886 research outputs found
Regular subalgebras and nilpotent orbits of real graded Lie algebras
For a semisimple Lie algebra over the complex numbers, Dynkin (1952)
developed an algorithm to classify the regular semisimple subalgebras, up to
conjugacy by the inner automorphism group. For a graded semisimple Lie algebra
over the complex numbers, Vinberg (1979) showed that a classification of a
certain type of regular subalgebras (called carrier algebras) yields a
classification of the nilpotent orbits in a homogeneous component of that Lie
algebra. Here we consider these problems for (graded) semisimple Lie algebras
over the real numbers. First, we describe an algorithm to classify the regular
semisimple subalgebras of a real semisimple Lie algebra. This also yields an
algorithm for listing, up to conjugacy, the carrier algebras in a real graded
semisimple real algebra. We then discuss what needs to be done to obtain a
classification of the nilpotent orbits from that; such classifications have
applications in differential geometry and theoretical physics. Our algorithms
are implemented in the language of the computer algebra system GAP, using our
package CoReLG; we report on example computations
Iterative character constructions for algebra groups
We construct a family of orthogonal characters of an algebra group which
decompose the supercharacters defined by Diaconis and Isaacs. Like
supercharacters, these characters are given by nonnegative integer linear
combinations of Kirillov functions and are induced from linear supercharacters
of certain algebra subgroups. We derive a formula for these characters and give
a condition for their irreducibility; generalizing a theorem of Otto, we also
show that each such character has the same number of Kirillov functions and
irreducible characters as constituents. In proving these results, we observe as
an application how a recent computation by Evseev implies that every
irreducible character of the unitriangular group \UT_n(q) of unipotent
upper triangular matrices over a finite field with elements is
a Kirillov function if and only if . As a further application, we
discuss some more general conditions showing that Kirillov functions are
characters, and describe some results related to counting the irreducible
constituents of supercharacters.Comment: 22 page
Naive boundary strata and nilpotent orbits
We study certain real Lie-group orbits in the compact duals of Mumford-Tate
domains, verifying a prediction made in [Green, Griffiths, Kerr; Mumford-Tate
domains: their geometry and arithmetic] and determining which orbits contain a
limit point of some period map. A variety of examples are worked out for the
groups SU(2,1), Sp_4, and G_2.Comment: 57 pages, 34 figure
Integrability of Supergravity Black Holes and New Tensor Classifiers of Regular and Nilpotent Orbits
In this paper we apply in a systematic way a previously developed integration
algorithm of the relevant Lax equation to the construction of spherical
symmetric, asymptotically flat black hole solutions of N=2 supergravities with
symmetric Special Geometry. Our main goal is the classification of these
black-holes according to the H*-orbits in which the space of possible Lax
operators decomposes, H* being the isotropy group of scalar manifold
originating from time-like dimensional reduction of supergravity from D=4 to
D=3 dimensions. The main result of our investigation is the construction of
three universal tensors, extracted from quadratic and quartic powers of the Lax
operator, that are capable of classifying both regular and nilpotent H* orbits
of Lax operators. Our tensor based classification is compared, in the case of
the simple one-field model S^3, to the algebraic classification of nilpotent
orbits and it is shown to provide a simple and practical discriminating method.
We present a detailed analysis of the S^3 model and its black hole solutions,
discussing the Liouville integrability of the corresponding dynamical system.
By means of the Kostant-representation of a generic Lie algebra element, we
were able to develop an algorithm which produces the necessary number of
hamiltonians in involution required by Liouville integrability of generic
orbits. The degenerate orbits correspond to extremal black-holes and are
nilpotent. We analyze these orbits in some detail working out different
representatives thereof and showing that the relation between H* orbits and
critical points of the geodesic potential is not one-to-one. Finally we present
the conjecture that our newly identified tensor classifiers are universal and
able to label all regular and nilpotent orbits in all homogeneous symmetric
Special Geometries.Comment: Analysis of nilpotent orbits in terms of tensor classifiers in
section 8.1 corrected. Table 1 corrected. Discussion in section 11 extende
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