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Highest-Weight Theory for Truncated Current Lie Algebras
Let g denote a Lie algebra over a field of characteristic zero, and let T(g)
denote the tensor product of g with a ring of truncated polynomials. The Lie
algebra T(g) is called a truncated current Lie algebra, or in the special case
when g is finite-dimensional and semisimple, a generalized Takiff algebra. In
this paper a highest-weight theory for T(g) is developed when the underlying
Lie algebra g possesses a triangular decomposition. The principal result is the
reducibility criterion for the Verma modules of T(g) for a wide class of Lie
algebras g, including the symmetrizable Kac-Moody Lie algebras, the Heisenberg
algebra, and the Virasoro algebra. This is achieved through a study of the
Shapovalov form.Comment: 42 pages. An extract from the author's PhD thesis. See also:
http://www.maths.usyd.edu.au/u/benw
Representations of Truncated Current Lie Algebras
Let g denote a Lie algebra, and let T(g) denote the tensor product of g with
a ring of truncated polynomials. The Lie algebra T(g) is called a truncated
current Lie algebra. The highest-weight representation theory of T(g) is
developed, and a reducibility criterion for the Verma modules is described.Comment: 5 pages. A summary of the article 'Highest-Weight Theory for
Truncated Current Lie Algebras' published on the arxi
Imaginary highest-weight representation theory and symmetric functions
Affine Lie algebras admit non-classical highest-weight theories through
alternative partitions of the root system. Although significant inroads have
been made, much of the classical machinery is inapplicable in this broader
context, and some fundamental questions remain unanswered. In particular, the
structure of the reducible objects in non-classical theories has not yet been
fully understood. This question is addressed here for affine sl(2), which has a
unique non-classical highest-weight theory, termed "imaginary". The reducible
Verma modules in the imaginary theory possess an infinite descending series,
with all factors isomorphic to a certain canonically associated module, the
structure of which depends upon the highest weight. If the highest weight is
non-zero, then this factor module is irreducible, and conversely. This paper
examines the degeneracy of the factor module of highest-weight zero. The
intricate structure of this module is understood via a realization in terms of
the symmetric functions. The realization permits the description of a family of
singular (critical) vectors, and the classification of the irreducible
subquotients. The irreducible subquotients are characterized as those modules
with an action given in terms of exponential functions, in the sense of Billig
and Zhao.Comment: 23 page
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