By Raphael Rouquier

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Extra info for Hall Algebras

Example text

S, let Ffj. be the generator corresponding to Hj. One has V ^ ' C VHi ; in case of a strict inequality denote by qj = Hks, where the product is over all indices 6 such that ms divides \Hj\ but doesn't divide \Hj\. (In order to have a unique notation, for SQ one has t = 0). 3. 1) S Q is an isomorphism except for the cases k — I -f 1, with I — 1 or 2. ,s. 2) If k < I —I, let IQ = (mi : . . 's of these numbers. F) = (m0r/l0)degE(F). Thus, E r is always one to one and S r is onto only if rriQr = IQ, with n > 0 if k = / — 1.

1 . Trivial invariant part, t h e case p > 1. T h e o r e m 3 . 1 . / / k = I -f 1 — 2/9, p > 1, iij = kjirij for j = 1 , . . , ra — p = n, rij multiple of mr , r = n + 1 , . . , ra, then i Jb+2m^ f^ ) 2Z P> 1 (nfci)mo/mm» P = 1 (0,ifn = 0 , r a = l ) . 2 that two extensions may differ by a multiple of (IIA:j)rao/m m , however the following explicit construction will be needed. Let (/o,<£o) the map D e an Y extension of (1,0) to [0,1] x BQ with norm 1 and of degree d. Consider S^EQUIVARIANT D E G R E E 43 fd(t,x0,z)=(^^{fo-^^---A^m3-ejznJmnt(i-t)(Ro-\^\),--^ + (1,0,0) where j runs from 1 to r?.

Once the extension to the ball C is performed, one extends for ip € [0, 27r]r by using the action of the group 7 , namely: f(t,x0,\zP\,z,lkz) = e2"'*/P/(iia:o,kpl,e-2"*/Pi,z) for*/|z| G A. 0i|^|e-«f-i^~ie-if-i^i) by the construction of / , thus / = e^/P/^^^j^i^e-^/P^^e-^/PS) is well defined. Furthermore = e l >/(t,a:o,2). D. Note that since lLp is the maximal group which leaves zp real and positive then this construction is compatible with the previous S -maps. Corollary 2 . 1 . If for all isotropy subgroups H of S then a S -map f : S IR /+1 \ {0} is trivial one has that dimV > W \ {0} has a non zero S -extension Thus, if k < I, n £ + 2 m ( S /+2n ) - 0.