13,173 research outputs found
Homology of Gaussian groups
We describe new combinatorial methods for constructing an explicit free
resolution of Z by ZG-modules when G is a group of fractions of a monoid where
enough least common multiples exist (``locally Gaussian monoid''), and,
therefore, for computing the homology of G. Our constructions apply in
particular to all Artin groups of finite Coxeter type, so, as a corollary, they
give new ways of computing the homology of these groups
The structure of the Kac-Wang-Yan algebra
The Lie algebra of regular differential operators on the circle
has a universal central extension . The invariant subalgebra
under an involution preserving the principal gradation
was introduced by Kac, Wang, and Yan. The vacuum -module
with central charge , and its irreducible quotient
, possess vertex algebra structures, and has a
nontrivial structure if and only if . We show that
for each integer , and are
-algebras of types and
, respectively. These results are formal
consequences of Weyl's first and second fundamental theorems of invariant
theory for the orthogonal group and the symplectic group
, respectively. Based on Sergeev's theorems on the invariant
theory of we conjecture that is of
type , and we prove this for . As an
application, we show that invariant subalgebras of -systems and
free fermion algebras under arbitrary reductive group actions are strongly
finitely generated.Comment: Final versio
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