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Using Cartan Subalgebras to Calculate Nilradicals and Levi Subalgebras of Lie Algebras W.A

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Using Cartan Subalgebras to Calculate Nilradicals and Levi Subalgebras of Lie Algebras W.A Journal of Pure and Applied Algebra 139 (1999) 25–39 www.elsevier.com/locate/jpaa View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Using cartan subalgebras to calculate nilradicals and Levi subalgebras of Lie algebras W.A. de Graaf School of Mathematical and Computational Sciences, University of St Andrews, St Andrews, Scotland, UK Received 15 February 1998; received in revised form 1 September 1998 Abstract In this paper we investigate the structure of a non-semisimple Lie algebra of characteristic 0, by using the action of a Cartan subalgebra. We give an algorithm for calculating the nilradical and an algorithm for ÿnding a Levi subalgebra. At the end of the paper these algorithms are put to practical tests. c 1999 Elsevier Science B.V. All rights reserved. MSC: 17-04; 17-08; 17B05; 17B30 1. Introduction In the theory of semisimple Lie algebras the action of a Cartan subalgebra is used to obtain information on the structure of the Lie algebra. This turns out to be a very powerful tool: semisimple Lie algebras of characteristic 0 can be completely classiÿed by the action of their Cartan subalgebras. For non-semisimple Lie algebras there is no structure theory available that is as detailed as the theory for semisimple Lie algebras. However, by Levi’s theorem, a non-semisimple Lie algebra is the direct sum of a semisimple subalgebra and a solvable ideal, called the radical. The radical contains a unique maximal nilpotent ideal, called the nilradical. So three subalgebras that reveal a large part of the structure of a Lie algebra are its nilradical, its solvable radical and a Levi subalgebra. Here we describe ways of using a Cartan subalgebra to explore the structure of a non-semisimple Lie algebra. In Section 3 an algorithm for ÿnding the nilradical is E-mail address: [email protected] (W.A. de Graaf) 0022-4049/99/$ – see front matter c 1999 Elsevier Science B.V. All rights reserved. PII: S0022-4049(99)00007-9 26 W.A. de Graaf / Journal of Pure and Applied Algebra 139 (1999) 25–39 described. And in Section 4 it is shown how we can use a Cartan subalgebra in the calculation of a Levi subalgebra. These sections are preceded by a section on Fitting and primary decompositions, which will be needed in the sequel. Finally Section 5 gives a practical evaluation of the algorithms. It is shown that (at least on the inputs considered) the algorithms proposed here run markedly faster than the previously known ones. If we want to compute with a Lie algebra, then we must ÿnd a presentation for it that can be handled by a computer. There are various ways of dealing with this problem (see [4]). Here we restrict our attention to the one most commonly used, namely the presentation by a table of structure constants. If L is of dimension n over k the ÿeld F, then it has a basis {x1;:::;xn}. It follows that there are constants cij ∈ F for 1 ≤ i; j; k ≤ n such that Xn k [xi;xj]= cij xk : k=1 k These cij are called structure constants; they determine the Lie multiplication com- pletely. Throughout this paper we assume that the input to our algorithms consists of a Lie algebra given by an array of n3 structure constants. Furthermore, the ÿeld F will always be of characteristic 0 (typically F = Q). For the deÿnitions of the basic concepts related to Lie algebras we refer to [9,12]. However, here we recall that for a Lie algebra L, the adjoint map, adL : L → End(L), is deÿned by adL x(y)=[x; y] for x; y ∈ L. Often we write ad x in place of adL x. Furthermore, if L is a Lie algebra of linear transformations of a vector space, then by L∗ we denote the associative algebra (with 1) generated by L. In particular (ad L)∗ is the associative algebra generated by all transformations ad x for x ∈ L. Let L be a Lie algebra over a ÿeld F of characteristic 0. We note that algorithms for calculating a Cartan subalgebra of L are known (see, e.g, [2,8,17]). Also there is an algorithm for calculating [U; V ] where U; V ⊂ L are subspaces of L [2]. Furthermore, an algorithm for ÿnding the solvable radical of L is given in [2]. In the sequel we will often refer to [12] for the theoretical background. In this book the chapters are enumerated by Roman numerals. A reference of the form “[12], Theorem V:3” will mean Theorem 3 of Chapter V of [12]. 2. Fitting and primary decompositions Let L be a ÿnite dimensional Lie algebra with Cartan subalgebra H.By[H m;L]we denote the subspace spanned by the elements [h1; [h2; [ :::;[hm;x] :::]]], where hi ∈ H for 1 ≤ i ≤ m and x ∈ L. Then [H m;L] ⊃ [H m+1;L], and therefore there is an integer c¿0 such that [H c;L]=[H c+1;L]. Lemma 1 (Fitting decomposition). Let H be a Cartan subalgebra of L; and let c¿0 be an integer such that [H c;L]=[H c+1;L]; then L decomposes as L = H ⊕ [H c;L]: W.A. de Graaf / Journal of Pure and Applied Algebra 139 (1999) 25–39 27 Proof. This follows from [12], Theorem II:4 together with Proposition III:1. The decomposition of Lemma 1 is called the Fitting decomposition of L with respect to H. Let H be a given Cartan subalgebra of L, then by repeatedly forming the product of subspaces, we can calculate the series [H m;L]. This gives us an algorithm for ÿnding the Fitting decomposition of L with respect to H. A Cartan subalgebra of a semisimple Lie algebra is diagonalisable and the Lie alge- bra decomposes as a direct sum of simultaneous eigenspaces. For non-semisimple Lie algebras this is replaced by the following result (see [12], Theorem II:6). Lemma 2. Let K be a nilpotent Lie algebra of linear transformations of the vector space V. Then V possesses a decomposition into a direct sum of subspaces V = V1 ⊕···⊕Vs (1) such that the following conditions hold: 1:Vi is invariant under K; 2: for every A ∈ K; the minimum polynomial of the restriction of A to Vi is a power of an irreducible polynomial (for 1 ≤ i ≤ s); 3: for each pair of subspaces Vi and Vj (i 6= j) there is an A ∈ K such that the minimum polynomials of the restrictions of A to Vi and Vj are powers of di erent polynomials. Moreover; the decomposition (1) is unique upto a permutation of the Vi. We call the decomposition (1) the primary decomposition of V with respect to K. The subspaces Vi are called the primary components of V with respect to K. (Typically the nilpotent Lie algebra K of the lemma is adL H for a Cartan subalgebra H of L.) In Section 4 we will need an algorithm for calculating the primary decomposition relative to a commutative Lie algebra K such that K ∗ is a torus. Here we give an algorithm for this task. We recall that a torus is a commutative associative algebra that has no nilpotent elements other than 0. Let K be a commutative Lie algebra of linear transformations of the ÿnite dimen- sional vector space V , such that the associative algebra K ∗ is a torus. An element x ∈ K ∗ is said to be a splitting element if x generates K. There are various algorithms known for computing splitting elements (see, e.g., [5], Section 5:5 of [10]). By the following lemma this also gives an algorithm for computing the primary decomposition of V with respect to K. Lemma 3. Let K be a Lie algebra of linear transformations such that K∗ is a torus. ∗ Let x be a splitting element in K . Let f = f1 ···fs be the factorisation of the minimum polynomial of x into irreducible factors. Set V0(fi(x)) = {v ∈ V |fi(x)v =0}; 28 W.A. de Graaf / Journal of Pure and Applied Algebra 139 (1999) 25–39 then V = V0(f1(x)) ⊕···⊕V0(fs(x)) is the primary decomposition of V with respect to K. ∗ ∗ Proof. First of all it is clear that the V0(fi(x)) are invariant under K since K is ∗ commutative. Let Ki be the algebra generated by the restrictions of y ∈ K to V0(fi(x)). Then Ki is generated by the restriction of x to V0(fi(x)). But the minimum polynomial of this map is the irreducible polynomial fi. Hence Ki is a ÿeld. As a consequence all elements of Ki have a minimum polynomial that is irreducible. 3. The Nilradical Let L be a ÿnite dimensional Lie algebra. Then by [12], Proposition I:6, L contains a unique maximal nilpotent ideal. This ideal is called the nilradical of L and denoted by NR(L). In the literature several algorithms for calculating the nilradical of a Lie algebra L of characteristic 0 have been proposed. A ÿrst algorithm for this task was presented in [2]. First the solvable radical R of L is calculated. Then a ag 0 = R0 ⊂ R1 ⊂···⊂Rn = R is constructed such that Ri is an ideal of R and dim Ri = i. The nilradical is the set of all elements x ∈ R such that [x; Ri+1] ⊂ Ri. A second algorithm was described in [14]. This algorithm calculates the nilradical by approximating it from below by smaller ideals.
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