7,141 research outputs found
The invariants of the Clifford groups
The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not
3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an
extraspecial group of order 2^(1+2m) extended by an orthogonal group). This
group and its complex analogue CC_m have arisen in recent years in connection
with the construction of orthogonal spreads, Kerdock sets, packings in
Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs.
In this paper we give a simpler proof of Runge's 1996 result that the space
of invariants for C_m of degree 2k is spanned by the complete weight
enumerators of the codes obtained by tensoring binary self-dual codes of length
2k with the field GF(2^m); these are a basis if m >= k-1. We also give new
constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix
[2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power
of M, and C_m is the automorphism group of this tensor power. Also, if C is a
binary self-dual code not generated by vectors of weight 2, then C_m is
precisely the automorphism group of the complete weight enumerator of the
tensor product of C and GF(2^m). There are analogues of all these results for
the complex group CC_m, with ``doubly-even self-dual code'' instead of
``self-dual code''.Comment: Latex, 24 pages. Many small improvement
Invariants for E_0-semigroups on II_1 factors
We introduce four new cocycle conjugacy invariants for E_0-semigroups on II_1
factors: a coupling index, a dimension for the gauge group, a super product
system and a C*-semiflow. Using noncommutative It\^o integrals we show that the
dimension of the gauge group can be computed from the structure of the additive
cocycles. We do this for the Clifford flows and even Clifford flows on the
hyperfinite II_1 factor, and for the free flows on the free group factor
. In all cases the index is 0, which implies they have trivial
gauge groups. We compute the super product systems for these families and,
using this, we show they have trivial coupling index. Finally, using the
C*-semiflow and the boundary representation of Powers and Alevras, we show that
the families of Clifford flows and even Clifford flows contain infinitely many
mutually non-cocycle-conjugate E_0-semigroups.Comment: 51 page
The Arason invariant of orthogonal involutions of degree 12 and 8, and quaternionic subgroups of the Brauer group
Using the Rost invariant for torsors under Spin groups one may define an
analogue of the Arason invariant for certain hermitian forms and orthogonal
involutions. We calculate this invariant explicitly in various cases, and use
it to associate to every orthogonal involution with trivial discriminant and
trivial Clifford invariant over a central simple algebra of even co-index a
cohomology class of degree 3 with coefficients. This invariant
is the double of any representative of the Arason invariant; it vanishes
when the algebra has degree at most 10, and also when there is a quadratic
extension of the center that simultaneously splits the algebra and makes the
involution hyperbolic. The paper provides a detailed study of both invariants,
with particular attention to the degree 12 case, and to the relation with the
existence of a quadratic splitting field.Comment: A mistake pointed out by A. Sivatski in Section 5.3 has been
corrected in the new version of this preprin
Eta invariants and the hypoelliptic Laplacian
The purpose of this paper is to give a new proof of results of Moscovici and
Stanton on the orbital integrals associated with eta invariants on compact
locally symmetric spaces. Moscovici and Stanton used methods of harmonic
analysis on reductive groups. Here, we combine our approach to orbital
integrals using the hypoelliptic Laplacian, with the introduction of a rotation
on certain Clifford algebras. Probabilistic methods play an important role in
establishing key estimates. In particular, we construct the proper It{\^o}
calculus associated with certain hypoelliptic diffusions
A noncommutative framework for topological insulators
We study topological insulators, regarded as physical systems giving rise to
topological invariants determined by symmetries both linear and anti-linear.
Our perspective is that of noncommutative index theory of operator algebras. In
particular we formulate the index problems using Kasparov theory, both complex
and real. We show that the periodic table of topological insulators and
superconductors can be realised as a real or complex index pairing of a
Kasparov module capturing internal symmetries of the Hamiltonian with a
spectral triple encoding the geometry of the sample's (possibly noncommutative)
Brillouin zone.Comment: 32 pages, final versio
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