11,327 research outputs found
On sequences associated to the invariant theory of rank two simple Lie algebras
We study two families of sequences, listed in the On-Line Encyclopedia of Integer Sequences (OEIS), which are associated to invariant theory of Lie algebras. For the first family, we prove combinatorially that the sequences A059710 and A108307 are related by a binomial transform. Based on this, we present two independent proofs of a recurrence equation for A059710, which was conjectured by Mihailovs. Besides, we also give a direct proof of Mihailovs' conjecture by the method of algebraic residues. As a consequence, closed formulae for the generating function of sequence A059710 are obtained in terms of classical Gaussian hypergeometric functions. Moreover, we show that sequences in the second family are also related by binomial transforms
Invariant Differential Operators for Non-Compact Lie Groups: Parabolic Subalgebras
In the present paper we start the systematic explicit construction of
invariant differential operators by giving explicit description of one of the
main ingredients - the cuspidal parabolic subalgebras. We explicate also the
maximal parabolic subalgebras, since these are also important even when they
are not cuspidal. Our approach is easily generalised to the supersymmetric and
quantum group settings and is necessary for applications to string theory and
integrable models.Comment: 44 pages; V2: important addition in Section 3 and misprints
corrected; more corrections in Section 3; v3-v6: various corrections; v7:
corrections in (11.7),(11.9),(11.11), and correspondingly in the Appendix;
v8: added dimensions of N-factors where missing; v9: added missing case in
11.37; v10: corrected misprint in 11.17; v11: added missing case in 11.37;
v12: typos corrected in (11.7),(11.9
On certain modules of covariants in exterior algebras
We study the structure of the space of covariants for a
certain class of infinitesimal symmetric spaces
such that the space of invariants is an exterior algebra with
. We prove that they are free modules over
the subalgebra of rank . In addition we
will give an explicit basis of . As particular cases we will recover same
classical results. In fact we will describe the structure of , the space of the equivariant matrix
valued alternating multilinear maps on the space of (skew-symmetric or
symmetric with respect to a specific involution) matrices, where is the
symplectic group or the odd orthogonal group. Furthermore we prove new
polynomial trace identities.Comment: Title changed. Results have been generalised to other infinitesimal
symmetric space
Small Orbits
We study both the "large" and "small" U-duality charge orbits of extremal
black holes appearing in D = 5 and D = 4 Maxwell-Einstein supergravity theories
with symmetric scalar manifolds. We exploit a formalism based on cubic Jordan
algebras and their associated Freudenthal triple systems, in order to derive
the minimal charge representatives, their stabilizers and the associated
"moduli spaces". After recalling N = 8 maximal supergravity, we consider N = 2
and N = 4 theories coupled to an arbitrary number of vector multiplets, as well
as N = 2 magic, STU, ST^2 and T^3 models. While the STU model may be considered
as part of the general N = 2 sequence, albeit with an additional triality
symmetry, the ST^2 and T^3 models demand a separate treatment, since their
representative Jordan algebras are Euclidean or only admit non-zero elements of
rank 3, respectively. Finally, we also consider minimally coupled N = 2, matter
coupled N = 3, and "pure" N = 5 theories.Comment: 40 pages, 9 tables. References added. Expanded comments added to
sections III. C. 1. and III. F.
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