Heisenberg Lie Color Algebras

In this article, we give a sufficient condition for a Lie color algebra to be complete. The color derivation algebra Der(?) and the holomorph L of finite dimensional Heisenberg Lie color algebra ? graded by a torsion-free abelian group over an algebraically closed field of characteristic zero are determined. We prove that Der(?) and Der(L) are simple complete Lie color algebras, but L is not a complete Lie color algebra.     

色李代数中的李定理

作者证明了无挠群上的色李代数的李定理, 同时, 也给出例子说明 无挠性是必要的     

Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Let \(\mathfrak g\) be a semisimple Lie algebra over a field \(\mathbb K\), \(\text{char}\left( \mathbb{K} \right)=0\), and \(\mathfrak g_1\) a subalgebra reductive in \(\mathfrak g\). Suppose that the restriction of the Killing form B of \(\mathfrak g\) to \(\mathfrak g_1 \times \mathfrak g_1\) is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra \(\mathfrak h_1\) of \(\mathfrak g_1\) there is a unique Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) containing \(\mathfrak h_1\); ( 2) \(\mathfrak g_1\) is self-normalizing in \(\mathfrak g\); ( 3) The B-orthogonal \(\mathfrak p\) of \(\mathfrak g_1\) in \(\mathfrak g\) is simple as a \(\mathfrak g_1\)-module for the adjoint representation. We give some answers to this natural question: For which pairs \((\mathfrak g,\mathfrak g_1)\) do ( 1), ( 2) or ( 3) hold? We also study how \(\mathfrak p\) in general decomposes as a \(\mathfrak g_1\)-module, and when \(\mathfrak g_1\) is a maximal subalgebra of \(\mathfrak g\). In particular suppose \((\mathfrak g,\sigma )\) is a pair with \(\mathfrak g\) as above and σ its automorphism of order m. Assume that \(\mathbb K\) contains a primitive m-th root of unity. Define \(\mathfrak g_1:=\mathfrak g^{\sigma}\), the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) \((\mathfrak g,\mathfrak g_1)\) satisfies ( 1); (b) For m prime, \((\mathfrak g,\mathfrak g_1)\) satisfies ( 2).     

Lie Algops and Simple Lie Algebras

ABSTRACT

Pairs (A, L) with A a commutative algebra and L a Lie algebra acting on A by derivations, called Lie algops, are studied as algebraic structures over arbitrary fields of arbitrary characteristic. Lie algops possess modules and tensor products—and are considered with respect to a central simple theory.

The simplicity problem of determining the faithful unital simple Lie algops ( A, L ) is of interest since the corresponding Lie algebras AL are usually simple (Jordan, 2000 Jordan , D. A. ( 2000 ). On the simplicity of Lie algebras of derivations of commutative algebras . J. Algebra 228 : 580 – 585 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). For locally finite Lie algops, and up to purely inseparable descent, this problem reduces by way of closures to the closed central simplicity problem of determining those which are closed central simple.

The simplicity and representation theories for locally nilpotent separably triangulable unital Lie algops are of particular interest because they relate to the problems of classifying simple Lie algebras of Witt type and their representations. Of these, the simplicity theory reduces to that of Jordan Lie algops.

The main Theorems 7.3 and 7.4 reduce the simplicity and representation theories for Jordan Lie algops to the simplicity and representation theories for simple nil and toral Lie algops.     

Semisimple Lie Algebras of Differential Operators

Starting from the commutation relations in a complex semisimple Lie algebra , one may obtain a space of vector fields on Euclidean space such that and are isomorphic when is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of is a Borel subalgebra . Furthermore, one may adjoin to the vector fields in multiplication operators to obtain an -parameter family of distinct presentations of as spaces of differential operators, where is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of can be presented in this way. All of this is carried out explicitly for arbitrary . In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.     

Nonlinear Maps on Simple Lie Algebras Preserving Lie Products

Let 𝔤 be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero. It is proved in this article that a bijective map ? on 𝔤 preserves Lie products if and only if it is a composition of a Lie algebra automorphism and a bijective map extended by an automorphism of the base field.     

On the Quantum KPRV Determinants for Semisimple and Affine Lie Algebras

Let g be a semisimple or affine Lie algebra and U _q (g) its quantized enveloping algebra. Extending earlier work, the KPRV determinant for an admissible integrable U _q (g) module V relative to a parabolic subalgebra pg is defined and shown to be nonzero. These determinants had previously been evaluated for g semisimple and p a Borel subalgebra. The present results can be used to extend this to g affine as will be shown in a subsequent publication.For a parabolic subalgebra the evaluation of these determinants is much more difficult. For appropriate overalgebras of the primitive quotients of the enveloping algebra U(g) defined by one-dimensional representations of p, these determinants had been calculated for g semisimple. However the quantum case is interesting because it is unnecessary to pass to overalgebras and besides for U(g):g affine, it is not even clear how these determinants should be defined. Here for g semisimple, the degrees of the determinants are computed and shown to depend on being the same type of functions as in the enveloping algebra case; yet in a different fashion. Some special cases (in type A ₄) are computed explicity. Here, as in the Borel case, the determinants take a remarkably simple form and notably can be expressed as a product of linear factors. However compared to the enveloping algebra case one finds additional factors corresponding to what are called quantum zeros and whose origin remains unknown.     

Extremal Properties of Bases for Representations of Semisimple Lie Algebras

Let be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of with weight basis . The supporting graph P of is defined to be the directed graph whose vertices are the elements of and whose colored edges describe the supports of the actions of the Chevalley generators on V. Four properties of weight bases are introduced in this setting, and several families of representations are shown to have weight bases which have or are conjectured to have each of the four properties. The basis can be determined to be edge-minimizing (respectively, edge-minimal) by comparing P to the supporting graphs of other weight bases of V. The basis is solitary if it is the only basis (up to scalar changes) which has P as its supporting graph. The basis is a modular lattice basis if P is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of sl(n, ) serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of sp(2n, ) and the irreducible one-dimensional weight space representations of any semisimple Lie algebra, are shown to be solitary and edge-minimal and to have modular lattice supports. Tools developed here are used to construct uniformly the irreducible one-dimensional weight space representations. Similar results for certain irreducible representations of the odd orthogonal Lie algebra o(2n + 1, ), the exceptional Lie algebra G ₂, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.     

Coset Decomposition for Semisimple Hopf Algebras

The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given.     

Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ? on 𝔤 is said to preserve solvability in both directions if ?, as well as ?^?1, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ? on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.     

Embedding of Lie Algebras in Isogeneralized Structurable Algebras

In a previous paper, the first-named author introduced generalized structurable algebras, while the second-named author introduced the isotopies of Lie algebras. In this paper, we combine the two analyses, submit the notion of isogeneralized structural algebras, and show that they include Lie algebras, all their axiom-preserving generalizations of graded, supersymmetric or isotopic type, as well as numerous other algebras.     

A new characterization of semisimple Lie algebras

Using Casimir elements, we characterize the semisimple Lie algebras among the quadratic Lie algebras. This characterization gives, in particular, a generalization of a consequence of Cartan's second criterion.

     

Flat Nonunimodular Lorentzian Lie Algebras

A flat Lorentzian Lie algebra is a left symmetric algebra endowed with a symmetric bilinear form of signature (?, +,…, +) such that left multiplications are skew-symmetric. In geometrical terms, a flat Lorentzian Lie algebra is the Lie algebra of a Lie group with a left-invariant Lorentzian metric with vanishing curvature. In this article, we show that any flat nonunimodular Lorentzian Lie algebras can be obtained as a double extension of flat Riemannian Lie algebras. As an application, we give all flat nonunimodular Lorentzian Lie algebras up to dimension 4.     

算子李代数的一些性质

算子群作为群的推广,算子群在群论里有许多应用.类似地,作为算子群和李代数的推广,算子李代数将会有许多应用.给出了算子李代数的一些性质,得到了算子李代数半单性的充分必要条件.同时得到算子李代数半单性与非退化killing型的关系.     

Triple Homomorphisms of Perfect Lie Algebras

Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([x, [y, z]]) = [f(x), [f(y), f(z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms.     

The Isomorphism Problem for Universal Enveloping Algebras of Lie Algebras

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ≅ U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H. Presented by D. Passman