268 research outputs found
Quantum affine Cartan matrices, Poincare series of binary polyhedral groups, and reflection representations
We first review some invariant theoretic results about the finite subgroups
of SU(2) in a quick algebraic way by using the McKay correspondence and quantum
affine Cartan matrices. By the way it turns out that some parameters
(a,b,h;p,q,r) that one usually associates with such a group and hence with a
simply-laced Coxeter-Dynkin diagram have a meaningful definition for the
non-simply-laced diagrams, too, and as a byproduct we extend Saito's formula
for the determinant of the Cartan matrix to all cases. Returning to invariant
theory we show that for each irreducible representation i of a binary
tetrahedral, octahedral, or icosahedral group one can find a homomorphism into
a finite complex reflection group whose defining reflection representation
restricts to i.Comment: 19 page
On the Classification of Automorphic Lie Algebras
It is shown that the problem of reduction can be formulated in a uniform way
using the theory of invariants. This provides a powerful tool of analysis and
it opens the road to new applications of these algebras, beyond the context of
integrable systems. Moreover, it is proven that sl2-Automorphic Lie Algebras
associated to the icosahedral group I, the octahedral group O, the tetrahedral
group T, and the dihedral group Dn are isomorphic. The proof is based on
techniques from classical invariant theory and makes use of Clebsch-Gordan
decomposition and transvectants, Molien functions and the trace-form. This
result provides a complete classification of sl2-Automorphic Lie Algebras
associated to finite groups when the group representations are chosen to be the
same and it is a crucial step towards the complete classification of
Automorphic Lie Algebras.Comment: 29 pages, 1 diagram, 9 tables, standard LaTeX2e, submitted for
publicatio
Arbitrarily large families of spaces of the same volume
In any connected non-compact semi-simple Lie group without factors locally
isomorphic to SL_2(R), there can be only finitely many lattices (up to
isomorphism) of a given covolume. We show that there exist arbitrarily large
families of pairwise non-isomorphic arithmetic lattices of the same covolume.
We construct these lattices with the help of Bruhat-Tits theory, using Prasad's
volume formula to control their covolumes.Comment: 9 pages. Syntax corrected; one reference adde
Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups
We study the symmetries of generalized spacetimes and corresponding phase
spaces defined by Jordan algebras of degree three. The generic Jordan family of
formally real Jordan algebras of degree three describe extensions of the
Minkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation,
Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) and
SO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simple
Jordan algebras of degree three correspond to extensions of Minkowskian
spacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra
(2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthal
triple systems defined over these Jordan algebras describe conformally
covariant phase spaces. Following hep-th/0008063, we give a unified geometric
realization of the quasiconformal groups that act on their conformal phase
spaces extended by an extra "cocycle" coordinate. For the generic Jordan family
the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are
given. The minimal unitary representations of the quasiconformal groups F_4(4),
E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our
earlier work hep-th/0409272.Comment: A typo in equation (37) corrected and missing titles of some
references added. Version to be published in JHEP. 38 pages, latex fil
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