Uniserial Representations of Vec(R) with a Single Casimir Eigenvalue Page: I
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Kuhns, Nehemiah. Uniserial Representations of Vec((JR) with a Single Casimir
Eigenvalue. Doctor of Philosophy (Mathematics), May 2018, 41 pp., 4 numbered
references.
In 1980 Feigin and Fuchs classified the length 2 bounded representations of Vec(R),
the Lie algebra of polynomial vector fields on the line, as a result of their work on the
cohomology of Vec(R). This dissertation is concerned mainly with the uniserial
(completely indecomposable) representations of Vec(R) with a single Casimir eigenvalue
and weights bounded below. Such representations are composed of irreducible
representations with semisimple eo action, bounded weight space dimensions, and weights
bounded below. These are known to be the tensor density modules with lowest weight X,
for any non-zero complex number X, and the trivial module C, with Vec(R) actions wA and
7c, respectively. Our proofs are cohomology arguments involving H1(Vec(R), Hom(V1,V2))
for irreducible representations V1 and V2. These results classify the finite length uniserial
extensions, with a single Casimir eigenvalue, of admissible irreducible Vec(R)
representations with weights bounded below. In almost every case there is at most one
uniserial representation with a given composition series. However, in the case of an odd
length extension with composition series {wi1, , 1, ...W,, 71}, there is a one-parameter
family of extensions. We also give preliminary results on uniserial representations of the
Virasoro Lie algebra.
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Kuhns, Nehemiah. Uniserial Representations of Vec(R) with a Single Casimir Eigenvalue, dissertation, May 2018; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc1157652/m1/2/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .