80 research outputs found
Experimental evidence for the occurrence of E8 in nature and the radii of the Gosset circles
A recent experimental discovery involving the spin structure of electrons in
a cold one-dimensional magnet points to a validation of a Zamolodchikov model
involving the exceptional Lie group . The model predicts 8 particles and
predicts the ratio of their masses. I.e., the vertices of the 8-dimensional
Gosset polytope identifies with the 240 roots of . Under the 2-D (Peter
McMullen) projection of the polytope, the image of the vertices are arranged in
8 concentric circles, here referred to as the Gosset circles. The Gosset
circles are understood to correspond to the 8 masses in the model, and it is
understood that the ratio of their radii is the same as the ratio of the
corres-ponding conjectural masses. A ratio of the two smallest circles (read 2
smallest masses) is the golden number. The conjectures have been now validated
experimentally, at least for the first five masses. The McMullen projection
generalizes to any complex simple Lie algebra whose rank is greater than 1. The
Gosset circles also generalize, using orbits of the Coxeter element. Using
results in a 1959 paper of mine, I found some time ago a very easily defined
operator whose spectrum is exactly the squares of the radii of these
generalized Gosset circles. As a confirmation, in the case, using only
the eigenvalues of a suitable multiple of , Vogan computed the ratio of the
. Happily these agree with the corresponding ratio of the Zamolodchikov
masses. The operator is written as a sum of rank 1 operators,
parameterized by the points in the extended Dynkin diagram. Involved in this
expansion are the coefficients of the highest root. Recalling the McKay
correspondence, in the case, the , together with 1, are the
dimensions of the irreducible representations of the binary icosahedral group.Comment: 19 pAGES, PLAIN TE
Center of U(n), Cascade of Orthogonal Roots, and a Construction of Lipsman-Wolf
Let be a complex simply-connected semisimple Lie group and let
\g=\hbox{\rm Lie}\,G. Let \g = \n_- +\hh + \n be a triangular decomposition
of \g. One readily has that \hbox{\rm Cent}\,U(\n) is isomorphic to the
ring S(\n)^{\n} of symmetric invariants. Using the cascade of
strongly orthogonal roots, some time ago we proved (see [K]) that S(\n)^{\n}
is a polynomial ring where is the cardinality of
. The authors in [LW] introduce a very nice representation-theoretic
method for the construction of certain elements in S(\n)^{\n}. A key lemma in
[LW] is incorrect but the idea is in fact valid. In our paper here we modify
the construction so as to yield these elements in S(\n)^{\n} and use the [LW]
result to prove a theorem of Tony Joseph.Comment: 10 pages, plain.tex; key words: cascade of orthogonal roots, Borel
subgroups, nilpotent coadjoint action, dedicated to Joe Wolf; msc keywords:
Representation theory, invariant theor
Minimal coadjoint orbits and symplectic induction
Let be an integral symplectic manifold and let be a
quantum line bundle, with connection, over having as curvature.
With this data one can define an induced symplectic manifold where . It is
then shown that prequantization on becomes classical Poisson bracket on
. We consider the possibility that if is the coadjoint
orbit of a Lie group then is the coadjoint orbit of some
larger Lie group . We show that this is the case if is a non-compact
simple Lie group with a finite center and is the maximal compact subgroup
of . The coadjoint orbit arises (Borel-Weil) from the action of on
\p where \g= \k +\p is a Cartan decomposition. Using the Kostant-Sekiguchi
correspondence and a diffeomorphism result of M. Vergne we establish a
symplectic isomorphism where is a non-zero minimal "nilpotent" coadjoint orbit of
. This is applied to show that the split forms of the 5 exceptional Lie
groups arise symplectically from the symplectic induction of coadjoint orbits
of certain classical groups.Comment: 38 pages, plain te
A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem
We give a branching law for subgroups fixed by an involution. As an
application we give a generalization of the Cartan-Helgason theorem and a
noncompact analogue of the Borel-Weil theorem.Comment: 60 pages, plain te
The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group
Let be a semisimple Lie group and let \g =\n_- +\hh +\n be a triangular
decomposition of \g= \hbox{Lie}\,G. Let \b =\hh +\n and let be Lie
subgroups of corresponding respectively to \hh,\n and \b. We may
identify \n_- with the dual space to \n. The coadjoint action of on
\n_- extends to an action of on \n_-. There exists a unique nonempty
Zariski open orbit of on \n_-. Any -orbit in is a maximal
coadjoint orbit of in \n_-. The cascade of orthogonal roots defines a
cross-section \r_-^{\times} of the set of such orbits leading to a
decomposition X = N/R\times \r_-^{\times}. This decomposition, among other
things, establishes the structure of S(\n)^{\n} as a polynomial ring
generated by the prime polynomials of -weight vectors in S(\n)^{\n}. It
also leads tothe multiplicity 1 of weights in S(\n)^{\n}.Comment: 24 pages, plain te
Cent U(n) and a construction of Lipsman-Wolf
Let be a complex simply-connected semisimple Lie group and let \g=
\hbox{\rm Lie}\,G. Let \g = \n_- +\hh + \n be a triangular decomposition of
\g. The authors in [LW] introduce a very nice representation theory idea for
the construction of certain elements in . A key lemma in
[LW] is incorrect but the idea is in fact valid. In our paper here we modify
the construction so as to yield the desired elements in \hbox{\rm
cent}\,U(\n).Comment: 14 pages in plain te
Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra
If is a complex simple Lie algebra, and does not exceed the
dual Coxeter number of , then the k coefficient of the power of the Euler product may be given by the dimension of a subspace
of defined by all abelian subalgebras of of
dimension . This has implications for all the coefficients of all the powers
of the Euler product. Involved in the main results are Dale Peterson's
theorem on the number of abelian ideals in a Borel subalgebra of
, an element of type and my heat kernel formulation of
Macdonald's -function theorem, a set of special highest
weights parameterized by all the alcoves in a Weyl chamber (generalizing Young
diagrams of null -core when ), and the homology
and cohomology of the nil radical of the standard maximal parabolic subalgebra
of the affine Kac-Moody Lie algebra.Comment: 44 pages, plain te
The Coxeter element and the branching law for the finite subgroups of SU(2)
Let be a finite subgroup of SU(2) and let be the unitary dual of . The unitary dual of
SU(2) may be written where . For
and let be the multiplicity of in
. Then we collect this branching data in the formal power series,
. One shows that there exists a
polynomial and known positive integers (independent of ) such
that . The problem is the determination
of the polynomial . If is such that is the trivial
representation, then it is classical that for a known integer
. The problem reduces to case where is nontrivial. The McKay
correspondence associates to a complex simple Lie algebra \g of type
A-D-E. We explicitly determine for using the orbits of
a Coxeter element on the set of roots of . Mysteriously the
polynomial has arisen in a completely different context in some papers
of Lusztig. Also Rossmann has recently shown that the polynomial
yields the character of .Comment: 13 pages, plain.te
Generalized Amitsur-Levitski theorem and equations for sheets in a reductive complex Lie algebra
I connect an old result of mine on a Lie algebra generalization of the
Amitsur-Levitski theorem with equations for sheets in a reductive Lie algebra
and with recent results of Kostant-Wallach on the variety of singular elements
in a reductive Lie algebra.Comment: 17 pages, key words: Amitsur-Levitski theorem, invariant theory,
polynomial identities, sheets, abelian ideals and solvable Lie algebras,
representation theory, enveloping algebras, nilpotent cone, Lie algebra
cohomolog
Dirac Cohomology for the Cubic Dirac Operator
Using the cubic Dirac operator and an extended notion of Dirac cohomology we
generalize results of Huang-Pandzic, which appeared in JAMS (Sept. 6, 2001
electronic) on a conjecture of D. VoganComment: 35 pages, plain te
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