Abstract
Let \({\mathfrak {g}}\) be an infinite-dimensional Lie algebra and G be the algebraic completion of its module. Using a geometric interpretation in terms of sewing two Riemann spheres with a number of marked points, we introduce a multiplication between elements of two spaces \(\mathcal {M}^k_m({\mathfrak {g}}, G)\) and \(\mathcal {M}^n_{m'}({\mathfrak {g}}, G)\) of meromorphic functions depending on a number of formal complex parameters \((x_1, \ldots , x_k)\) and \((y_1, \ldots , y_n)\) with specific analytic and symmetry properties, and associated with a \({\mathfrak {g}}\)-valued series. These spaces form a chain–cochain complex with respect to a boundary–coboundary operator. The main result of the paper shows that the multiplication is defined by an absolutely convergent series and takes values in the space \(\mathcal {M}^{k+n}_{m+m'}({\mathfrak {g}}, G).\)
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References
Androulidakis, I., Skandalis, G.: A Baum–Connes conjecture for singular foliations. Ann. K-Theory 4(4), 561–620 (2019)
Bazaikin, Ya. V., Galaev, A.S.: Losik classes for codimension one foliations. J. Inst. Math. Jussieu 21(4), 1391–1419 (2022)
Bazaikin, Ya.V., Galaev, A.S., Gumenyuk, P.: Non-diffeomorphic Reeb foliations and modified Godbillon–Vey class. Math. Z. 300(2), 1335–1349 (2022)
Bazaikin, Ya.V., Galaev, A.S., Zhukova, N.I.: Chaos in Cartan foliations. Chaos 30(10), 103116 (2020)
Beltia, D., Grundling, H., Neeb, K.-H.: Covariant representations for possibly singular actions on \(C^*\)-algebras. Diss. Math. 549, 1–94 (2020)
Bu, Q., Zhu, S.: The orthogonal Lie algebra of operators: ideals and derivations. J. Math. Anal. Appl. 489(1), 124134 (2020)
Carrillo Rouse, P., So, B.K.: K-theory and index theory for some boundary groupoids. Results Math. 75(4), Paper No. 172, 20 pp (2020)
Crainic, M., Moerdijk, I.: \(\check{{\rm C}}\)ech–De Rham theory for leaf spaces of foliations. Math. Ann. 328(1–2), 59–85 (2004)
Elworthy, K.D.: Infinite-dimensional degree theory and Ramer’s finite co-dimensional differential forms. Q. J. Math. 72(1–2), 571–602 (2021)
Francesco, Ph., Mathieu, P., Senechal, D.: Conformal Field Theory. Graduate Texts in Contemporary Physics (1997)
Fuks, D.B.: Cohomology of Infinite-Dimensional Lie Algebras. Contemporary Soviet Mathematics, Consultant Bureau, New York (1986)
Gui, B.: Categorical extensions of conformal nets. Commun. Math. Phys. 383(2), 763–839 (2021)
Gupta, V.P., Jain, R., Talwar, B.: On closed Lie ideals of certain tensor products of \(C^*\)-algebras II. Math. Nachr. 293(1), 101–114 (2020)
Hirshberg, I., Wu, J.: The nuclear dimension of \(C^*\)-algebras associated to topological flows and orientable line foliations. Adv. Math. 386, Paper No. 107798, 56 pp (2021)
Huang, Y.-Zh.: A cohomology theory of grading-restricted vertex algebras. Commun. Math. Phys. 327(1), 279–307 (2014)
Isidro, J.M.: Jordan Triple Systems in Complex and Functional Analysis. Mathematical Surveys and Monographs, vol. 243. American Mathematical Society, Providence (2019)
Li, X., Renault, J.: Cartan subalgebras in \(C^*\)-algebras. Existence and uniqueness. Trans. Am. Math. Soc. 372(3), 1985–2010 (2019)
Liu, H.: Smooth crossed product of minimal unique ergodic diffeomorphisms of a manifold and cyclic cohomology. J. Topol. Anal. 11(3), 739–751 (2019)
Oliva-Maza, J.: On Lie group representations and operator ranges. Proc. Am. Math. Soc. 149(10), 4317–4329 (2021)
Samoilenko, A.M., Prikarpatskii, Ya.A., Blekmor, D., Prikarpatskii, A.K.: Theory of multidimensional Delsarte–Lions transmutation operators. I. (Ukrainian) Ukrain. Mat. Zh. 70(12), 1660–1695 (2018); translation in Ukr. Math. J. 70(12), 1913–1952 (2019)
Samoilenko, A.M., Prikarpatskii, Ya.A., Blekmor, D., Prikarpatskii, A.K.: Theory of multidimensional Delsarte–Lions transmutation operators. II. (Ukrainian) Ukrain. Mat. Zh. 71(6), 808–839 (2019); translation in Ukr. Math. J. 71(6), 921–955 (2019)
Takhtajan, L.A.: Etudes of the resolvent. (Russian) Uspekhi Mat. Nauk 75(1)(451), 155–194 (2020); translation in Russ. Math. Surv. 75(1), 147–186 (2020)
Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. Pure Math. 19, 459–566 (1989)
Yamada, A.: Precise variational formulas for abelian differentials. Kodai Math. J. 3, 114–143 (1980)
Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9(1), 237–302 (1996)
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The second author is supported by the Academy of Sciences of the Czech Republic (RVO 67985840).
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Communicated by M. S. Moslehian.
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Levin, D., Zuevsky, A. The extension of cochain complexes of meromorphic functions to multiplications. Adv. Oper. Theory 8, 56 (2023). https://doi.org/10.1007/s43036-023-00277-7
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DOI: https://doi.org/10.1007/s43036-023-00277-7