Algorithm LeviSubalgebra
4.15 Using Cartan subalgebras to compute Levi subalgebras subalgebras
In this section we put Theorem 4.14.3 to use. By this theorem any Car- tan subalgebra H of a Lie algebra L of characteristic 0 contains a C a r t a n subalgebra H1 of a certain Levi subalgebra of L. By L e m m a 4.14.1, adLH1 consists of semisimple linear transformations. Also if we suppose the solv- able radical R to be nilpotent (which by Lemma 4.13.1 we can do without loss of generality), then adLx is nilpotent for x C R. Hence every semisimple element of a d H "comes from" a Levi subalgebra of L. Here we show how we can go after these semisimple elements directly.
In the sequel we let L be a Lie algebra over the field F of characteristic 0, not equal to its solvable radical R. Throughout we assume that R is nilpotent.
D e f i n i t i o n 4.15.1 A commutative subalgebra T of L is said to be toral if d i m T = d i m a d L T and the associative algebra (adLT)*, generated by 1 together with adLT, is commutative and semisimple.
The first condition is included to avoid calling the centre of a Lie algebra toral.
4.15 Using C a r t a n subalgebras to compute Levi subalgebras 137 P r o p o s i t i o n 4 . 1 5 . 2 Let K be a Levi subalgebra of L. Let H1 be a Caftan subalgebra of K . Then H1 is a toral subalgebra of L. Put H = CL(H1), the centralizer of H1 in L. Then H is a Caftan subalgebra of L, and H = H1 G CR(H1).
P r o o f . L e m m a 4.14.1 together with Root fact 6 imply t h a t HI is a commu- tative subalgebra of L such that adLh is a semisimple for h C H1. Also since K does not contain elements of the centre of L we have that the adjoint representation of L is faithful on HI, hence H1 is a total subalgebra of L.
Let x C CL(H1), then x = y + r, where y C K and r E R. So for h C HI we have 0 = [h, x] = [h, y] + [h, r]. But the first element is in K and the second in R. Therefore [h, y] = [h, r] = 0 for all h C H1. Since H1 is a C a r t a n subalgebra of K we have y E HI. It follows that CL(H1) = HI 9 CR(H1).
As in the proof of L e m m a 4.14.2 we see that H = CL(H1) = Lo(H1).
Hence by Proposition 3.2.5 we have NL(H) = H. Furthermore, let H k denote the k-th term of the lower central series of H. Then
H k - [H1 | CR(H1), [H1 | CR(H1),"" , [H1 @ CR(H1), H1 9 C R ( H 1 ) ] ' " ]]
= [CR(H1), [CR(H1),'." [CR(H1), C R ( H 1 ) ] ' " ]] -- CR(H1) k C R k, and hence H is nilpotent. Now by L e m m a 3.2.2, H is a C a f t a n subalgebra
of L. C]
C o r o l l a r y 4 . 1 5 . 3 Let H be a Caftan subalgebra of L, then H contains a Cartan subalgebra H1 of a Levi subalgebra of L. Furthermore, for any such H1 we have that H = H1 @ CR(H1).
P r o o f . The first assertion follows from Theorem 4.14.3. Set H ~ = H1 @ CR(H1). By Proposition 4.15.2, H ~ is a C a f t a n subalgebra of L. We have to show that H ~ = H. As in the proof of Lemma 4.14.2 we have that H' = C L ( H 1 ) = Lo (adhl) for a certain hi C H1. Hence hi is a regular element. So H and H I have a regular element in common and hence they
are equal. D
Now we consider the problem of calculating a toral subalgebra inside a given C a r t a n subalgebra H of L. The next proposition yields a way of doing this. Furthermore it states that a maximal toral subalgebra of H is
"almost" (possibly modulo elements of the centre of L), equal to a C a r t a n subalgebra of a Levi subalgebra of L.
138 Lie algebras with non-degenerate Killing form P r o p o s i t i o n 4 . 1 5 . 4 Let H be a Caftan subalgebra of L. Let T be maximal (with respect to inclusion) among all total subalgebras of L contained in H.
Then there is a Levi subalgebra K of L and a Caftan subalgebra H1 of K such that adLH1 = a d L T and H1 C H. Let x C H and let adLx = s + n be the Jordan decomposition of adLx. Then there is an h C H such that adLh = s.
P r o o f . By Corollary 4.15.3 there is a Levi s u b a l g e b r a K of L having a C a r t a n s u b a l g e b r a H1 such t h a t H1 C H . F u r t h e r m o r e , H = HI G CR(H1).
Let t C T and write t = h + r where h E H1 and r E CR(H1). T h e n [h, r] = 0 and adt = adh + a d r . Now adh is semisimple and a d r is nilpotent (because the radical R is nilpotent). So adt = adh + a d r is the J o r d a n decomposition of adt. But adt is semisimple and hence a d r = 0. So T consists of elements h + r for h C HI and r lies in the centre of L. Now since T is m a x i m a l and c o m m u t e s with H1 it follows t h a t for all h C H1 there is an r in the centre of L such t h a t h + r C T. We conclude t h a t a d T = adH1.
For the last s t a t e m e n t write x = h + r, where h C H1 and r C R. Again we have t h a t adx = adh + a d r is the J o r d a n decomposition of adx. So by the uniqueness of the J o r d a n decomposition we infer t h a t s = adh. D
On the basis of P r o p o s i t i o n 4.15.4 we formulate an algorithm:
A l g o r i t h m ToralSubalgebra
Input: a Lie algebra L of characteristic 0 such t h a t its solvable radical is nilpotent.
Output: a toral s u b a l g e b r a T of L such t h a t a d T - adH1 for a C a r t a n s u b a l g e b r a H1 of a Levi s u b a l g e b r a K .
Step 1 Set R "=SolvableRadical(L), H "=CartanSubalgebraBigField(L), and C "=Centre(H).
Step 2 Let { h i , . . . , hm} be a basis of C. For 1 _< i _< m c o m p u t e the J o r d a n decomposition adLhi - adLsi + adLni of adLhi; where si, ni C H . Step 3 Let T be the span of all si for 1 __ i _< m. R e t u r n T.
C o m m e n t s : Let T be m a x i m a l among all toral subalgebras of L con- tained in H . Let H1 C H be a C a r t a n subalgebra of a Levi subalge- b r a of L such t h a t adLH1 - a d L T (cf. P r o p o s i t i o n 4.15.4). T h e n since H - H1 @ CR(H1) (Corollary 4.15.3) we have t h a t HI is contained in the centre of H . T h e same holds for T because a d L T = adLH1. By P r o p o s i t i o n 4.15.4 we see t h a t the elements si, n~ exist. We use the a l g o r i t h m Jordan- Decomposition (see Section A.2) to c o m p u t e the J o r d a n decomposition of
4.15 Using Cartan subalgebras to compute Levi subalgebras 139 adLhi. Then by solving a system of linear equations, we find si. (Note that si is not necessarily unique.) Now since adLs~ and adLni are polynomials in adLhi without constant term we see that si, ni commute with everything that commutes with hi. In particular si, ni lie in the centre of H.
We show that the span of the si is a maximal toral subalgebra contained in H. Let s E H be such that adLs is semisimple. Write s = h + r where h E HI and r C CR (H1). Then in the same way as in the proof of Proposition 4.15.4 we see that adLr -- O. Write h - }-~im__l c~ihi for some c~i C F. Then adLs -- ~-~im=l (~iadLhi - ~ i (~iadLsi + ~-~i (~iadLni. The first summand is a sum of commuting semisimple transformations and hence semisimple itself.
Similarly the second summand is nilpotent. Furthermore they commute with each other. Hence this is the Jordan decomposition of adLs and it follows that adL(}--~.i c~ini) = 0. So adLs = adLs' for some s' E T. It follows that s r T implies that the subalgebra generated by T together with s contains elements of the centre and is therefore not toral. The conclusion is that T is a maximal total subalgebra of H.
Proposition 4.15.5 Let H be a Caftan subalgebra of L. Let T and K be as in Proposition ~.15.~. Let
L - L 1 G " ' G L n
be the (collected) primary decomposition of L relative to T. Write Li = Vi G Ri, where Ri - R N Li and Vi is a complementary subspace. Then there is a basis of K consisting of elements of the form vi + ri, where vi E V} and ri C Ri.
P r o o f . Let K = K1 • . . . G Km be the primary decomposition of K with respect to T. Also R = R 1 G . . "@Rn is the primary decomposition of R with respect to T. Adding these decompositions, and taking the subspaces Ki and Rj such that the minimum polynomials of the restrictions of adt to Ki and Rj are equal for all t C T, together, we obtain the (collected) primary decomposition of L relative to T. Hence, if Li is not contained in R, then Li = Kki G Ri and it is seen that V} has a basis consisting of elements of the form wi + ri, where wi C Kk~ and ri E Ri, and the result follows. [::]
Let T, Li, Vi, Ri be as in Proposition 4.15.5. Let H1 be a Cartan subalgebra of a Levi subalgebra K of L such that adLH1 - a d L T . We note the following:
The centralizer CL (T) occurs among the primary components of L. It contains H1. The root spaces of the Levi subalgebra K relative to H1
140 Lie algebras with non-degenerate Killing form are contained in the other primary components. Furthermore, these root spaces generate K (this follows from Root fact 11). Suppose that L1 - C L ( T ) . If for 2_< i_< n w e h a v e that R/ = 0 w h e n e v e r V/ ~ 0, then it follows that the subalgebra generated by the ~ is a Levi subalgebra.
9 If we are not so fortunate, then in the equation systems (4.16), we can reduce the number of variables. We start with a basis x 1 , . . . , xm of a complement in L to R, consisting of elements of the spaces ~ . Then t to in the iteration of the algorithm LeviSubalgebra we add elements r i xi. But by Proposition 4.15.5, we can take these elements from R k where k is such that xi E Vk.
Now we formulate an algorithm for calculating a Levi subalgebra of L, in the case where the radical R is nilpotent. By Lemma 4.13.1, this also gives an algorithm for the general case.
A l g o r i t h m
LeviSubalgebra
Input" a finite-dimensional Lie algebra L of characteristic 0 such that the solvable radical SR(L) is nilpotent.
Output" a basis of a Levi subalgebra of L.
Step 1 Set T "=ToralSubalgebra(L).
Step 2 Calculate the primary decomposition L1 @... @ Ls of L with respect to adT.
Step 3 Calculate spaces ~ and R4 C R such that Li - ~ 9 Ri. If R i = 0 for all i such that Y~ ~ 0, then return the subalgebra generated by the
Step 4 Take bases of the t~ together to obtain a basis x 1 , . . . , x m of a com- plement in L to R. Iteratively calculate the equation systems (4.16), where the r i t are taken from Rk, where k is such that xi E Vk. Solve the systems and return the resulting subalgebra.
C o m m e n t : for the calculation of the primary decomposition of L with respect to T we can proceed exactly as in the algorithm PrimaryDecompo- sitionSmallField; the randomly chosen element x in Step 3 is either decom- posing or good with high probability. We leave the details to the reader.
E x a m p l e 4.15.6 Let L be the 6-dimensional Lie algebra of Example 4.13.2.
Then x3 is a non-nilpotent element and H = L0(adx3) is a commutative
4.16 Notes 141 subalgebra, and hence it is a Cartan subalgebra (cf. the algorithm Cartan-
SubalgebraBigField). Now H is spanned by xa, x6; and xa spans a maximal torus T. The primary decomposition of L with respect to T reads
L = H @ L1 @ L2 G L4 9 L5,
where Li is spanned by xi for i = 1, 2, 4, 5. Tile primary components not contained in the radical are L2 and L4. We have that L2NR = L4NR = 0 and hence it immediately follows that they are contained in a Levi subalgebra of L. So the subalgebra K they generate is a Levi subalgebra. It is spanned by x2, x4, xa - 89
4 . 1 6 N o t e s
The Lie algebra of Table 4.4 is a symmetry algebra admitted by the heat equation (see [10], w
The proofs in Section 4.10 (for Lie algebras over modular fields) follow [76]. Theorem 4.14.3 is a result of Chevalley; for the proof we followed [19].
For a different approach see
[34].
The notion of splitting element appears in [27] (see also Appendix A).
There they are elements of a semisimple associative algebra with a maximal number of eigenvalues. Since splitting elements of Cartan subalgebras in semisimple Lie algebras satisfy the analogous property, we have adopted the same terminology. Also elements of an associative algebra with a reducible minimum polynomial are called decomposable in [28]. In Section 4.11 we use the term decomposing to denote the same property.
The algorithm in Section 4.13 for calculating a Levi subalgebra using the derived series of the solvable radical is from [71]. The algorithm LeviSubal- gebraBykCSeries is taken from [36]. In this paper it is also proved that this algorithm runs in polynomial time (since the left hand side of (4.16) does not depend on the output of the previous round, it can be proved that the coefficients of the solutions do not blow up). The algorithm for calculating a Levi subalgebra using the information carried by a Cartan subalgebra is taken from
[34].
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