# Download Nilpotent Lie Groups:: Structure and Applications to by Roe W. Goodman PDF

There such t h a t 1 <_ p <_ r-1 is d e f i n e d by Fp(x,y) = p-I z mk ~p-k ( x ' y ) k=l - ~o(mkX'mp-k y) (F z = O) Remarks 1.

For the case of an n-parameter commutative group of diagonalizable automorphisms of a n i l p o t e n t Lie algebra, cf. Favre [1]. g. in Bourbaki [1], Hochschild [I~, and Jacobson [1]. 3 The construction given here is the dual to the construction of B i r k h o f f [ I ] . The l i n e a r dual space to the enveloping algebra of isomorphic to the formal power series functions on ~ a n n i h i l a t o r of the ideal degree < n . Jn+l ~ is canonically (cf. Dixmier [23]),and the of Birkhoff is the space Pn of polynomials of For the analogue of the B i r k h o f f construction in the case of a solvable Lie algebra, cf.

X 7} , and a s i x - d i m e n s i o n a l commutative o p e r a t o r on W. is obtained by s e t t i n g 1 < n < 7 25 This f i l t r a t i o n is o f l e n g t h seven, and i t is obvious t h a t r e l a t i v e to t h i s filtration gr(g) = ~ (a,O,O,d) w i t h the parameters writing Ix,y] a and = ~(x,y), d , arbitrary. Letting Vn = span (Xn) we see t h a t the decomposition (I) of ~ , and i n t o a sum o f homogeneous terms is given by ~o(Xl,Xn) = Xn+ 1 , ~o(Xl,X7) = 0 ~o(X2,X3) = ax 5 ~o(X2,X4) = ax 6 ~o(X2,X5) = ( a - d ) x 7 ~o(X3,X4) = dx 7 ~1(x2,x3) = bx 6 , ~1(x2,x4) = bx 7 ~ 2 ( x 2 ' x 3 ) = cx7 ' u3 = ' " = u6 = 0 ( A l l e n t r i e s not o b t a i n a b l e by skew-symmetry are z e r o ) .