6,406 research outputs found
Maximal closed subroot systems of real affine root systems
We completely classify and give explicit descriptions of the maximal closed
subroot systems of real affine root systems. As an application we describe a
procedure to get the classification of all regular subalgebras of affine Kac
Moody algebras in terms of their root systems
Twisted Demazure modules, fusion product decomposition and twisted Q--systems
In this paper, we introduce a family of indecomposable finite-dimensional
graded modules for the twisted current algebras. These modules are indexed by
an -tuple of partitions \bxi=(\xi^{\alpha})_{\alpha\in R^+} satisfying
a natural compatibility condition. We give three equivalent presentations of
these modules and show that for a particular choice of \bxi these modules
become isomorphic to Demazure modules in various levels for the twisted affine
algebras. As a consequence we see that the defining relations of twisted
Demazure modules can be greatly simplified. Furthermore, we investigate the
notion of fusion products for twisted modules, first defined in \cite{FL99} for
untwisted modules, and use the simplified presentation to prove a fusion
product decomposition of twisted Demazure modules. As a consequence we prove
that twisted Demazure modules can be obtained by taking the associated graded
modules of (untwisted) Demazure modules for simply-laced affine algebras.
Furthermore we give a semi-infinite fusion product construction for the
irreducible representations of twisted affine algebras. Finally, we prove that
the twisted -sytem defined in \cite{HKOTT02} extends to a non-canonical
short exact sequence of fusion products of twisted Demazure modules
Borel-de Siebenthal theory for affine reflection systems
We develop a Borel-de Siebenthal theory for affine reflection systems by
classifying their maximal closed subroot systems. Affine reflection systems
(introduced by Loos and Neher) provide a unifying framework for root systems of
finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras,
and extended affine Lie algebras. In the special case of nullity toroidal
Lie algebras, we obtain a one-to-one correspondence between maximal closed
subroot systems with full gradient and triples , where is a
prime number, is a -tuple of integers in the interval and
is a Hermite normal form matrix with determinant . This
generalizes the result of Dyer and Lehrer in the setting of affine Lie
algebras
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