7 research outputs found
The Painleve Property, W Algebras and Toda Field Theories associated with Hyperbolic Kac-Moody Algebras
We show that the Painlev\'{e} test is useful not only for probing
(non-)integrability but also for finding the values of spins of conserved
currents (W currents) in Toda field theories (TFTs). In the case of the TFTs
based on simple Lie algebras the locations of resonances are shown to give
precisely the spins of conserved W currents. We apply this test to TFTs based
on strictly hyperbolic Kac-Moody algebras and show that there exist no
resonances other than that at n=2, which corresponds to the energy-momentum
tensor, indicating their non-integrability. We also check by direct calculation
that there are no spin-3 nor -4 conserved currents for all the hyperbolic TFTs
in agreement with the result of our Painlev\'{e} analysis.Comment: 27 pages, latex, a uuencoded file for a table availabl
On the Imaginary Simple Roots of the Borcherds Algebra
In a recent paper (hep-th/9703084) it was conjectured that the imaginary
simple roots of the Borcherds algebra at level 1 are its only
ones. We here propose an independent test of this conjecture, establishing its
validity for all roots of norm . However, the conjecture fails for
roots of norm -10 and beyond, as we show by computing the simple multiplicities
down to norm -24, which turn out to be remakably small in comparison with the
corresponding multiplicities. Our derivation is based on a modified
denominator formula combining the denominator formulas for and
, and provides an efficient method for determining the imaginary
simple roots. In addition, we compute the multiplicities of all roots
up to height 231, including levels up to and norms -42.Comment: 14 pages, LaTeX2e, packages amsmath, amsfonts, amssymb, amsthm,
xspace, pstricks, longtable; substantially extended, appendix with new
root multiplicities adde
On the fundamental representation of Borcherds algebras with one imaginary simple root
Borcherds algebras represent a new class of Lie algebras which have almost
all the properties that ordinary Kac-Moody algebras have, and the only major
difference is that these generalized Kac-Moody algebras are allowed to have
imaginary simple roots. The simplest nontrivial examples one can think of are
those where one adds ``by hand'' one imaginary simple root to an ordinary
Kac-Moody algebra. We study the fundamental representation of this class of
examples and prove that an irreducible module is given by the full tensor
algebra over some integrable highest weight module of the underlying Kac-Moody
algebra. We also comment on possible realizations of these Lie algebras in
physics as symmetry algebras in quantum field theory.Comment: 8 page