Loops in \(\mathrm{SU}(2)\), Riemann surfaces, and factorization. I. (English) Zbl 1337.22012
The paper under review extends the previous work of the second named author [J. Funct. Anal. 260, No. 8, 2191–2221 (2011; Zbl 1217.22019)], where it was proved that a loop \(g: S^1\to\mathrm{SU}(2)\) has a triangular factorization if and only if \(g\) has a root subgroup factorization. Generalizations are obtained in the present paper, when the unit disk and the sphere (double of the disk) are replaced by a based compact Riemann surface and its double respectively. The authors show that a \(mathrm{SU}(2)\) valued multiloop having an analogue of a root subgroup factorization, viewed as a transition function, defines a semistable holomorphic \(\mathrm{SL}(2,\mathbb{C})\) bundle. The authors use the ideas of generalized Fourier-Laurent expansions developed by I. M. Krichever and S. P. Novikov [J. Geom. Phys. 5, No. 4, 631–661 (1988; Zbl 0707.17015)] to obtain a linear triangular factorization. They present some calculations of determinants for spin Toeplitz operators in the scalar case. They also announce that a number of explicit calculations for elliptic and hyperelliptic surfaces will appear in the second part of the paper.
Reviewer: Dongwen Liu (Zhejiang)
MSC:
| 22E67 | Loop groups and related constructions, group-theoretic treatment |
| 47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
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