100 research outputs found
A survey of Heisenberg categorification via graphical calculus
In this expository paper we present an overview of various graphical
categorifications of the Heisenberg algebra and its Fock space representation.
We begin with a discussion of "weak" categorifications via modules for Hecke
algebras and "geometrizations" in terms of the cohomology of the Hilbert
scheme. We then turn our attention to more recent "strong" categorifications
involving planar diagrammatics and derived categories of coherent sheaves on
Hilbert schemes.Comment: 23 pages; v2: Some typos corrected and other minor improvements made;
v3: Some small errors corrected; v4: Code corrected to fix problem with
missing arrows on some diagram
Towers of graded superalgebras categorify the twisted Heisenberg double
We show that the Grothendieck groups of the categories of finitely-generated
graded supermodules and finitely-generated projective graded supermodules over
a tower of graded superalgebras satisfying certain natural conditions give rise
to twisted Hopf algebras that are twisted dual. Then, using induction and
restriction functors coming from such towers, we obtain a categorification of
the twisted Heisenberg double and its Fock space representation. We show that
towers of wreath product algebras (in particular, the tower of Sergeev
superalgebras) and the tower of nilcoxeter graded superalgebras satisfy our
axioms. In the latter case, one obtains a categorification of the quantum Weyl
algebra.Comment: 29 pages; v2: Minor changes (corrected typos, etc.) throughou
A survey of equivariant map algebras with open problems
This paper presents an overview of the current state of knowledge in the
field of equivariant map algebras and discusses some open problems in this
area.Comment: 18 pages. v2: Minor correction
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