Killing Form Research Articles

On Lie algebras with only inner derivations

It is the purpose of this article to give a characterization of abelian by nilpotent Lie algebras which have only inner derivations. The motivation has been provided by two recent papers [3, 41 which deal with a similar problem in group theory. The Lie algebra case allows some modifications and simplifications as compared to the group theory case. We begin with a brief summary about Lie algebras which have only inner derivations and also about Lie algebras which must possess outer derivations. If the algebra has nondegenerate Killing form, then all derivations are inner [6, p. 741, while if the algebra is nilpotent, then there exist outer derivations [S]. This latter result has been extended in a variety of ways (see [ 131 and the bibliography given there). In particular, we men- tion that if the algebra is solvable and has no outer derivations, then its center must be 0 [2, 121. On the other hand there are examples of Lie algebras with nonzero center and only inner derivations (see [9]). Also a general method for constructing algebras with only inner derivations is described in [7]. In another direction, Schenkman’s famous derivation tower theorem [lo] asserts that an algebra with zero center can be sub- invariantly embedded in an algebra with only inner derivations. Schenkman’s paper also provides bounds on this embedding which are actually met as is shown in [8]. All Lie algebras considered here are finite dimensional over a field. Let

Almost paracontact and parahodge structures on manifolds

In this paper we study the paracomplex analogues of almost contact structures, and we introduce and study the notion of parahodge structures on manifolds. In particular, we construct new examples of paracomplex manifolds and we find all simply connected parahermitian symmetric coset spaces, which are the adjoint orbits of noncompact simple Lie groups, with parahodge structures induced by the Killing forms. This is done by (i) observing that a version of the results of A. Morimoto [4] on almost contact structures can be formulated and proved for almost paracontact structures, and by (ii) the methods of geometric quantization [3] applied to parahermitian symmetric triples [1] in conjunction with results of [7]. Two of the main results are Theorem 2.5 (which ties together the above structures) and Corollary 3.9.

Track formulas and derivations in simple jordan pairs

In finite dimensional Lie algebras, Jordan algebras, and other algebraic structures the study of derivations has been facilitated by having a nontrivial trace formula on hand (see for example [∗]) . Tuere is no common pattern in proving these trace formulas, they all depend on the underlying structures. In this note we derive such a trace formula for finite dimensional central simple Jordan pairs. We use it to determine all derivations the Killing form and the dimensions of the derivation algebras of the Jordan pairs. Dur primary tool is a Trace Reduction Formula.

KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank

Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.

Local BRST symmetry and the geometry of gauge-fixing

Extended BRST transformations are constructed for arbitrary gauge theories by consideration of their algebra alone. Dynamics are introduced by writing the most general lagrangian which is globally extended-BRST invariant and provides a gauge-fixing term of the fork 1 2 F αγ αβF β . It is shown that all F α ≡ γ αβ F β must be variational derivatives of one universal scalar function F proportional to the squares of the classical gauge fields, while γ αβ, in principle still arbitrary, is interpreted as the generalized Killing form. A superspace formulation of Yang-Mills theory is used to promote the BRST symmetry to a local symmetry.

Rational Pontryagin classes and Killing forms

Rational Pontryagin classes and Killing forms

Invariant volumes of compact groups

Invariant volumes are calculated for all compact simple Lie groups. The measure is defined by means of the Riemannian metric, induced in the group by the Killing form, so the total normalisation is fixed. The coset space aspects are discussed.

Una propiedad de las álgebras de Lie semisimples con forma de Killing indefinida

real connected Lie Groups may be or not the reunion of their uniparametric subgroups. The aim of this note is to study a characterization of the Lie Groups such that are the reunion of their uniparametric subgroups . To begin with, it is proved that if G is a real connected semisimple Lie Group and the Killing form on the Lie Algebra of G is an indefinide form, G is not the reunion of their uniparametric subgroups. It is proved the next result : Let G be a real connected Lie Group . G is the reunion of their uniparametric subgroups if and only if rad(G) is exponential and the Killing form on the Lie Algebra of G/rad(G) is a definide form (positive or negative).

Yano vector fields and conformally related spaces

We discuss some geometrical properties of Killing forms of order three and exhibit a procedure for constructing spaces admitting such tensors. As an example we prove that all conformally flat spaces admitting such a tensor are related to spaces of constant curvature.

K-spaces of maximal rank

We consider a special type of K-space, i.e., almost-Hermitian manifolds whose fundamental form is a Killing form. The K-spaces of this type are characterized by the fact that their dimension is equal to the rank of the covariant derivative of the structure form. A number of the properties of such spaces are established (they are Einstein, compact, have finite fundamental group, etc.). It is proved that every K-space is locally equivalent to a product of a K-space of maximal rank and a Kahler manifold. The K-spaces with zero holomorphic sectional curvature are studied.

The Killing form for graded Lie algebras

A Killing form with nice symmetry and invariance properties is constructed for an arbitrary graded Lie algebra. When this form is nondegenerate, Casimir operators can be constructed, and the graded Lie algebra possesses properties analogous to those possessed by semisimple ordinary Lie algebras.

The manifold of the Lagrangean subspaces of a symplectic vector space

Let (E, a) be a real 2n-dimensional symplectic vector space with symplectic form a, i.e., a is a nondegenerate skew-symmetric bilinear form on E. Then an n-dimensional subspace λ of E will be called a Lagrangean subspace if alλ = 0 holds. The set Λ(E) of all Lagrangean subspaces of (E, a) has a structure of n(n + l)-dimensional compact connected regular algebraic variety. If we put A\λ): = [μ 6 Λ(E) I dim (λ Π μ) = k} for λ £ Λ(E), then Λ°U) is a cell (i.e., diίϊeomorphic to i r ( n + 1 ) / 2 ) for any λ e Λ(E). Moreover Σ tf): = |J*2>i Λ*W) is an algebraic subvariety of Λ(E), and defines an oriented cycle of codimension one, whose Poincare dual is a generator of H(Λ(E), Z) = Z and defines the Maslov-Arnold index [1], [3], [4]. This index plays an important role in the proof of Morse index theorem in the calculus of variations [4]. In the present note, we shall give a differential geometric characterization of 2 U)> i ^ ? by introducing an appropriate riemannian metric on Λ{E) we shall show that 2 (λ) is the cut locus of some μ e Λ(E) and Λ°(X) is the interior set of μ. In fact, take a basis {eu fά] (1 , and ao((p, q), (p q')): = (q, p'y — </?, q'y. We put ^ 0: = {(p,0)|p e R } and μ0: = {(0, q)\qe R } which are of course Lagrangean subspaces. Then the (real representation of) unitary group U(n) naturally acts on Λ(ή): = A(R) transitively, and its isotropic subgroup at Λo is given by O(n). Thus Λ(ή) is diffeomorphic to U(n)/O(n). Now M = U(n)/O(n) has a structure of a compact symmetric space whose riemannian structure comes from the Killing form of the Lie algebra of U{ή). In the present note we shall determine the cut locus and the first conjugate locus of a point of M, from which we may prove the assertion mentioned above. For compact simply connected symmetric spaces, it is known that the cut locus and the first conjugate locus of any point coincide with each other (see [2]). Note that 7Γχ(M) = Z for our manifold M = U(n)/O(n). Finally we shall determine all closed geodesies of M and calculate their intersection number with the oriented cycle U**iΛ*(n).

A proof of Blattner's conjecture

Let G be a connected, semisimple Lie group. In Harish-Chandra's work on the Plancherel formula, the discrete series of irreducible unitary representations [5] plays a crucial role. Roughly speaking, besides the discrete series itself, the representations which enter the Plancherel decomposition of L2(G) can all be built up from discrete series representations of subgroups of G. Among the various irreducible unitary representations, the discrete series representations are therefore of particular importance. The main technique in the representation theory of compact Lie groups is to investigate the restriction of any given representation to a maximal torus. Similarl~r to understand the structure of an irreducible representation of a noncompact semisimple Lie group, one may try to determine its restriction to a maximal compact subgroup. Blattner's conjecture predicts how the discrete series representations should break up under the action of a maximal compact subgroup. It formally resembles the formula for the multiplicity of a weight for finite-dimensional representations. A companion statement to Blattner's conjecture, analogous to the theorem of the highest weight, characterizes discrete series representations, up to infinitesimal equivalence, in terms of their restriction to a maximal compact subgroup [13]. Once and for all, we make the assumption that G contains a compact Cartan subgroup. Exactly in this situation G has a non-empty discrete series [5]. Let K be a maximal compact subgroup, with HcK. The Lie algebras of G, K, H will be written as go, ~0, bo, and their complexifications as g, [, b. A non-zero root of (g, b) is called compact or noncompact, depending on whether or not its root space lies in L We denote the sets of compact and noncompact roots by, respectively, q~ and 4"; ~ = ~c w 4" is thus the setof all non-zero roots of (g, b). The Killing form of g induces an inner product ( , ) on ib*, the space of linear functions on b which assume imaginary values on b0An element ~teib* is said to be singular if it is perpendicular to at least one ct~q~, and nonsingular otherwise. The differentials of the characters of H form a lattice A cib*, which contains ~b. Since

On the structure of simple pseudo Lie algebras and their invariant bilinear forms

By definition simple pseudo Lie algebras do not contain any nontrivial ideal. We show that ’’graded simplicity’’ implies ’’simplicity’’ and discuss the uniqueness of invariant bilinear forms on a simple pseudo Lie algebra. A lot of examples of simple pseudo Lie algebras is given together with their invariant bilinear forms. Under certain general assumptions we derive that the Lie algebra ℊ contained in a simple pseudo Lie algebra 𝒶 is reductive. Assuming that ℊ is reductive, we prove that the ’’adjoint representation of ℊ in the odd subspace of 𝒶’’ is completely reducible with at most two irreducible components. Finally we show that the pseudo Lie algebras with nondegenerate ’’generalized Killing form’’ are direct products of simple pseudo Lie algebras.

Global Solvability of the Casimir Operator

Let G be a semisimple Lie group, g its Lie algebra. The Killing form on g is a nondegenerate symmetric bilinear form which induces an isomorphism a: g* -. g. The Casimir operator is the unique biinvariant differential operator on G which annihilates constants and whose principal symbol is the pullback of the Killing form under the isomorphism a. Let C denote the Casimir operator and C-(G) the space of infinitely differentiable functions on G. THEOREM. If G is a non-compact semisimple Lie group with finite center, f e C-(G), then there is a u e C-(G) such that Cu = f. Since G is not compact, C is not elliptic but ultrahyperbolic. Analogous questions have been considered by S. Helgason [3] and A. Cerezo and F. Rouviere [1]. Helgason proves local solvability of all biinvariant differential operators and Cerezo and Rouviere prove global solvability of the Casimir operator on complex semisimple Lie groups. The latter authors prove that the imaginary part of the complex Casimir operator on SL(2, C) is not globally solvable. This shows that not every biinvariant operator is globally solvable. Henceforward G will denote a group satisfying the hypotheses of the theorem and g will denote the Lie algebra of left invariant vector fields on G. Proof. The remainder of the paper is devoted to a proof of this theorem. In spite of its length the proof is simple in outline. A result from functional analysis and Hormander's propagation of singularities theorem can be combined to show that it suffices to demonstrate three things.

Classification of some 2-graded lie algebras

We continue the study of the structure of 2-graded Lie algebras (see next section for the definition) which was begun by Hochschild [6] and Hochschild and the author [4]. AU semi-simple 2-graded Lie algebras over an algebraically closed field of characteristic 0 have been determined in [4]. In this paper we introduce the analog of the Killing form for 2-graded Lie algebras. AS in the case of ordinary Lie algebras, this form plays an important role in the structure theory. For semi-simple 2-graded Lie algebras this form is non-degenerate although the converse is not true. Semi-simplicity of 2-graded Lie algebras is preserved under both extension and restriction of scalars. Our main result is the classification of indecomposable 2-graded Lie algebras L over an algebraically closed field of characteristic 0 under the condition that the zero component Lo of L be simple (Theorem 8). There are seven infinite series of such algebras. This classification is accomplished in two steps. The first step is the classification of those 2-graded Lie algebras L which are simple and have Lo simple (Theorem 6). In the proof of this theorem we need the following fact: If‘ L, is a simple Lie algebra then the multiplicity 172 r) (resp. nzl ) of L, in the symmetric power S2(Lo) (resp. the exterior power A-(Lo)) is nzo = I if Lo =: rl ,t (112 2),

Simple supersymmetries

Two infinite families of simple graded Lie algebras (GLA’s) over the complex numbers are described: the special linear algebras SL(m‖n) [whose Bose sector is the direct sum of a one-dimensional algebra with the ordinary Lie algebra SL(m) ×SL(n)], and the orthosymplectic algebras OSp(2r‖s) [with Bose sector Sp(2r) ×O(s)]. The GLA’s of physics fit into these two families either directly or via Inönü–Wigner contraction. These algebras along with further exceptional GLA’s constitute all the classical GLA’s, i.e., all GLA’s with a nondegenerate metric (not necessarilly Killing) form. The existence of infinite families of hyperexceptional GLA’s, i.e., of GLA’s that are simple but not classical is pointed out.

On Derivations of Lie Algebras

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

The asymptotic expansion of the zeta-function of a compact semisimple Lie group

If G G is a connected, simply connected, semisimple Lie group with metric given by the negative of the Killing form and zeta-function Z ( t ) Z(t) , then \[ Z ( t ) = Vol ⁡ G ( 4 π t ) dim ⁡ G / 2 exp ⁡ | δ | 2 t + exponentially small error as t ↓ 0. Z(t) = \frac {{\operatorname {Vol} G}} {{{{(4\pi t)}^{\dim G/2}}}}\exp |\delta {|^2}t + {\text {exponentially small error as }}t \downarrow 0. \]

Enhanced social docility in male hooded rats by dermal cautery of the vibrissal pads

Ablation of the vibrissal pads in rats causes subsequent deposition of scar tissue with little or no regrowth of the vibrissae. Cauterized and intact mature male Long Evans rats were tested for shockelicited fighting, mouse killing, and colony intrusion forms of laboratory-induced aggression. The results revealed that only conspecific social fighting is blocked by ablation of the major vibrissal follicles. Although no significant group differences were noted in tests for mouse killing, shock-elicited paired fighting and territorial defense against a strange intruder were minimal in cauterized groups. The results emphasize the importance of specific sensory experience in reference to distinct forms of aggressive responding and support a new experimental technique for further investigation of sensory interactions with sources of aggressive behavior.