{ij}}\) should satisfy certain homogeneous equations. In turn, these equations determine a projective variety X R . We compute the first homology group of an irreducible component M of X R , under some assumptions on M. We look into algebraic properties of graded Lie algebras \({R^{(\varepsilon)}}\) where \({\varepsilon\in X_R}\).

A Generalization of Extended Affine Lie Algebras

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.

Actions of Lie Superalgebras on Reduced Rings

In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk for finite dimensional restricted Lie algebras. Our main result is

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].

Finite-dimensional solvable Lie algebras generated by normal operators are commutative

It is proved that any finite-dimensional solvable Lie algebra generated by normal operators is commutative, and some other related results are given.

On the Dimensions of Commutative Subalgebras and Subgroups of Nilpotent Algebras and Lie Groups of Class 2

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.

On rational isomorphisms of Lie algebras

Let \(\mathfrak{n}\) be a finite-dimensional noncommutative nilpotent Lie algebra for which the ring of polynomial invariants of the coadjoint representation is generated by linear functions. Let \(\mathfrak{g}\) be an arbitrary Lie algebra. We consider semidirect sums \(\mathfrak{n} \dashv _\rho \mathfrak{g}\) with respect to an arbitrary representation ρ: \(\mathfrak{g}\) → der \(\mathfrak{n}\) such that the center z\(\mathfrak{n}\) of \(\mathfrak{n}\) has a ρ-invariant complement.

A Comparison of Deformations and Geometric Study of Varieties of Associative Algebras

The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. In each case we describe the corresponding notions of degeneration and rigidity. We illustrate these notions by examples and give some general properties. The last part of this work shows how these notions help in the study of varieties of associative algebras. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general approach called global deformation follows from a general theory by Schlessinger and was developed by A. Fialowski in order to deform infinite-dimensional nilpotent Lie algebras. In a nonstandard framework, M. Goze introduced the notion of perturbation for studying the rigidity of finite-dimensional complex Lie algebras. All these approaches share the common fact that we make an “extension” of the field. These theories may be applied to any multilinear structure. In this paper, we will be dealing with the category of associative algebras.

Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials

Hecke proved that the theta series of a positive definite even unimodular lattice is a polynomial of the well-known Essenstein series E4(z) and the Ramanujan series Δ24(z). A natural question is what kind of polynomials in E4(z) and Δ24(z) could be the theta series of positive definite even unimodular lattices. In this paper, we find two combinatorial identities on the theta series of the root lattices of the finite-dimensional simple Lie algebras of type D4n and the cosets in their integral duals, in terms of E4(z) and Δ24(z). Using these two identities, we prove that three families of weighted symmetric polynomials of two fixed families of polynomials of E4(z) and Δ24(z) are the theta series of certain positive definite even unimodular lattices, obtained by gluing finitely many copies of the root lattices of the finite-dimensional simple Lie algebras of type D2n. The results also show that the full permutation groups are the hidden symmetry of the theta series of certain unimodular lattices.

The Cartan-Type Modular Lie Superalgebra KO

ABSTRACT The finite-dimensional odd contact Lie superalgebras KO(n, n + 1, t ) over a field of prime characteristic are studied, where n is a positive integer and t is an n-tuple of non-negative integers. In particular, it is proven that KO(n, n + 1, t ) is simple and has no non-singular associative bilinear forms. Moreover, an explicit description of the derivation superalgebra of KO(n, n + 1, t ) is given, and as a consequence it is shown that the outer derivation superalgebra of KO(n, n + 1, t ) is Abelian of dimension . Communicated by K. Misra.

Number of representations providing noiseless subsystems

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free (DF) subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function ${f}_{d}(n)$ which is the fraction of all $d$-dimensional quantum systems which preserve $n$ bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.

ON THE PRIMARY DECOMPOSITION THEOREM OF MODULAR LIE SUPERALGEBRAS

This gives some identities of associative Lie superalgebras and some properties of modular Lie superalgebras. Furthermore, the primary decomposition theorem of modular Lie superalgebras is shown. It is well known that the primary decomposition theorem of modular Lie algebras has played an important role in the classification of the finite-dimensional simple modular Lie algebras (see [5, 6]). Analogously, the primary decomposition theorem of modular Lie superalgebras may play an important role in the open classification of the finite dimensional simple modular Lie superalgebras.

On graphs and Lie rings

From a finite oriented graph Γ, finite-dimensional graded nilpotent Lie rings \(\mathfrak{l}\left( \Gamma \right)\)(Γ) and \(\mathfrak{g}\left( \Gamma \right)\)(Γ) are naturally constructed; these rings are related to subtrees and connected subgraphs of Γ, respectively. Diverse versions of these constructions are also suggested. Moreover, an embedding of Lie rings of the form \(\mathfrak{l}\left( \Gamma \right)\)(Γ) in the adjoint Lie rings of finite-dimensional associative rings (also determined by the graph Γ) is indicated.

On Depth-Three Graded Lie Algebras of Characteristic Three with Classical Reductive Null Component#

We consider the finite-dimensional transitive irreducible graded Lie algebras of depth three over an algebraically closed field of characteristic three with classical reductive component L 0, and show that the L 0-module L −1 in such algebras must be restricted. This result contrasts with the depth-one case, in which non-restricted modules L −1 can occur, but is similar to the depth-two case, where L −1 is necessarily restricted.