dorsal/arxiv
View SchemaLoop algebras, gauge invariants and a new completely integrable system
| Authors | M. Quinn, S. F. Singer |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9606008 |
| URL | https://arxiv.org/abs/solv-int/9606008 |
Abstract
One fruitful motivating principle of much research on the family of integrable systems known as ``Toda lattices'' has been the heuristic assumption that the periodic Toda lattice in an affine Lie algebra is directly analogous to the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper shows that the analogy is not perfect. A discrepancy arises because the natural generalization of the structure theory of finite-dimensional simple Lie algebras is not the structure theory of loop algebras but the structure theory of affine Kac-Moody algebras. In this paper we use this natural generalization to construct the natural analog of the nonperiodic Toda lattice. Surprisingly, the result is not the periodic Toda lattice but a new completely integrable system on the periodic Toda lattice phase space. This integrable system is prescribed purely in terms of Lie-theoretic data. The commuting functions are precisely the gauge-invariant functions one obtains by viewing elements of the loop algebra as connections on a bundle over $S^1$.
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"abstract": "One fruitful motivating principle of much research on the family of\nintegrable systems known as ``Toda lattices\u0027\u0027 has been the heuristic assumption\nthat the periodic Toda lattice in an affine Lie algebra is directly analogous\nto the nonperiodic Toda lattice in a finite-dimensional Lie algebra. This paper\nshows that the analogy is not perfect. A discrepancy arises because the natural\ngeneralization of the structure theory of finite-dimensional simple Lie\nalgebras is not the structure theory of loop algebras but the structure theory\nof affine Kac-Moody algebras. In this paper we use this natural generalization\nto construct the natural analog of the nonperiodic Toda lattice. Surprisingly,\nthe result is not the periodic Toda lattice but a new completely integrable\nsystem on the periodic Toda lattice phase space. This integrable system is\nprescribed purely in terms of Lie-theoretic data. The commuting functions are\nprecisely the gauge-invariant functions one obtains by viewing elements of the\nloop algebra as connections on a bundle over $S^1$.",
"arxiv_id": "solv-int/9606008",
"authors": [
"M. Quinn",
"S. F. Singer"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Loop algebras, gauge invariants and a new completely integrable system",
"url": "https://arxiv.org/abs/solv-int/9606008"
},
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