Download e-book for kindle: Module Categories of Analytic Groups by Andy R. Magid

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Thus, R(G) = C[t9 T 1 ]. We also have that [t]r = (t) and [ r 1 ] r = . 15), we get all G-modules from {[t]T, [t'1]^ by submodules of tensor products and direct sums of [t]r and [t~x\. In the notation of Example A, if raeZ, then if m > 0 , C[m] = [>]fm and if m < 0 , C[m] = [^~1]f~m, and every G-module is a direct sum of some C[m]s. Example Bf Let G = C (additive group). We first list some representative functions on G. Let t:G^C be given by r(x) = jc. If yeG, (fy)(x) = t(y + x) = y + x, so f y = y + f.

Now let u . . ,vn. Then fiti =fi(x(Vj)) = i{xmiVj), which is x™* if / = / and zero otherwise. Thus, ftj = 8ijtm'. 9)(a), (ft) and (c) show that / = IaIft7/I,/, Thus, / is a polynomial, with complex coefficients, in t and f"1. We already know that t is representative and a similar calculation shows that t~l • y = y'" 1 for all y in G, so T 1 is representative. Thus, R(G) = C[t9 T 1 ]. We also have that [t]r = (t) and [ r 1 ] r = . 15), we get all G-modules from {[t]T, [t'1]^ by submodules of tensor products and direct sums of [t]r and [t~x\.

It is clear from Examples A and B that if V and W are G-modules and f\V-*W a G-homomorphism, then /(Vi)cWi' for each / and fN = N'f where N, Nr are the degree 1 endomorphisms of V and W. Thus we can sum up by saying that Mod(G) is the category of graded vector spaces with degree 1 endomorphisms. We will determine its tensor automorphisms now. We want to consider some special objects in Mod(G): Wn[i\ for / e Z and n nonnegative, is the n-dimensional vector space with basis j c o , . . , *»-i, grading such that xk has degree k + /, and degree 1 endomorphism N(xk) = xk+i (where xn = 0).