Integrable systems from Poisson reductions of generalized Hamiltonian torus actions

Abstract We develop a set of sufficient conditions for guaranteeing that an integrable system with a symmetry group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> on a manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> descends to an integrable system on a dense open subset of the quotient Poisson space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> . The higher dimensional phase space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> carries a bivector <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:mrow> </mml:math> yielding a bracket on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>K</mml:mi> </mml:msup> </mml:mrow> </mml:math> is a Poisson algebra. The unreduced system on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> is supposed to possess ‘action variables’ that generate a proper, effective action of a group of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and descend to action variables of the reduced system. In view of the form of the group and since <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:mrow> </mml:math> could be a quasi-Poisson bivector, we say that we work with a generalized Hamiltonian torus action. The reduced systems are in general superintegrable owing to the large set of invariants of the proper Hamiltonian action of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">U</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . We present several examples and apply our construction for solving open problems regarding the integrability of systems obtained previously by reductions of master systems on doubles of compact Lie groups: the cotangent bundle, the Heisenberg double and the quasi-Poisson double. Furthermore, we offer numerous applications to integrable systems living on moduli spaces of flat connections, using the quasi-Poisson approach.