BANACH ALGEBRAS ASSOCIATED TO LAX PAIRS

作者：Glazebrook James F^*

来源：Reports on Mathematical Physics, 2015, 75(2): 279-301.

DOI：10.1016/s0034-4877(15)30008-2

• Banach algebra
• Banach homogeneous space
• Jordan-Lie-Banach algebra
• Lax pair
• vertex operator
• bosonization
• state space
• Burchnall-Chaundy C*-algebra
• CAR algebra
• GNS representation
• Sato correspondence
• Plucker embedding

Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall-Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand Naimark Segal construction. Instrumental is the Plucker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan Lie Banach algebras structures can be compared. Keywords: Banach algebra, Banach homogeneous space, Jordan Lie Banach algebra, Lax pair, vertex operator, bosonization, state space, Burchnall Chaundy C*-algebra, CAR algebra, GNS representation, Sato correspondence, Plucker embedding.

• 出版日期2015-4