Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction with grade 1 regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra G(X)C( lambda , lambda -1) are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group W(G) of the simple Lie algebra G. A representative w epsilon W(G) of a regular conjugacy class can be lifted to an inner automorphism of G given by w=exp (2i pi ad I0/m), where I 0 is the defining vector of an sl2 subalgebra of G. The grading is then defined by the operator d(m,I0)=m lambda (d/d lambda )+ad I0 and any grade 1 regular element Lambda from the Heisenberg subalgebra associated with (w) takes the form Lambda =(C++ lambda C-), where (I0, C-)=-(m-1)C- and C+ is included in an sl2 subalgebra containing I 0. The largest eigenvalue of adI0 is (m-1) except for some cases in F4, E6,7,8. We explain how these Lie algebraic results follow from known results and apply them to construct integrable systems. If the largest ad I0 eigenvalue is (m-1), then using any grade 1 regular element from the Heisenberg subalgebra associated with (w) we can construct a KdV system possessing the standard W-algebra defined by I 0 as its second Poisson bracket algebra. For G a classical Lie algebra, we derive pseudo-differential Lax operators for those non-principal KdV systems that can be obtained as discrete reductions of KdV systems related to gln. Non-Abelian Toda systems are also considered.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)