Gel’fand, I. M.; Cherednik, I. V. The abstract Hamiltonian formalism for the classical Yang-Baxter bundles. (English. Russian original) Zbl 0536.58006 Russ. Math. Surv. 38, No. 3, 1-22 (1983); translation from Usp. Mat. Nauk 38, No. 3(231), 3-21 (1983). Let \(I_{\alpha}\) be generators of the Lie algebra \(gl_ n\) and L be an element of \(gl_ n((\lambda))\) depending on x: \(L(\lambda;x)=\sum_{\alpha}u_{\alpha}(\lambda;x)I_{\alpha}, u_{\alpha}=\sum^{+\infty}_{i=-k}u^ i_{\alpha}(x)\lambda^ i,\) where k are arbitrary nonnegative integers. The Poisson bracket for f and g, which are functionals of \(u^ i_{\alpha}\), is defind by \[ \{f,g\}=\sum_{\alpha,\beta;i,j}\int \delta f/\delta u^ i_{\alpha}(x)\delta u^ j_{\beta}(y)H^{ij}_{\alpha b}(x,y)dxdy \] where \[ H^{ij}_{\alpha \beta}=H^{ij}_{\alpha b}((u^ m_{\gamma});x,y) \] are generalized functions of x and y depending on finitely many \(u^ m_{\gamma}\). The authors indicate that to ensure oneself that the above operation does define a Poisson bracket for any f and g one needs only to verify it for \(u^ i_{\alpha}\). For arbitrary solutions of the (triangle) Yang-Baxter equation and an L in its general form the authors construct the Poisson bracket which satisfies the so- called Faddeev relation \[ \sum_{\alpha,\beta}\{u_{\alpha}(\lambda;x),u_{\beta}(\mu;y)\}I_{\alpha}\otimes I_{\beta}=[L(\lambda;x)\otimes 1+1\otimes L(\mu;y),r(\lambda,\mu)]\delta(x-y) \] where \(r(\lambda\),\(\mu)\) is an x- independent matrix function of \(\lambda\) and \(\mu\) with its value in the tensor product \(gl_ n\otimes gl_ n\), and 1 denotes the identity matrix. Given L and r the authors solve the above Faddeev relation for \(H^{ij}_{\alpha \beta}(x,y)=\{u^ i_{\alpha}(x),u^ j_{\beta}(y)\}\) under some conditions on \(r(\lambda\),\(\mu)\), and show that the Hamiltonian conditions on \(H^{ij}_{\alpha \beta}\) for arbitrary L are equivalent to the classical triangular equation for \(r(\lambda\),\(\mu)\). Furthermore they construct a hierarchy of Hamiltonians which are involutory with respect to the above Poisson bracket. The nonlinear differential equations corresponding to these Hamiltonians are also presented. Reviewer: G.Tu Cited in 4 Documents MSC: 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 17B45 Lie algebras of linear algebraic groups 81T08 Constructive quantum field theory Keywords:Lax equations; Poisson bracket; Faddeev relation PDFBibTeX XMLCite \textit{I. M. Gel'fand} and \textit{I. V. Cherednik}, Russ. Math. Surv. 38, No. 3, 1--22 (1983; Zbl 0536.58006); translation from Usp. Mat. Nauk 38, No. 3(231), 3--21 (1983) Full Text: DOI