By I.G. Macdonald

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20]); there, the generalization of manifolds studied is in terms of the commutative ring of smooth functions defined on a manifold. Here, we are looking instead at a generalization of the structure given by the smooth functions defined with domain an open subset of an arbitrary Hausdorff, complete, locally convex topological vector space and with range a fixed manifold. In section 2 we give details on our construction of a diffeological Lie group A(A), corresponding to a generalized symmetrized Cartan matrix A with a possibly countably infinite number of rows and columns such that the rows and columns are uniformly £2 bounded.

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Math. vol. 2, (1981), 365-397. 23. J. S. 3, (1974), 503-505. 24. L. Pontrjagin, Topological groups, Princeton University Press, fifth printing, 1958. 25. T. Robart, Groupes de Lie de dimension infinie. second et troisieme theoremes de Lie. ce, C. R. Acad. Sci. Paris, Serie I 322 (1996), 1071-1074. 26. T. Robart, Sur I'integrabilite des sous-algebres de Lie en dimension infinie, Can. J. , vol. 49 (4), 1997 pp. 820-839. 27. T. Robart, On Milnor's regularity and the path functor for the class of infinite dimensional Lie algebras of CBH type, Algebras, Groups and Geometries 16, 1998.

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