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Additional resources for Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman
In the supercase, the group r acts on SIItDO(R) via its action on Rand on D. 1) transforms D into (cz+d+o()D. The ring SIItDO(R)f denotes the r-invariant elements of SIItDO(R). We have the filtration SIItDO(R)~, k E Z, inherited from the filtration of SIItDO(R). 8) for the supercase is the sequence, which is split short exact by Theorem 3 below, involving supermodular forms of weight k (for all parities of k) o ~ SIItDO(R)~k_l ~ SIItDO(R)~k ~ SMk(r) ~ o. 4) The analogue of Proposition 1 for supermodular forms is as follows.
7. Supermodular Forms and Superpseudodifferential Operators We work on the supercomplex plane ((;111 with local coordinate (z, (), where (2 = 0, and with the canonical supersymmetric (SUSY) structure given by the maximal non-integrable structure distribution of rank 011 generated by the vector field D = + satisfying D2 = This is, up to isomorphism, the unique such SUSY structure extendible to p111. ) If (z, () is another local coordinate defining the same SUSY structure, then D = J. jj where J = D( () is the superanalogue of the usual Jacobian.
5). The considerations of Section 3-6 about the multiplications of modular forms induced by the multiplications of various kinds of automorphic llfDO's can be generalized in the more or less obvious way (thus an automorphic super-llfDO of mixed weight is just a super-llfDO which is multiplied on the left and on the right by some powers of J (A) under the action of A E r), and the arguments given in the last section can also be generalized using the superversion of the non-commutative residue map given in [MRJ.