1994/10/31 by F. Delduc, L. Feher
Physics and Astronomy · #hep-th
paper · pdf · doi:10.1088/0305-4470/28/20/016
published as J.Phys.A28:5843-5882,1995 · 44 pages, ENSLAPP-L-493/94, substantial revision, SWAT-95-77. (use OLATEX (preferred) or LATEX)
arxiv created 1995/06/08 · arxiv updated 2010/11/01
Generalized KdV hierarchies associated by Drinfeld-Sokolov reduction to grade one regular semisimple elements from non-equivalent Heisenberg subalgebras of a loop algebra \G⊗\bf C[λ,λ-1] are studied. The graded Heisenberg subalgebras containing such elements are labelled by the regular conjugacy classes in the Weyl group \bf W(\G) of the simple Lie algebra \G. A representative w∈ \bf W(\G) of a regular conjugacy class can be lifted to an inner automorphism of \G given by w=exp(2iπ\rm ad I0/m), where I0 is the defining vector of an sl2 subalgebra of \G.The grading is then defined by the operator dm,I0=mλd\over dλ + \rm ad I0 and any grade one regular element Λ from the Heisenberg subalgebra associated to [w] takes the form Λ= (C+ +λC-), where [I0, C-]=-(m-1) C- and C+ is included in an sl2 subalgebra containing I0. The largest eigenvalue of \rm 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 \rm ad I0 eigenvalue is (m-1), then using any grade one regular element from the Heisenberg subalgebra associated to [w] we can construct a KdV system possessing the standard \W-algebra defined by I0 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.