304 research outputs found
Highlights of Symmetry Groups
The concepts of symmetry and symmetry groups are at the heart of several
developments in modern theoretical and mathematical physics. The present paper
is devoted to a number of selected topics within this framework: Euclidean and
rotation groups; the properties of fullerenes in physical chemistry; Galilei,
Lorentz and Poincare groups; conformal transformations and the Laplace
equation; quantum groups and Sklyanin algebras. For example, graphite can be
vaporized by laser irradiation, producing a remarkably stable cluster
consisting of 60 carbon atoms. The corresponding theoretical model considers a
truncated icosahedron, i.e. a polygon with 60 vertices and 32 faces, 12 of
which are pentagonal and 20 hexagonal. The Carbon 60 molecule obtained when a
carbon atom is placed at each vertex of this structure has all valences
satisfied by two single bonds and one double bond. In other words, a structure
in which a pentagon is completely surrounded by hexagons is stable. Thus, a
cage in which all 12 pentagons are completely surrounded by hexagons has
optimum stability. On a more formal side, the exactly solvable models of
quantum and statistical physics can be studied with the help of the quantum
inverse problem method. The problem of enumerating the discrete quantum systems
which can be solved by the quantum inverse problem method reduces to the
problem of enumerating the operator-valued functions that satisfy an equation
involving a fixed solution of the quantum Yang--Baxter equation. Two basic
equations exist which provide a systematic procedure for obtaining completely
integrable lattice approximations to various continuous completely integrable
systems. This analysis leads in turn to the discovery of Sklyanin algebras.Comment: Plain Tex with one figur
Maximal Subgroups of Compact Lie Groups
This report aims at giving a general overview on the classification of the
maximal subgroups of compact Lie groups (not necessarily connected). In the
first part, it is shown that these fall naturally into three types: (1) those
of trivial type, which are simply defined as inverse images of maximal
subgroups of the corresponding component group under the canonical projection
and whose classification constitutes a problem in finite group theory, (2)
those of normal type, whose connected one-component is a normal subgroup, and
(3) those of normalizer type, which are the normalizers of their own connected
one-component. It is also shown how to reduce the classification of maximal
subgroups of the last two types to: (2) the classification of the finite
maximal -invariant subgroups of center-free connected compact simple
Lie groups and (3) the classification of the -primitive subalgebras of
compact simple Lie algebras, where is a subgroup of the corresponding
outer automorphism group. In the second part, we explicitly compute the
normalizers of the primitive subalgebras of the compact classical Lie algebras
(in the corresponding classical groups), thus arriving at the complete
classification of all (non-discrete) maximal subgroups of the compact classical
Lie groups.Comment: 83 pages. Final versio
Spectral transfer morphisms for unipotent affine Hecke algebras
In this paper we will give a complete classification of the spectral transfer
morphisms between the unipotent affine Hecke algebras of the various inner
forms of a given quasi-split absolutely simple algebraic group, defined over a
non-archimidean local field and split over an unramified extension
of . As an application of these results, the results of [O4] on the
spectral correspondences associated with such morphisms and some results of
Ciubotaru, Kato and Kato [CKK] we prove a conjecture of Hiraga, Ichino and
Ikeda [HII] on the formal degrees and adjoint gamma factors for all unipotent
discrete series characters of unramified simple groups of adjoint type defined
over .Comment: 61 pages; We explained the comparison with Lusztig's parameterization
of unipotent representations in more detai
On the structure of axial algebras
Axial algebras are a recently introduced class of non-associative algebra
motivated by applications to groups and vertex-operator algebras. We develop
the structure theory of axial algebras focussing on two major topics: (1)
radical and simplicity; and (2) sum decompositions.Comment: 27 page
Depth and the local Langlands correspondence
Let G be an inner form of a general linear group over a non-archimedean local
field. We prove that the local Langlands correspondence for G preserves depths.
We also show that the local Langlands correspondence for inner forms of special
linear groups preserves the depths of essentially tame Langlands parameters.Comment: The proof of Lemma 3.2 in the first version contained a mistake. In
the second version the analysis of depth for inner forms of SL_n was extended
and paragraph 2.4 was simplifie
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