By Masaki Kashiwara, Takahiro Kawai

**Read or Download Algebraic Analysis. Papers Dedicated to Professor Mikio Sato on the Occasion of his Sixtieth Birthday, Volume 1 PDF**

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**Extra info for Algebraic Analysis. Papers Dedicated to Professor Mikio Sato on the Occasion of his Sixtieth Birthday, Volume 1**

**Example text**

Type construction, starting from a given positive linear functional W on the tensor ^-algebra sé of test-functions generated by sdx. From this viewpoint (initiated by the Wightman reconstruction theorem [2], and deeply developed by H. Borchers [3]), $? can be considered as isomorphic to the (closure of the) quotient-space sé/369 if ffi denotes the null-ideal of W. T. e. on its vacuum expectation values or Green functions) or by conditions on the subspace £$ of the algebra sé; for example the "locality property," which states that [A(f), A(g)] = 0 for all couples (/, g) whose inverse Fouriertransforms / , g have "acausal supports," may be expressed by saying that the corresponding elements f®g-g®f of sé belong to £8.

But these may very well not lead to any new Riemann surfaces. Instead, the situation may be like the genus-one case and will produce either the same free theory as the chiral Potts case or produce an interacting parafermion theory just as the eight-vertex model produces an interacting fourfermion theory. Indeed, the major unsolved conceptual problem at this stage is the following: can one expect to find a system of commuting transfer matrices for any Riemann surface, or are there special properties, such as a large group of automorphisms, which a Riemann surface must possess before it can produce a set of commuting transfer matrices?

We call Ch(P) the characteristic set of P ([4], [12]). 2) Theorem 2. bundle T*X. 1). Theorem 3. Let x be a point in X. If there exists £e T*X -{0} such that the set {λ e C*; (x, \£) e Ch(P)} is contained in the right half-plane Re λ > 0, then P is locally solvable at x. In [4], we have constructed a solution of Pu=f directly by using the symbol of inverse of P. Here we prove Theorem 3 as a corollary of the following theorem which has been suggested by Prof. Kashiwara. 22 Takashi Aoki Theorem 4.