On spherical nilpotent orbits and beyond

Construction process for simple Lie algebras

Left-Symmetric Structures on Simple Modular Lie Algebras

CHAPTER 4 - CARTAN'S CRITERION

According to a theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the poset of nilpotent orbits, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type A2k−1. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper.In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an SL2(C)-variety whose normalization is A2, an Sp4(C)-variety whose normalization is A4, and a two-dimensional variety whose normalization is the simple surface singularity A3. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the nilpotent cone at a point in the subregular orbit.

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

Geometric quantization of real minimal nilpotent orbits

We consider a complex simple Lie algebra $${\mathfrak{g}}$$ , with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct $${\mathfrak{g}}$$ -invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra $${{\rm U}(\mathfrak{g})}$$ . This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters.

Symmetry reduction for the Kadomtsev-Petviashvili equation using its infinite dimensional invariance group

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo $H$-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type $B_n$ with $|2|$-grading do not contain non-Heisenberg pseudo $H$-type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types $A_n$, $C_n$ and $D_n$ provide such a possibility. Among exceptional algebras only $F_4$ and $E_6$ contain non-Heisenberg pseudo $H$-type Lie algebras as their negative part of $|2|$-grading. An analogous question addressed to real simple graded Lie algebras is more difficult, and we give results revealing the main differences with the complex situation.

In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra \(\mathcal{D}\) and the seven-dimensional central simple commutative algebra \(\mathcal{C}\). We prove that every local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is a derivation, and every 2-local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also a derivation. We also prove that every local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is an automorphism, and every 2-local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also an automorphism.

Preliminary group classification of the nonlinear differential–difference equations

In [10] Patera and Sharp conceived a new relation, subjoining, between semisimple Lie algebras. Our objective in this paper is twofold. Firstly, to lay down a mathematical formalization of this concept for arbitrary Lie algebras. Secondly, to give a complete classification of all maximal subjoinings between Lie algebras of the same rank, of which many examples were already known to the above authors.The notion of subjoining is a generalization of the subalgebra relation between Lie algebras. To give an intuitive idea of what is involved we take a simple example. Suppose is a complex simple Lie algebra of type B2. Let be a Cartan subalgebra of and Δ the corresponding root system. We have the standard root diagramInside B2 there lies the subalgebra A1 × A1 which can be identified with the sum of and the root spaces corresponding to the long roots of B2.

Let G be a simply connected semisimple algebraic group with Lie algebra g, let G(0) subset of G be the symmetric subgroup defined by an algebraic involution sigma and let g(1) subset of g be the isotropy representation of G(0). Given an abelian subalgebra a of g contained in g(1) and stable under the action of some Borel subgroup B-0 subset of G(0), we classify the B-0-orbits in a and characterize the sphericity of G(0)a. Our main tool is the combinatorics of sigma-minuscule elements in the affine Weyl group of g and that of strongly orthogonal roots in Hermitian symmetric spaces.

In this lecture we indicate how to complete the last step in the process outlined at the beginning of Part II: to take our knowledge of the classification and representation theory of complex algebras and groups and deduce the corresponding statements in the real case. We do this in the first section, giving a list of the simple classical real Lie algebras and saying a few words about the corresponding groups and their (complex) representations. The existence of a compact group whose Lie algebra has as complexification a given semisimple complex Lie algebra makes it possible to give another (indeed, the original) way to prove the Weyl character formula; we sketch this in §26.2. Finally, we can ask in regard to real Lie groupsGa question analogous to one asked for the representations of fmite groups in §3.5: which of the complex representationsVofGactually come from real ones. We answer this in the most commonly encountered cases in §26.3. In this final lecture, proofs, when we attempt them, are generally only sketched and may require more than the usual fortitude from the reader.KeywordsIrreducible RepresentationSpin RepresentationReal FormCartan SubalgebraExterior PowerThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An inner ideal of a Lie algebra L over a commutative ring k is a k-submodule B of L such that [ B [ B L ] ] ⊆ B [B[BL]] \subseteq B . This paper investigates properties of inner ideals and obtains results relating ad-nilpotent elements and inner ideals. For example, let L be a simple Lie algebra in which D y 2 = 0 D_y^2 = 0 implies y = 0 y = 0 , where D y {D_y} denotes the adjoint mapping determined by y. If L satisfies the descending chain condition on inner ideals and has proper inner ideals, then L contains a subalgebra S = ⟨ e , f , h ⟩ S = \langle e,f,h\rangle , isomorphic to the split 3-dimensional simple Lie algebra, such that D e 3 = D f 3 = 0 D_e^3 = D_f^3 = 0 . Lie algebras having such 3-dimensional subalgebras decompose into the direct sum of two copies of a Jordan algebra, two copies of a special Jordan module, and a Lie subalgebra of transformations of the Jordan algebra and module. The main feature of this decomposition is the correspondence between the Lie and the Jordan structures. In the special case when L is a finite dimensional, simple Lie algebra over an algebraically closed field of characteristic p > 5 p > 5 this decomposition yields: Theorem. L is classical if and only if there is an x ≠ 0 x \ne 0 in L such that D x p − 1 = 0 D_x^{p - 1} = 0 and if D y 2 = 0 D_y^2 = 0 implies y = 0 y = 0 . The proof involves actually constructing a Cartan subalgebra which has 1-dimensional root spaces for nonzero roots and then using the Block axioms.