371 research outputs found
Dimensions of Imaginary Root Spaces of Hyperbolic Kac--Moody Algebras
We discuss the known results and methods for determining root multiplicities
for hyperbolic Kac--Moody algebras
Bilateral identities of the Rogers-Ramanujan type
We derive by analytic means a number of bilateral identities of the
Rogers-Ramanujan type. Our results include bilateral extensions of the
Rogers-Ramanujan and the G\"ollnitz-Gordon identities, and of related
identities by Ramanujan, Jackson, and Slater. We give corresponding results for
multiseries including multilateral extensions of the Andrews-Gordon identities,
of Bressoud's even modulus identities, and other identities. The here revealed
closed form bilateral and multilateral summations appear to be the very first
of their kind. Given that the classical Rogers-Ramanujan identities have
well-established connections to various areas in mathematics and in physics, it
is natural to expect that the new bilateral and multilateral identities can be
similarly connected to those areas. This is supported by concrete combinatorial
interpretations for a collection of four bilateral companions to the classical
Rogers-Ramanujan identities.Comment: 25 page
On the number of representations providing noiseless subsystems
This paper studies the combinatoric structure of the set of all
representations, up to equivalence, of a finite-dimensional semisimple Lie
algebra. This has intrinsic interest as a previously unsolved problem in
representation theory, and also has applications to the understanding of
quantum decoherence. We prove that for Hilbert spaces of sufficiently high
dimension, decoherence-free subspaces exist for almost all representations of
the error algebra. For decoherence-free subsystems, we plot the function
which is the fraction of all -dimensional quantum systems which
preserve bits of information through DF subsystems, and note that this
function fits an inverse beta distribution. The mathematical tools which arise
include techniques from classical number theory.Comment: 17 pp, 4 figs, accepted for Physical Review
On the Combinatorial Structure of Primitive Vassiliev Invariants, III - A Lower Bound
We prove that the dimension of the space of primitive Vassiliev invariants of
degree n grows - as n tends to infinity - faster than Exp(c Sqrt(n)) for any c
< Pi Sqrt (2/3).
The proof relies on the use of the weight systems coming from the Lie algebra
gl(N). In fact, we show that our bound is - up to multiplication with a
rational function in n - the best possible that one can get with gl(N)-weight
systems.Comment: 11 pages, 12 figure
Ramanujan's "Lost Notebook" and the Virasoro Algebra
By using the theory of vertex operator algebras, we gave a new proof of the
famous Ramanujan's modulus 5 modular equation from his "Lost Notebook" (p.139
in \cite{R}). Furthermore, we obtained an infinite list of -identities for
all odd moduli; thus, we generalized the result of Ramanujan.Comment: To appear in Comm. Math. Phy
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