By Matthew Witten
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Extra resources for Hyperbolic Partial Differential Equations. Modern Applied Mathematics and Computer Science
We now examine 2-cocycles on some of the Lie algebras Lh(n). Recall that a bilinear differential operator w(X, Y) on a Lie algebra L is called a (generalized) 2-cocycle if w(X, Y) w(y, X) and w(X, [7, Z]) + w(Y, [Z, X]) + w(Z, [X, K]) ~ 0, VX, Y,ZEL, were a ~ 0 means that a E S 5 Im 6, (or 2 , Im(A5 - 1) in the discrete case). 9 Let X. = (1, . . , 1). ). Then, w,- is a 2-cocycle on Lh(n). 10) =Sx + div(y) )'(3) C)'0) ^ ^ so that w, is skew-symmetric. Next, O-0M3) [x, y] \ /z = 2 [X, Ylwjj + 2 (*/^ ~ Yi»j)> /Z klk + c p .
11L) = 2 ^ ( v , [ X a , Xv] + 2 ^*a(^ (v) )X v - 2 g(v)Xv(5^)Xa. 11), we find that 2 sX(g™)Xa = 2 s™Xa(g™)Xv, so that sX = 25(a)XCT. In other words, a = id. 11 ■ _ 0 = 1©0. 9 it follows that 0(D„) C Dn. 10. 12 Let a,:L —» End V, be two representations, / = 1,2, and let a: V, —» V2 be a homomorphism. Then l © a : L x a V, -* L xCTV2 is a homomorphism of Lie algebras if and only if a intertwines the representations cr, and cr2, i-eo-2(X)a - aa,(X), VX G L. 12) 404 B. A. KUPERSHMIDT Proof. Let us write a instead of 1 © a.
1), there exists at least one X,- ^ 1. 6, we can move this X7 into the first place. So, let X, = X ¥^ 1. Set p = p,, and define i|i; = p / P \ 0^ G k, 7 = 2 , . . , m. 10) from coordinates (M, p, p 2 , . . , p,M) to (M, p, i|/2, . . 6), we get, for7 = 2, . . , m: % = 2 [7/P,P7'",(^vP ~ Xpd,) + P7'(^vP, - X /P/ d v)](|^) = X frA ~ [^ + ^(x - W A M ^ l (7-3) Thus, we transformed the index X in #x(«) from X = (X; X2, . . , X,M) into X' = (X; . . \j + y,(K - 1), . . ). 4) In particular, choosing 7, = X//(l - X), we obtain X' = (X; 0, .
Hyperbolic Partial Differential Equations. Modern Applied Mathematics and Computer Science by Matthew Witten