Classification and Identification of Lie Algebras

Front Matter
	Copyright
	Contents
	Preface
	Acknowledgements
Part 1. General Theory
	Chapter 1. Introduction and Motivation
	Chapter 2. Basic Concepts
		2.1. Definitions
		2.2. Levi theorem
		2.3. Classification of complex simple Lie algebras
		2.4. Chevalley cohomology of Lie algebras
	Chapter 3. Invariants of the Coadjoint Representation of a Lie Algebra
		3.1. Casimir operators and generalized Casimir invariants
		3.2. Calculation of generalized Casimir invariants using the infinitesimal method
			3.2.1. Formulation of the problem and number of functionally independent solutions.
			3.2.2. Method of characteristics for a single PDE.
			3.2.3. Solution of the system of PDEs (3.2).
		3.3. Calculation of generalized Casimir invariants by the method of moving frames
Part 2. Recognition of a Lie Algebra Given by Its Structure Constants
	Chapter 4. Identification of Lie Algebras through the Use of Invariants
		4.1. Elementary invariants
		4.2. More sophisticated invariants
	Chapter 5. Decomposition into a Direct Sum
		5.1. General theory and criteria
		5.2. Algorithm
		5.3. Examples
	Chapter 6. Levi Decomposition. Identification of the Radicaland Levi Factor
		6.1. Original algorithm
		6.2. Modified algorithm
		6.3. Examples
	Chapter 7. The Nilradical of a Lie Algebra
		7.1. General theory
		7.2. Algorithm
		7.3. Examples
		7.4. Identification of the nilradical using the Killing form
Part 3. Nilpotent, Solvable and Levi Decomposable Lie Algebras
	Chapter 8. Nilpotent Lie Algebras
		8.1. Maximal Abelian ideals and their extensions
		8.2. Classification of low-dimensional nilpotent Lie algebras
	Chapter 9. Solvable Lie Algebras and Their Nilradicals
		9.1. General structure of a solvable Lie algebra
		9.2. General procedure for classifying all solvable Lie algebras with a given nilradical
		9.3. Upper bound on the dimension of a solvable extension of a given nilradical
		9.4. Particular classes of nilradicals and their solvable extensions
		9.5. Vector fields realizing bases of the coadjoint representation of a solvable Lie algebra
	Chapter 10. Solvable Lie Algebras with Abelian Nilradicals
		10.1. Basic structural theorems
		10.2. Decomposability properties of the solvable Lie algebras
			10.2.1. Nilradicals of minimal dimension.
		10.3. Solvable Lie algebras with centers of maximal dimension
		10.4. Solvable Lie algebras with one nonnilpotent element and an n-dimensional Abelian nilradical
		10.5. Solvable Lie algebras with two nonnilpotent elements and n-dimensional Abelian nilradical
		10.6. Generalized Casimir invariants of solvable Lie algebras with Abelian nilradicals
			10.6.1. General form of the generalized Casimir invariants and their number.
			10.6.2. Diagonal structure matrices.
			10.6.3. Casimir invariants in the case f = 1.
			10.6.4. Casimir invariants for low-dimensional nilradicals.
			10.6.5. Summary.
	Chapter 11. Solvable Lie Algebras with Heisenberg Nilradical
		11.1. The Heisenberg relations and the Heisenberg algebra
		11.2. Classification of solvable Lie algebras with nilradical h(m)
			11.2.1. Basic classification theorem.
		11.3. The lowest dimensional case m = 1
		11.4. The case m = 2
		11.5. Generalized Casimir invariants
	Chapter 12. Solvable Lie Algebras with Borel Nilradicals
		12.1. Outer derivations of nilradicals of Borel subalgebras
		12.2. Solvable extensions of the Borel nilradicals NR(b(g))
			12.2.1. Solvable extensions of the Borel nilradicals of maximal dimension.
			12.2.2 Solvable extensions of the split real form of the nilradical NR(b(g))
			12.2.3. Solvable extensions of the Borel nilradicals of less than maximal dimension.
			12.2.4. Solvable extensions of dimension n_{NR}+1.
		12.3. Solvable Lie algebras with triangular nilradicals
			12.3.1. The structure of the algebras of strictly upper triangular matrices and their derivations.
			12.3.2. Illustration of the procedure for low dimensions.
		12.4. Casimir invariants of nilpotent and solvable triangular Lie algebras
			12.4.1. Invariants of nilpotent triangular Lie algebras.
			12.4.2. Invariants of the solvable triangular Lie algebras.
			12.4.3. Casimir invariants of solvable extensions of t(4).
			12.4.4. General results.
			12.4.5. Conclusions.
	Chapter 13. Solvable Lie Algebras with Filiform and Quasifiliform Nilradicals
		13.1. Classification of solvable Lie algebras with the model filiform nilradical n_{n,1}
			13.1.1. Nilpotent algebra n_{n,1}.
			13.1.2. Construction of solvable Lie algebras with the nilradical n_{n,1}.
			13.1.3. Standard forms of solvable Lie algebras with the nilradical n_{n,1}.
			13.1.4. The generalized Casimir invariants of n_{n,1} and of its solvable extensions.
		13.2. Classification of solvable Lie algebras with the nilradical n_{n,2}
			13.2.1. Nilpotent algebra n_{n,2} and its structure.
			13.2.2. Construction of solvable Lie algebras with the nilradical n_{n,2}.
			13.2.3. Generalized Casimir invariants of n_{n,2} and of its solvable extension.
		13.3. Solvable Lie algebras with other filiform nilradicals
		13.4. Example of an almost filiform nilradical
			13.4.1. Automorphisms and derivations of the nilradical n_{n,3}.
			13.4.2. Construction of solvable Lie algebras with the nilradical n_{n,3}.
			13.4.3. Dimension n = 6.
			13.4.4. Dimension n = 5.
		13.5. Generalized Casimir invariants of nn,3 and of its solvable extensions
	Chapter 14. Levi Decomposable Algebras
		14.1. Levi decomposable algebras with a nilpotent radical
		14.2. Levi decomposable algebras with nonnilpotent radicals
		14.3. Levi decomposable algebras of low dimensions
			14.3.1. Levi extensions of Abelian radicals.
			14.3.2. Levi extensions of non-Abelian radicals with Abelian nilradicals.
			14.3.3. Levi decomposable algebras with non-Abelian nilradicals.
Part 4. Low-Dimensional Lie Algebras
	Chapter 15. Structure of the Lists ofLow-Dimensional Lie Algebras
		15.1. Ordering of the lists
		15.2. Computer-assisted identification of a given Lie algebra
	Chapter 16. Lie Algebras up to Dimension 3
		16.1. One-dimensional Lie algebra
		16.2. Solvable two-dimensional Lie algebra with the nilradical n_{1,1}
		16.3. Nilpotent three-dimensional Lie algebra
		16.4. Solvable three-dimensional Lie algebras with the nilradical 2n_{1,1}
		16.5. Simple three-dimensional Lie algebras
	Chapter 17. Four-Dimensional Lie Algebras
		17.1. Nilpotent four-dimensional Lie algebra
		17.2. Solvable four-dimensional algebras with the nilradical 3n_{1,1}
		17.3. Solvable four-dimensional Lie algebras with the nilradical n_{3,1}
		17.4. Solvable four-dimensional Lie algebras with the nilradical 2n_{1,1}
	Chapter 18. Five-Dimensional Lie Algebras
		18.1. Nilpotent five-dimensional Lie algebras
		18.2. Solvable five-dimensional Lie algebras with the nilradical 4n_{1,1}
		18.3. Solvable five-dimensional Lie algebras with the nilradical n_{3,1} ⊕ n_{1,1}
		18.4. Solvable five-dimensional Lie algebras with the nilradical n_{4,1}
		18.5. Solvable five dimensional Lie algebras with the nilradical 3n_{1,1}
		18.6. Solvable five-dimensional Lie algebras with the nilradical n_{3,1}
	Chapter 19. Six-Dimensional Lie Algebras
		19.1. Nilpotent six-dimensional Lie algebras
		19.2. Solvable six-dimensional Lie algebras with the nilradical 5n_{1,1}
		19.3. Solvable six-dimensional Lie algebras with the nilradical n_{3,1} ⊕ 2n_{1,1}
		19.4. Solvable six-dimensional Lie algebras with the nilradical n_{4,1} ⊕ n_{1,1}
		19.5. Solvable six-dimensional Lie algebras with the nilradical n_{5,1}
		19.6. Solvable six-dimensional Lie algebras with the nilradical n_{5,2}
		19.7. Solvable six-dimensional Lie algebras with the nilradical n_{5,3}
		19.8. Solvable six-dimensional Lie algebras with the nilradical n_{5,4}
		19.9. Solvable six-dimensional Lie algebras with the nilradical n_{5,5}
		19.10. Solvable six-dimensional Lie algebra with the nilradical n_{5,6}
		19.11. Solvable six-dimensional Lie algebras with the nilradical 4n_{1,1}
		19.12. Solvable six-dimensional Lie algebras with the nilradical n_{3,1} ⊕ n_{1,1}
		19.13. Solvable six-dimensional Lie algebra with the nilradical n_{4,1}
	Bibliography
	Index
Book Presentation by Authors
	The monograph: Classification and Identification of Lie Algebras
	Why to be interested in identification and classification of Lie algebras?
	What can be found in our book
	Conclusions