Loop Groups, Discrete Versions of Some Classical Integrable by Percy Deift

By Percy Deift

The speculation of classical $R$-matrices presents a unified procedure to the certainty of so much, if no longer all, recognized integrable structures. This paintings, that is appropriate as a graduate textbook in the trendy thought of integrable platforms, offers an exposition of $R$-matrix conception by way of examples, a few outdated, a few new. specifically, the authors build non-stop types of a number of discrete platforms of the kind brought lately via Moser and Vesclov. within the framework the authors identify, those discrete platforms look as time-one maps of integrable Hamiltonian flows on co-adjoint orbits of applicable loop teams, that are in flip made out of extra primitive loop teams via classical $R$-matrix conception. Examples comprise the discrete Euler-Arnold most sensible and the billiard ball challenge in an elliptical area in $n$ dimensions. past result of Moser on rank 2 extensions of a hard and fast matrix may be integrated into this framework, which suggests particularly that many famous integrable systems---such because the Neumann process, periodic Toda, geodesic movement on an ellipsoid, etc.---can even be analyzed by way of this system.

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O o ^ i . 4) (note that we always have sgn ftm+i = 1, sgn Q _ m = — 1) and define a decomposition of 42 LOOP G R O U P S , INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 0 ft_ = sgn Q j < 0 0 + nO_=M+Ufi_=(T\E. 1 and for m — 2, + a_i ft. r_ m - ioo) , - ioo) 2 44 P. DEIFT, L, C. LI, AND C. T O M E I and the group operation is pointwise multiplication on E. Note that the case m = 0 reduces (essentially) to the group G of Section 2. The Lie-algebra £ associated to G is the set of all smooth functions X : E —• g£(N, W) such that (i)' X(X) and all its derivatives have limits at Xk i ioo for — m < k < m, (ii)' X(xk ± ioo) is diagonal with real entries at Xk + ioo, — m < k < m, (hi)' ("zero winding condition") m J2 (-l)m-hX(xk fc= —m m + ioo)= (-l)m~kX(xk-ioo), J2 k= — m and the bracket operation is taken pointwise on E.

A) = / + ( A ) , with f± G T T ± £ 2 ( S ) . As (det g)(-X) = AGS, det g(X) and as (det g)(X) > 0 for A G S 0 , the above "Vanishing Lemma" argument now applies and we conclude that det(I+fi±(X)) = 0, which is a contradiction. Hence I + JJL± are invertible. This completes the proof of our results, which we summarize in the following theorem: T h e o r e m 3 . 5 8 . 4°(A) g+(t,X)= = t f + ( t A ) ?

110) above. ))(A) ,M(t,A)] , M(0,A) = Afo(A) generated by the Ad* -invariant Hamiltonian H(A) = ]im [^ ti{A(\)logA(\)-A(\))^f rtoo J_ir on g*. 116) algorithm 2Z . 118) For all times t, M(£, A) has the form where M(t) is real and skew, and M(k) = Af* for all k £ ZZ . 115) follows from jR-matrix theory and can be verified directly. One obtains the formula and one needs dff-(t,A)_ „ , . _ 1! (t, A)" = (TT_ log M ( t , •) )(A) . 102), 1 by ^~ ' dia g° nalit 12121") y> """ ^ = -(7_(l)-h7o) . 119). 114) that (1 — \2)M(t, A) has an analytic contin- uation to Re A > 0 and to Re A < 0, and grows at most quadratically as A —• oo.

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