331 research outputs found
Poincare-Birkhoff-Witt Theorems
We sample some Poincare-Birkhoff-Witt theorems appearing in mathematics.
Along the way, we compare modern techniques used to establish such results, for
example, the Composition-Diamond Lemma, Groebner basis theory, and the
homological approaches of Braverman and Gaitsgory and of Polishchuk and
Positselski. We discuss several contexts for PBW theorems and their
applications, such as Drinfeld-Jimbo quantum groups, graded Hecke algebras, and
symplectic reflection and related algebras.Comment: 30 pages; survey article to appear in Mathematical Sciences Research
Institute Proceeding
Conformal Designs based on Vertex Operator Algebras
We introduce the notion of a conformal design based on a vertex operator
algebra. This notation is a natural analog of the notion of block designs or
spherical designs when the elements of the design are based on self-orthogonal
binary codes or integral lattices, respectively. It is shown that the subspaces
of fixed degree of an extremal self-dual vertex operator algebra form conformal
11-, 7-, or 3-designs, generalizing similar results of Assmus-Mattson and
Venkov for extremal doubly-even codes and extremal even lattices. Other
examples are coming from group actions on vertex operator algebras, the case
studied first by Matsuo. The classification of conformal 6- and 8-designs is
investigated. Again, our results are analogous to similar results for codes and
lattices.Comment: 35 pages with 1 table, LaTe
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