By Samiou E.
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Additional info for 2-step nilpotent Lie groups of higher rank
6) i: A 1 • ~A Let Mo(A1)Y lle in denotes FI y = = I in ~ (a i ^ b i) i=I M(AI)el But clearly the inclusion. Thus in A2AI rather since all terms of A2(i)y = x , where y = 0 implies x = 0 ; 45 the problem figure is reduced A 1 ~ (tl> x ... x (t k) We may assume whenever A2AI = 0 ti both orders are finite and and moment. y = 0 ; done. flO (b). 7. PROOF. Let M(Q) in = I . 6, A2(AI) x = 0 . follow. Mo(Q which for the 7"o(A1) ~ M ( A I ) e: N~ @: A tj ) projection. 8 PROPOSITION. tj I ~ m < n ~ k Mo(A2)z A2(A2 ) groups.
11 and = 0 . Con- ~*[e I] = [e l q 5: G - G 1 with = 0 ~I 5 = ~ . ab(e) . ,1): thus at el: ¢: N - A (a,5,1): = Ker Hence by P r o p o s i t i o n let be given. ab(e))v*) = O versely, extension e*(ab(e),A) (e) E x a c t n e s s . split extensions. 4) e I - e2 ab(e 1) - ab(e 2) is commutative. y*[ab(e2) ] = = 0pext(~,h)e*(e2,A2)f . 4) is commutative. ( ~2 ~ )* = h * ( ~ l )* * = h*~1~ * Indeed, = ~I* h*7 * = 28 3. The Schur Multiplicator and the Universal Now we define the Schur multiplicator nite) group Q and, likewise, homomorphism 7- We define M(Q) = RO[F,F])/[R,F] tation of Q.
A u ~ U , the naturallty is obvious. is G-linear. is G-llnear, follows from ® (ux k)'a = xklu-1 ~A Just is the this definition of a restriction does not depend on the choice of transversal, PROOF. Res ° [ 1 ; p. 1901. Assume ~a| . 5 DEFINITION. a) = ~A(ga) Assume IG:UI < - • Wlth ~ = ~z:Z - ZG @U Z as C°rn--~-l°T°rn (B'~): Hn (G'B)=T°rG(B'Z) " T°rG(B'ZG ~U Z) - H n ( U , V B ) corn=Ext( ~,A ). ~ : Hn(U,VA) - Ext;( ZG @U Z ,A ) - Ext;( Z,A )=Hn(G,A) .
2-step nilpotent Lie groups of higher rank by Samiou E.