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Non-Abelian Minimal Closed Ideals of Transitive Lie Algebras. (MN-25):

Non-Abelian Minimal Closed Ideals of Transitive Lie Algebras. (MN-25):

Copyright Date: 1981
Pages: 228
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  • Book Info
    Non-Abelian Minimal Closed Ideals of Transitive Lie Algebras. (MN-25):
    Book Description:

    The purpose of this book is to provide a self-contained account, accessible to the non-specialist, of algebra necessary for the solution of the integrability problem for transitive pseudogroup structures.

    Originally published in 1981.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-5365-6
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. [i]-[ii])
  2. Table of Contents
    (pp. [iii]-[iv])
  3. Preface
    (pp. [v]-[vi])
    Jack F. Conn
  4. Introduction
    (pp. 1-14)

    Transitive pseudogroups of local diffeomorphisms preserving geometric structures on manifolds have been studied by many authors; the origins of this subject are classical, and may be said to lie in the works of Sophus Lie and Élie Cartan. The structure of such a pseudogroup Γ acting on a manifold X is reflected in the structure of the Lie algebra of formal infinitesimal transformations of Γ, that is to say, those formal vector fields on X which are formal solutions to the linear partial differential equation which defines the infinitesimal transformations of Γ. The Lie algebras of formal vector fields obtained...

  5. 1. Preliminaries
    (pp. 15-64)

    Throughout this section, we denote by K an arbitrary field of characteristic zero which is endowed with the discrete topology.

    Let V be a Hausdorff topological vector space over K. Then V is said to be linearly compact if:

    (i) V is complete; and

    (ii) There exists a fundamental system {Vα} of neighborhoods of 0 in V such that each Vαis a vector subspace of finite codimension in V.

    An example of such a space is provided by the local algebra\[\text{F}=\text{K}[[{{\text x}_{1}},\ \ldots \ ,{{\text{x}}_{\text n}}]]\]of formal power series over K in n indeterminates x1, … , xn. Denote In by F0...

  6. 2. Derivations of Transitive and Simple Lie Algebras
    (pp. 65-94)

    Throughout this section, we denote by K a field of characteristic zero.

    Let L be a Lie algebra over K, and let A be a subalgebra of L. By a derivation of A into L, we shall mean a K-linear mapping\[\overline{\text{D}}:\text{A}\to \text{L}\ ,\]such that, for all ξ,$\eta \ \epsilon \ \text{A}$\[\overline{\text{D}}([\xi,\eta ])=[\overline{\text{D}}(\xi),\eta ]+[\xi,\overline{\text{D}}(\eta )]\ ;\]from the Jacobi identity in L, we see that if$\zeta\ \epsilon \ \text{L}$, then ad(ζ) is a derivation of A into L. When A = L, such a linear mapping$\overline{\text{D}}$is simply a derivation of L. Recall that, in the last section, we established the notation Der(R) for the Lie algebra...

  7. 3. Simple Algebras with Parameters
    (pp. 95-109)

    Let L be a linearly compact Lie algebra over a field K of characteristic zero. Suppose that V is a finite-dimensional vector space over K. We denote by F the linearly compact local algebra\[\text{F}=\text{F}\{{{\text{V}}^{*}}\}\]of formal power series on V, and we set\[{{\text{F}}^{l}}={{\text{F}}^{l}}\{{{\text V}^{*}}\}\ ,\quad \quad \quad \text{for}\ l\ge -1\ .\]Recall that the completed tensor product$\text{L}{{\hat{\otimes }}_{\text K}}\ \text{F}$, defined in Section one, is a linearly compact vector space over K. There is a natural structure of Lie algebra on the dense subspace$\text{L}{{\otimes }_{\text{K}}}\ \text{F}$of$\text{L}{{\hat{\otimes }}_{\text K}}\ \text{F}$, defined by setting\[\caption {(3.1)} [\xi \otimes \text{f},\eta \otimes \text{g}]=[\xi,\eta ]\otimes (\text{fg})\ ,\]for all ξ,$\eta \ \epsilon \ \text{L}$and f,$\text{g}\ \epsilon \ \text{F}$. One verifies easily that there is a...

  8. 4. Closed Ideals of Transitive Lie Algebras
    (pp. 110-151)

    To increase the intelligibility of this section, we have reproduced, in Lemma 4.1 through Corollary 4.3, certain technical results from the paper ([11]) of Guillemin.

    Let L be a linearly compact Lie algebra over a field K of characteristic zero, and let I be a non-zero closed ideal of L. According to (ii) of Lemma 1.6, there exists a closed ideal J of I which is properly contained in I and strictly maximal. Since J is closed in L, the normalizer NL(J) of J in L is a closed subalgebra of L. The first series of lemmas of this section...

  9. 5. Minimal Closed Ideals of Complex Type
    (pp. 152-216)

    Throughout this section, if V is a linearly compact topological vector space over${\rm I\!R}$, we denote by\[{{\text{V}}_{\mathbb{C}}}=\mathbb{C}{{\otimes }_{\rm{I\!R}}}\text{V}\]the complexification of V; then${{\text{V}}_{\mathbb{C}}}$, is a linearly compact complex topological vector space.

    Let L be a linearly compact Lie algebra over${\rm I\!R}$, and let I be a non-abelian minimal closed ideal of L. Recall that, by Corollary 4.3, there exists a unique maximal closed ideal J of I, and the quotient

    R = I/J

    is a non-abelian simple transitive Lie algebra. According to Proposition 1.4, the commutator ring KRof R is a field which is a...

  10. References
    (pp. 217-220)
  11. Back Matter
    (pp. 221-222)