Algorithm ProductSpace
1.13 Restricted Lie algebras
by setting x - ~ - x 3, where ~ denotes the coset v + W. T h e space V / W is called the quotient module of V and W.
T h e L - m o d u l e V is called irreducible if it has no s u b m o d u l e s other t h a n 0 a n d V itself. It is called completely reducible if V is a direct s u m of irreducible L-modules.
D e f i n i t i o n 1 . 1 2 . 3 Let V be a finite-dimensional L-module.
L-submodules
O - Vo c V1 c . . . c V~+~ - V
A series of
such that the L-modules ~ + l / V i are irreducible for 0 < i < n is called a composition series of V with respect to the action of L.
L e m m a 1 . 1 2 . 4 Let V be a finite-dimensional L-module.
composition series.
Then V has a
P r o o f . T h e p r o o f is by induction on dim V. T h e s t a t e m e n t is trivial if dim V = 0 or if V is irreducible. Now suppose V is not irreducible and dim V > 0, and assume t h a t the result holds for all L - m o d u l e s of dimen- sion less t h a n dim V. Let W be a m a x i m a l p r o p e r s u b m o d u l e of V (such a s u b m o d u l e exists because V is not irreducible). T h e n by induction W has a composition series. To this series we add V a n d we o b t a i n a composition series of V ( V / W is irreducible since W is a m a x i m a l p r o p e r s u b m o d u l e ) . [:]
1.13 Restricted Lie algebras
In this section all algebras are defined over the field F of characteristic p > O .
Let A be an algebra and let d be a derivation of A. We use the Leibniz formula (1.11), with n - p. Since all coefficients (~) are 0 for 1 _< k _< ; - 1 we see t h a t
d p (xy) - d p (x)y + x d p (y).
T h e conclusion is t h a t d p is again a derivation of A.
Now let L be a Lie algebra over F . T h e n for x C L we have t h a t a d x is a derivation of L. So (adz) p is also a derivation of L, a n d it may h a p p e n t h a t this derivation is again of the form ady for a y E L. Lie algebras L with the p r o p e r t y t h a t this h a p p e n s for all elements x of L are called restricted.
If L is a restricted Lie algebra, then there is a m a p fp : L -+ L such t h a t (adx) p = adfp(X). Now let c: L --+ C ( L ) be any m a p from L into its centre.
T h e n gp = fp + c is also a m a p such t h a t (adx) p = adgp(X). So if C ( L ) ~ 0,
30 Basic constructions
1.13 Restricted Lie algebras 31 Now let x, y E L, where L is a restricted Lie algebra such t h a t C ( L ) - O.
T h e n by setting T - 1 in (1.16) a n d taking a - adx a n d b = a d y we get p-1
+ = + + dy)
i=1 p-1
= adfp(X) + a d f p ( y ) + E a d s i ( x , y) i=1
p-1
= + A(Y) + Z y))
i=1
So since C ( L ) - 0 we see t h a t f p ( x + y ) - f p ( X ) + f p ( y ) + E i = I p-1 8i(X, y). Fur- thermore, for a E F a n d x C L we have ( a d a x ) p - aPadfp(X) - a d a P f p ( x ) . Hence f p ( a X ) - aPfp(X). Now in a restricted Lie algebra we consider maps fp t h a t have the same properties as the m a p fp in the case where L has no centre.
D e f i n i t i o n 1 . 1 3 . 2 Let L be a restricted Lie algebra over F . fp " L -+ L is called a p-th power mapping, or p - m a p if
T h e n a map
1. (adx) p - a d f p ( x ) f o r all x C L,
2. f p ( x + y) -- fp(X) + f p ( y ) + E i = I p-1 si(x, y) f o r x, y C L,
J . -
Let L be a restricted Lie algebra over F . If it is clear from the context which p - t h power m a p fp we mean, then we also write x p instead of fp(X).
E x a m p l e 1 . 1 3 . 3 Let A be an associative algebra over the field F of char- acteristic p > 0. T h e n since A is associative, we can raise its elements to the p - t h power. Set L - ALie, the associated Lie algebra. T h e n for a C L we have a p e L. F u r t h e r m o r e , by (1.14) it follows t h a t (ada)P(b) = ad(aP)(b).
So L is restricted. Also by (1.16), where we set T = 1, we have t h a t a ~-~ a p is a p - t h power mapping. As a consequence, any subalgebra K of L t h a t is closed u n d e r the m a p a ~ a p is a u t o m a t i c a l l y restricted; a n d the restriction of a ~ a p to K is a p - t h power mapping.
Now let A be an algebra over F and Der(A) the Lie algebra of all derivations of L. As seen at the start of this section, d p E Der(A) for all d E Der(A). Hence Der(A) is a subalgebra of E n d ( A ) L i e closed u n d e r the p - t h power map. So Der(A) is a restricted Lie algebra.
32 Basic constructions Let f be a bilinear form on the vector space V. Let L S be the Lie algebra defined by (1.3). Let a E L S, then f ( a P v , w) - ( - 1 ) P f ( v , aPw) - - f ( v , aPw) so t h a t a p C L S. As a consequence L I is a restricted Lie algebra relative to the m a p a ~ a p.
Suppose we are given a Lie algebra L over F . In principle, when checking whether L is restricted or not, we would have to check whether (adx) p is an inner derivation of L for all x C L. However, by the next result it is enough to check this only for the elements of a basis of L. Furthermore, a-priori it is not clear whether any restricted Lie algebra has a p-th power mapping.
Also this is settled by the next proposition.
P r o p o s i t i o n 1 . 1 3 . 4 L e t L be a L i e algebra over the field F of c h a r a c t e r i s t i c p > O. L e t { x l , x 2 , . . . } be a (possibly i n f i n i t e ) basis o f L . S u p p o s e that there are yi C L f o r i > 1 such that ( a d x i ) p - adyi. T h e n L has a u n i q u e p - t h p o w e r m a p p i n g x ~-~ x p such that x i - yi. p
P r o o f . We prove uniqueness. Let fp 9 L ~ L and gp 9 L -+ L be two p-th power mappings such t h a t f p ( X i ) - gp(Xi) - Yi for i > 1. Set h = fp - gp.
T h e n from the definition of p-th power mapping we have h ( x + y) - h ( x ) + h ( y ) and h ( a x ) - a P h ( x ) , for x, y E L and a e F . Hence ker h is a subspace of L. But xi C ker h for all i _ 1. It follows t h a t h - 0 and fp - gp.
T h e proof of existence is more complicated; it needs the concept of uni- versal enveloping algebras. We defer it to Section 6.3. [::1 On the basis of the last proposition we formulate an algorithm for check- ing whether a given finite-dimensional Lie algebra is restricted.
A l g o r i t h m IsRestricted
Input: a finite-dimensional Lie algebra L defined over a field of characteristic p > 0 .
Output: true if L is restricted, false otherwise.
Step 1 Select a basis { X l , . . . , X n } of L and compute the matrices of a d x i for 1 _< i _< n. Let V be the vector space spanned by these matrices.
Step 2 for 1 <_ i _< n determine whether ( a d x i ) p C V . If one of these is not in V then r e t u r n false. Otherwise r e t u r n true.
Let L be a restricted Lie algebra. In order to calculate with p-th power mappings, we have to decide how to represent such a m a p p i n g on a com- puter. Proposition 1.13.4 gives a straightforward solution to this problem.
1.14 Extension of the ground field 33 We can describe a p-th power mapping by giving the images of the ele- ments of a basis of L. Then the image of an arbitrary element of L can be calculated using conditions 2 and 3 from Definition 1.13.2.
E x a m p l e 1.13.5 Let G be a finite p-group, and L be the corresponding Lie algebra over F = Fp as constructed in Section 1.4. Then L = V1 | Vs, where Vi is isomorphic to the factor group Gi via the isomorphism ai : Gi --+
Vi. Now let x E Vi and let g C hi(G) be such that T i ( g ) : X (where ~-i is as in Section 1.4). Then by Theorem 1.4.1, gP E ~pi(G). Now we set xp = .rpi(gp).
It can be shown that x p does not depend on the particular pre-image g chosen. Also it can be shown that for x C ~ we have ( a d x ) p - - adx p. This together with Proposition 1.13.4 shows that L is a restricted Lie algebra.
Now let G be the group generated by gl, g2, g3 of Example 1.4.5. Let L be the corresponding 3-dimensional Lie algebra over F2 with basis {el, e2, e3}.
Then e~ - ~-2(gl 2) - e3, and similarly e 2 - e3 and e 2 - 0. Furthermore, using the fact that
81(a,
b)--[b, a]
(see Example 1.13.1) we see that+ = + + - + + -
1.14 E x t e n s i o n of t h e g r o u n d field
Let L be a Lie algebra over the field F, and let/~ be an extension field of F.
Then the structure constants of L are also elements of/~, so we can define a second Lie algebra L that has the same structure constants as L, but is defined over/~. Formally we set L - L @g I~. This space consists of finite sums ~-~4 Yi @ )~i, where Yi E L and Ai C F. For @ we have the following rules:
(x + y ) | = x@)~ + y | A, x | (~ + # ) = x | ~ + x | ( a x ) | 1 6 3 1 7 4 f o r a C F ,
~(x | ~) = x | M .
Furthermore, the product in L is defined by [x @ )~, y @ #] - Ix, y] | ~#. Let { X l , . . . ,Xn} be a basis of L, then the rules for @ imply that every y C ], can be written as
n
Y = E A i ( x i | w h e r e ) ~ i C F f o r l < i _ < n .
i=1
34 Basic constructions It follows t h a t the elements X l | 1 , . . . , Xn | 1 form a basis of L. And the structure constants of L relative to this basis equal the structure constants of L relative to the basis {x 1 , . . . , Xn}.
Now let V, W be two subspaces of L; then V - V | and I/V - W @F/~
are subspaces of L. Let { V l , . . . , v s } and { W l , . . . , w t } be bases of V and W respectively. If we input L, V and W (with respective bases {xl | 1 , . . . , x n | { v 1 | , vs@l} and {w1@1,... , wt@l}) into the algorithm ProductSpace then, because the structure constants of L are equal to those of L, we see that this algorithm performs exactly the same operations as on the input L, V, W. (Solving linear equations with coefficients in F gives an answer with coefficients in F.) Hence
IV | -t~, W | if'] -- [g~ W] | -if'.
Let L (k) denote the k-th term of the derived series of L, then by the above discussion we have t h a t ],(k) _ L(k) | F. And the same holds for the terms of the lower central series.
Also, the algorithm Centre solves exactly the same set of linear equations on input L as on input L. Hence we see that C(L) - C(L) | b'. And generally, Ck(L) - Ck(L) @g F , where Ck(L) denotes the k-th t e r m of the upper central series of L.
1 . 1 5 F i n d i n g a d i r e c t s u m d e c o m p o s i t i o n
Let L be a finite-dimensional Lie algebra. Then L may or may not be the direct sum of two or more ideals. If this happens to be the case then the structure of the direct s u m m a n d s of L may be studied independently. To- gether they determine the structure of L. So the direct sum decomposition can be a very valuable piece of information. In this section we describe an algorithm to compute a decomposition of L as a direct sum of indecompos- able ideals (i.e., ideals t h a t are not direct sums of ideals themselves).
Suppose that L - I1 @/2 is the direct sum of two ideals I1,/2 and I1 is contained in the centre of L. Then I1 is called a central component of L.
First we give a m e t h o d for finding such a central component if it exists.
Let J1 be a complementary subspace in C(L) to C(L) N [L, L]. Then as -/1 is contained in the centre of L, it is an ideal of L. Let J2 be a complementary subspace in L to J1 containing [L, L]. Then
[L, J2] C [L, L] C J2
so t h a t J2 is an ideal of L. Furthermore L - -/1 | J2 and -/1 is central and
1.15 F i n d i n g a direct s u m d e c o m p o s i t i o n 35
J2 does not c o n t a i n a central c o m p o n e n t . T h e conclusion is t h a t J1 is a m a x i m a l central c o m p o n e n t .
Now we s u p p o s e t h a t C ( L ) C [L, L] (i.e., t h a t L does not have a central c o m p o n e n t ) a n d we t r y to d e c o m p o s e L as a direct s u m of ideals.
We recall t h a t M n ( F ) is the associative m a t r i x a l g e b r a consisting of all n x n m a t r i c e s over F (cf. E x a m p l e 1.1.4). F r o m A p p e n d i x A we recall t h a t a non-zero element e C M n ( F ) is called an i d e m p o t e n t if e 2 - e.
T w o i d e m p o t e n t s el a n d e2 are called orthogonal if el e2 = e2el = 0. A n i d e m p o t e n t is said to be p r i m i t i v e if it is not the s u m of two o r t h o g o n a l i d e m p o t e n t s .
P r o p o s i t i o n 1 . 1 5 . 1 P u t n - d i m L. The Lie algebra L is the direct s u m of k (non-zero) ideals I 1 , . . . , Ik if and only if the centralizer
CM~(F) (adL) -- {a e M n ( F ) l a . a d x - a d x . a f o r all x e L } contains k orthogonal idempotents e l , . . . , ek such that el + ' " + ek is the identity on L and Ir - erlr f o r r = 1 , . . . , k. Furthermore, the ideal Ir is indecomposable if and only if the corresponding idempotent er is primitive.
P r o o f . F i r s t we s u p p o s e t h a t
L = I1 @ . . . G Ik,
w h e r e Ir is a n o n - z e r o ideal of L for 1 __ r <_ k. For an element x C L we w r i t e x - Xl + . . . + Xk, where xr C It. Let er 9 L --+ L be defined by er(x) - Xr (i.e., er is the p r o j e c t i o n onto Ir). T h e n for y , z E L we have e r a d y ( z ) - er[y,z] - [y,Z]r - [ y , Zr] - a d y e r ( z ) . Hence er C C M , ( F ) ( a d L ) . Also it is clear t h a t eres - 0 for r # s, a n d t h a t el + ' " + ek -- 1.
Now let e l , . . . ,ek be k o r t h o g o n a l i d e m p o t e n t s in CMn(F)(adL) such t h a t e l + . . "+ek 1. T h e n set Ir -- erL for 1 _< r __ k. Since er 2 = er we have for x C Ir a n d y E L t h a t [y,x] - [ y , erX] - adyer(X) - e r a d y ( x ) - er[y,x]
so t h a t Ir is an ideal of L. Also
L - 1 9 L - (el + . . . + e k ) L -- I1 + ' " + Ik.
Now let x C Ir M Is. T h e n x = erX, a n d also esX = x so t h a t x = eserX -= O.
Hence L is the direct s u m of t h e Ir.
If er is not p r i m i t i v e then, as above, we see t h a t Ir is the direct s u m of ideals. F u r t h e r m o r e , if Ir is the direct s u m of ideals of Ir ( t h a t are t h e n a u t o m a t i c a l l y ideals of L), t h e n er is not primitive. [::]
36 Basic constructions T h e o r e m 1.15.2 L e t A be the a s s o c i a t i v e algebra C M ~ ( F ) ( a d L ) a n d p u t Q = A / R a d ( A ) . S u p p o s e t h a t L has no central c o m p o n e n t . T h e n O. is c o m m u t a t i v e .
P r o o f . We may suppose that the ground field F is algebraically closed.
Indeed, if F denotes the algebraic closure of F, then as seen in Section 1.14 we have C ( L ) | -F -- C ( L | F ) and [L | F , L | F--] -- [L, L] | F . Therefore also L | F has no central component. Also since a basis of C M , ~ ( F ) ( a d L ) can be calculated by solving a set of linear equations over F we have C M , ~ ( F ) ( a d L ) | -- C M , ( F ) | 1 7 4 And finally R a d ( A | F) - Rad(A) | F by Proposition A.1.3. Hence the result for L | F will imply the result for L.
By Theorem A.1.5 we may write A - S @ Rad(A), where S is a semisim- ple subalgebra of A isomorphic to Q. We suppose that S is not commutative.
By Theorem A.1.4, S decomposes as a direct sum of simple associative al- gebras that are full matrix algebras over division algebras over F. As F is algebraically closed, the only division algebra over F is F itself (for a proof of this we refer to [69]). Hence from our assumption that S is not commu- tative it follows that S contains a full matrix algebra of degree at least 2.
Now every full matrix algebra of degree >_ 2 contains a full matrix algebra of degree 2. It follows that S contains elements eij for 1 <_ i, j <_ 2 such that eijekl -- (Sjkeil. Then 1 E A decomposes as a sum of three orthogonal idempotents 1 = ell + e22 + (1 - ell - e22) and the first two of these are certainly non-trivial. So by Proposition 1.15.1, L decomposes as a direct sum of ideals
L - II (~ I2 @ I3 where I1 - e l l L and I2 - e22L. Now we have maps
f 2 1 " I 1 ~ h , and f12 "/2 > I1
defined by f21 ( x ) - e21x and fl2(Y) = e12y for x C I1 and y E /2. Since e21x =- e22e21x we see that f21 maps I1 i n t o / 2 , and likewise f12 maps /2 into 11. Furthermore f 1 2 ( f 2 1 ( x ) ) - e12e21x - e l l X - x for x C I1. So f12 and f21 are each others inverses. In particular they are non-singular linear maps.
Because L does not have a central component we have that I1 a n d / 2 are not commutative Lie algebras. So there exist x, y C I1 such that [x, y] 7t 0.
We calculate
f21 ([x, y]) - e21[x, y] - e 2 1 a d x ( y ) - a d x ( e 2 1 y ) - [x, e21y] - 0
1.15 Finding a direct sum decomposition 37 where last equality follows from [11,/2] - 0. But since f21 is non-singular this implies that Ix, y] - 0 and we have reached a contradiction. The con- clusion is that S (and hence Q) is commutative. [::1 A l g o r i t h m DirectSu m Decom position
Input" a finite-dimensional Lie algebra L.
Output: a list of indecomposable ideals of L such that L is their direct sum.
Step 1 Compute the centre C(L) and the derived subalgebra [L,L]. Com- pute a basis of a complement J in C(L) to C ( L ) N [L, L]. Let K be a complement to J in L such that K contains [L, L].
Step 2 Set n := dim(K) and compute the centralizer A - C M n ( F ) ( a d K K ).
Step 3 Compute Rad(A) and set Q "= A / R a d ( A ) . Compute the set of primitive orthogonal idempotents e l , . . . , ek in Q such that el + " " +
~ k - l c Q .
Step 4 Lift the idempotents ~1,... , ~k to a set of orthogonal idempotents e l , . . . , e k C A such that e l + - . . + e r - 1 C A. For 1 _< r_< k set I~ - e~K.
Step 5 Let { Y l , . . . , Y m } be a basis of J and for 1 _< i _< m let Ji be the ideal of L spanned by Yi. Return { J l , . 9 9 , Jm} U { I 1 , . . . , Ik}.
C o m m e n t s . If x l , . . . ,Xn is a basis of K, then
C M n ( F ) ( a d K ) -- {a C M n ( F ) l a. adKxi -- adKxi " a for 1 < i < n}, and hence CM~(F)(adK) can be calculated by solving a system of linear equations. For the calculations in the associative algebra A (calculation of the radical, of central idempotents and lifting them modulo the radical), we refer to Appendix A. We note that since Q is commutative, all idempotents in Q are central. Hence a set of primitive orthogonal idempotents with sum 1 can be found by the algorithm Centraildempotents (as described in Appendix A).
R e m a r k . A computationally difficult part of the algorithm DirectSumDe- composition is the calculation of the centralizer CM,,(F)(adK). The dimen- sion of this algebra may be substantially bigger than the dimension of K.
So it would be desirable to have an algorithm that uses Lie algebra methods only. To the best of our knowledge no such algorithm exists for the general case. However, for the special case of semisimple Lie algebras we do have
38 Basic constructions an algorithm that does not "leave" the Lie algebra (see Section 4.12).
1.16 N o t e s
The algorithms QuotientAlgebra, AdjointMatrix, Centralizer, Centre, Normal- izer, ProductSpace, Derivations can be found in [7]. We have followed [71]
for the algorithm DirectSumDecomposition.
Usually (cf., [48], [88]), a Lie algebra of characteristic p > 0 is called restricted if it has a p-th power mapping. However, due to Proposition 1.13.4 (which is taken from [48]), this is equivalent to our definition of restrictedness.