137,853 research outputs found
The lamplighter group of rank two generated by a bireversible automaton
We construct a 4-state 2-letter bireversible automaton generating the
lamplighter group of rank two. The action of the
generators on the boundary of the tree can be induced by the affine
transformations on the ring of formal power series over
.Comment: 18 pages, 2 figure
Lie Algebras and Growth in Branch Groups
We compute the structure of the Lie algebras associated to two examples of
branch groups, and show that one has finite width while the other, the
``Gupta-Sidki group'', has unbounded width. This answers a question by Sidki.
More precisely, the Lie algebra of the Gupta-Sidki group has Gelfand-Kirillov
dimension .
We then draw a general result relating the growth of a branch group, of its
Lie algebra, of its graded group ring, and of a natural homogeneous space we
call "parabolic space", namely the quotient of the group by the stabilizer of
an infinite ray. The growth of the group is bounded from below by the growth of
its graded group ring, which connects to the growth of the Lie algebra by a
product-sum formula, and the growth of the parabolic space is bounded from
below by the growth of the Lie algebra.
Finally we use this information to explicitly describe the normal subgroups
of the "Grigorchuk group". All normal subgroups are characteristic, and the
number of normal subgroups of index is odd and is asymptotically
The tensor product in the theory of Frobenius manifolds
We introduce the operation of forming the tensor product in the theory of
analytic Frobenius manifolds. Building on the results for formal Frobenius
manifolds which we extend to the additional structures of Euler fields and flat
identities, we prove that the tensor product of pointed germs of Frobenius
manifolds exists. Furthermore, we define the notion of a tensor product diagram
of Frobenius manifolds with factorizable flat identity and prove the existence
such a diagram and hence a tensor product Frobenius manifold. These diagrams
and manifolds are unique up to equivalence. Finally, we derive the special
initial conditions for a tensor product of semi--simple Frobenius manifolds in
terms of the special initial conditions of the factors.Comment: 41 pages, amslatex, uses xy-te
Irregular singularities in Liouville theory
Motivated by problems arising in the study of N=2 supersymmetric gauge
theories we introduce and study irregular singularities in two-dimensional
conformal field theory, here Liouville theory. Irregular singularities are
associated to representations of the Virasoro algebra in which a subset of the
annihilation part of the algebra act diagonally. In this paper we define
natural bases for the space of conformal blocks in the presence of irregular
singularities, describe how to calculate their series expansions, and how such
conformal blocks can be constructed by some delicate limiting procedure from
ordinary conformal blocks. This leads us to a proposal for the structure
functions appearing in the decomposition of physical correlation functions with
irregular singularities into conformal blocks. Taken together, we get a precise
prediction for the partition functions of some Argyres-Douglas type theories on
the four-sphere.Comment: 84 pages, 6 figure
The realization of input-output maps using bialgebras
The theory of bialgebras is used to prove a state space realization theorem for input/output maps of dynamical systems. This approach allows for the consideration of the classical results of Fliess and more recent results on realizations involving families of trees. Two examples of applications of the theorum are given
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