# Lie Algebras (Dover Books on Mathematics)

## Nathan Jacobson

Language: English

Pages: 352

ISBN: 0486638324

Format: PDF / Kindle (mobi) / ePub

Chapter 1 introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Cartan’s criterion and its consequences, and split semi-simple Lie algebras. Chapter 5, on universal enveloping algebras, provides the abstract concepts underlying representation theory. The basic results on representation theory are given in three succeeding chapters: the theorem of Ado-Iwasawa, classification of irreducible modules, and characters of the irreducible modules. In Chapter 9 the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic zero are determined. These results are applied in Chapter 10 to the problems of sorting out the simple Lie algebras over an arbitrary field. The reader, to fully benefit from this tenth chapter, should have some knowledge about the notions of Galois theory and some of the results of the Wedderburn structure theory of associative algebras.

Nathan Jacobson, presently Henry Ford II Professor of Mathematics at Yale University, is a well-known authority in the field of abstract algebra. His book,

*Lie Algebras,*is a classic handbook both for researchers and students. Though it presupposes knowledge of linear algebra, it is not overly theoretical and can be readily used for self-study.

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condition. Doklady Akad. Nauk S.S.R. (N.S.) 118 (1958), pp. 1074–1077. [5] On a problem of Burnside. Izvest. Akad. Nauk S.S.R., Ser. Mat. 23 (1959), pp. 3–34. KOSZUL, J. L. [1] Homologie et cohomologie des algèbres de Lie. Bull. Soc. Math. France 78 (1950), pp. 65–127. LANDHERR, W. [1] Dber einfache Liesche Ringe. Abhandl. Math. Sem. Univ. Hamburg. 11 (1935), pp. 41–64. [2] Liesche Ringe vom Typus A. Abhandl. Math. Sem. Univ. Hamburg. 12 (1938), pp. 200–241. LARDY, P. [1] Sur la

* = * and this will imply the theorem. We note first that * ≠ 0. Thus let W be a non-zero element of . Then by I, = {W}* is a subsystem and * = {w}*. Since {W}* is the set of polynomials in W with constant term 0, {W}* is nilpotent. Hence ∈ Ω. Since ≠ 0 it follows that * ≠ 0. This implies that the subspace spanned by all the vectors xB*, x ∈ , B* ∈ , is not 0. Also ≠ . For otherwise, any . If we use similar expressions for the xi we obtain in *. A repetition of this process gives in *. Since * is

we have where is a maximal submodule of . We have seen that there is only one such submodule. Hence it is clear that any two irreducible e-extreme modules with the same highest weight are isomorphic. The existence of an irreducible e-extreme module with highest weight Λ is clear also, for, the module satisfies these requirements. We summarize our main results in the following THEOREM 2. Let the notations be as in Theorem1 and let Λ(h) be a linear function on . Then there exists an irreducible

by all the elements [x1 …, xr] where 0 r n = dim and 0= Φ1. Then dim r = and C = Φ1 ⊕ 1⊕ 2 ⊕ … ⊕ n. Proof: If (u1, u2, …, un) is a basis for , then the skew symmetry and multilinearity of [x1, …, xr] imply that every element of r, r 1, is a linear combination of the elements [ui1, ui2, …, uir] where i1 < i2 < … < ir = 1, …, n. Now assume that the basis is orthogonal: (ui, uj) = 0 if i ≠ j. (It is well known that such bases exist.) The condition (ui, uj) = 0 and (27) implies that uiuj = —ujui.

matrices in the matrix algebra Φ3. 6. Some basic module operations The notion of a submodule of a module for a Lie or associative algebra is clear: is a subspace of closed under the composition by elements of the algebra. If is a submodule, then we obtain the factor module / which is the coset space / with the module compositions (x + )a = xa + , a in the algebra. If 1 and 2 are two modules for an associative or a Lie algebra, then the space 1 ⊕ 2 is a module relative to the composition (x1 +