Abstract
A nuclear ideal is an ideal contained in an ambient monoidal dagger category which has all of the structure of a compact closed category, except that it lacks identities. Intuitively, the identities are too “singular” to live in the ideal. Typical examples include the ideal of Hilbert-Schmidt maps contained in the category of Hilbert spaces, or the ideal of test functions contained in the category DRel of tame distributions on Euclidean space.
In this paper, we construct a category of tame formal distributions with coefficients in an associative algebra. We show that there is a formal analogue of the nuclear ideal constructed in DRel, and hence there is a partial trace operation on the category. By taking formal distributions with coefficients in the dual of a cocommutative Hopf algebra, we obtain a categorical generalization of the Borcherds’ notion of elementary vertex group. Furthermore, when considering the algebra of symmetric endomorphisms of an object in such a category, we obtain a vertex group in Borcherds’ sense. The nuclear ideal structure induces a partial trace operator on such vertex groups.
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Blute, R., Panangaden, P. (2010). Dagger Categories and Formal Distributions. In: Coecke, B. (eds) New Structures for Physics. Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12821-9_6
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DOI: https://doi.org/10.1007/978-3-642-12821-9_6
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