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1、arXiv:0712.3390v1 math.RA 20 Dec 2007C-SUPPLEMENTED SUBALGEBRAS OF LIE ALGEBRASDAVID A. TOWERSDepartment of Mathematics and Statistics Lancaster University Lancaster LA1 4YF England d.towerslancaster.ac.ukAbstractA subalgebra B of a Lie algebra L is c-supplemented in L if there is a subalgebra C of
2、L with L = B + C and B C BL, where BL is the core of B in L. This is analogous to the corresponding conceptof a c-supplemented subgroup in a finite group. We say that L is c- supplemented if every subalgebra of L is c-supplemented in L. We give here a complete characterisation of c-supplemented Lie
3、algebras overa general field.Mathematics Subject Classification 2000: 17B05, 17B20, 17B30, 17B50. Key Words and Phrases: Lie algebras, c-supplemented subalgebras, completely factorisable algebras, Frattini ideal, subalgebras of codi- mension one.1IntroductionThe concept of a c-supplemented subgroup
4、of a finite group was introduced by Ballester-Bolinches, Wang and Xiuyun in 2 and has since been studied1by a number of authors. The purpose of this paper is study the correspond- ing idea for Lie algebras. As we shall see, stronger results can be obtained in this context.Throughout L will denote a
5、finite-dimensional Lie algebra over a field F.If B is a subalgebra of L we define BL, the core (with respect to L) of B to be the largest ideal of L contained in B. We say that B is core-free in L if BL= 0. A subalgebra B of L is c-supplemented in L if there is a subalgebra C of L with L = B + C and
6、 B C BL. We say that L is c-supplemented if every subalgebra of L is c-supplemented in L. We shall give a completecharacterisation of c-supplemented Lie algebras over a general field. Following 4 we will say that L is completely factorisable if for every subalgebra B of L there is a subalgebra C suc
7、h that L = B+C and BC = 0. It turns out that c-supplemented Lie algebras are intimately related to the completely factorisable ones, and our results generalise some of those obtained in 4. Incidentally, it is claimed in 4 that if F has characteristic zero then L is completely factorisable if and onl
8、y if the Frattini subalgebra of every subalgebra of L is trivial. We shall see that this is false. If A and B are subalgebras of L for which L = A + B and A B = 0 we will write L = A+B; if, furthermore, A,B are ideals of L we write L = A B. The notation A B will indicate that A is a subalgebra of B,
9、 and A 1 and m is odd, then Lm() is simple and has only one subal- gebra of codimension one.(ii) If m 1 and m is even, then Lm() has a unique proper ideal of codimension one, which is simple, and precisely one other subalgebra of codimension one.6(iii) L1() has a basis u1,u0,u1 with multiplication u
10、1,u0 = u1+ 0u1(0 F,0= 0 if = 0), u1,u1 = u0,u0,u1 = u1.(iv) If F has characteristic different from two then L1()= L1(0)= sl2(F).(v) If F has characteristic two then L1()= L1(0) if and only if 0is a square in F.The above properties enable us to determine which of the algebras Lm() are c-supplemented.
11、Proposition 3.4 If L= Lm() then L is c-supplemented if and only L= L1(0) and F has characteristic different from two.Proof. Suppose that L= Lm() and L is c-supplemented, and let x L. Then there is a subalgebra M1of L such that L = Fx+M1, and FxM1 (Fx)L= 0, since Lm() has no one-dimensional ideals. C
12、hoose y M1. Then, similarly, there is a subalgebra M2of codimension one in L such that L = Fy+M2and M16= M2. Since L = M1+M2we have that M1M26= 0. Let z M1 M2. Then there is a subalgebra M3of codimension one in L such that L = Fz + M3, so L has at least three subalgebras of codimension one in L. It
13、follows from Theorem 3.3 that m = 1. Suppose that L 6= L1(0). Then F has characteristic two and 0is not a square in F. Since L is completely factorisable there is a two-dimensional subalgebra M of L such that L = Fu1+ M. It follows that M = F(u1+ u1)+F(u0+u1) for some , F. But then u1+u1,u0+u1 M sho
14、ws that 0= 2, a contradiction. A further straightforward calculation shows that if L= L1(0) and F has characteristic two, then Fu1is contained in every maximal subalgebra of L, and so has no c-supplement in L. Conversely, suppose that L= L1(0) and F has characteristic different from two. Then L= sl2
15、(F), by Theorem 3.3 (iv) and it is easy to check that L is c-supplemented.We can now determine the simple and semisimple c-supplemented Lie algebras.Corollary 3.5 If L is simple then L is c-supplemented if and only L= L1(0) and F has characteristic different from two.7Proof. Let L be simple and c-su
16、pplemented. Then L has a core-free maximal subalgebra of codimension one in L and so L= Lm(), by Theorem 3.2. The result now follows from Proposition 3.4.Notice, in particular, that sl2(F) is the only simple completely factoris-able Lie algebra over any field. However, this is not the only simple ele-mentary Lie algebra, even over a field of characteristic zero: over the realfield every compact
