# Lectures on the Icosahedron by Felix Klein, George Gavin Morrice

By Felix Klein, George Gavin Morrice

*(x + iy)*; an announcement and exam of the elemental challenge, with a view of its algebraic personality; and common theorems and a survey of the topic. the second one half explores the speculation of equations of the 5th measure and their historic improvement; introduces geometrical fabric; and covers canonical equations of the 5th measure, the matter of A's and Jacobian equations of the 6th measure, and the overall equation of the 5th measure. moment revised version with extra corrections.

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Let f(z)ezW[[z]]. 14) These identities also hold for more general f for which the series are well defined, such as f(z,z0)ezW[[z,z0]]. 15) Proof. 12). 13) is similar. 19) L(-l)L(l)n = L(l)nL(-l) - 2nL(l)n-1L{0) + n(n - l)L(l)11-1. 19) we have K-i)e^^(^L (-i)x: /(z)" nf (1) " n>0 v ^ /(z)*£(-l)£(l)" n! 20). 11). For convenience, we assume that v G V is homogeneous of weight n E 2£ : L(0)v = nv. 22) where w' and u> are arbitrary elements of W and W, respectively. 23) 48 IGOR B. L^L(0)(-z-2)nv, z'1) -y(i(i)e'I(1'(-r,)"»)r1) = Y{e,L(-1\-z-2)L^L(-\)v,z-1) + Y(2z-1ezL^L(0)(-z-2)L^v, z~x) - YiLiiy^i-z-^Wv^-1).

1 for n — 1 implies the assertion for n, with v =. 1. 8) which is a polynomial in z\ since [£(0), L(—1)] = L(—1). argument here is typical of the general case. 10) for v EV (recall t h at V is characterized as the space of linear functionals on V vanishing on all but finitely man y V( n )). 7). 7). 11) is of the form (v , Y ^ , 2 2 ) ^ ( ^ 3 , £3)1)? 12). 12) are equal, as desired. 6. 1. 2) a n d the Jacobi identity will follow. 4. 18) hold. Let vuv2,v3 G V a n d v' £ V'. 1) where of course Y(y\, ZQ -f z2) is to be expanded in nonnegative integral powers of z2.

5. 5). 2) for Y(v,z) (v G V) acting on W. If rationality and commutativity hold for (ui1,Ui2, Uis), where U\, 112 G V and U3 G W, and (i^h) is an arbitrary permutation of (123), then (W, Y) is a V-module. 2. Adjoint vertex operators and the contragredient module In order to extend the discussion of the previous section to duality for two module elements, we shall need the theory of adjoint operators. 1)). 4) VERTEX OPERATOR ALGEBRAS AND MODULES 45 for v E V, wf E W , w E W . ) T h e operator ( — z ~ 2 ) L ^ has the obvious meaning; it acts on a vector of weight n G Z as multiplication by ( — z ~ 2 ) n .