965 research outputs found
The local functors of points of Supermanifolds
We study the local functor of points (which we call the Weil-Berezin functor)
for smooth supermanifolds, providing a characterization, representability
theorems and applications to differential calculus
Highest weight Harish-Chandra supermodules and their geometric realizations
In this paper we discuss the highest weight -finite
representations of the pair consisting of ,
a real form of a complex basic Lie superalgebra of classical type
(), and the maximal compact subalgebra of
, together with their geometric global realizations. These
representations occur, as in the ordinary setting, in the superspaces of
sections of holomorphic super vector bundles on the associated Hermitian
superspaces .Comment: This article contains of part of the material originally posted as
arXiv:1503.03828 and arXiv:1511.01420. The rest of the material was posted as
arXiv:1801.07181 and will also appear in an enlarged version as subsequent
postin
SUSY structures, representations and Peter-Weyl theorem for
The real compact supergroup is analized from different perspectives
and its representation theory is studied. We prove it is the only (up to
isomorphism) supergroup, which is a real form of
with reduced Lie group , and a link with SUSY structures on is established. We describe a large family of complex semisimple
representations of and we show that any -representation
whose weights are all nonzero is a direct sum of members of our family. We also
compute the matrix elements of the members of this family and we give a proof
of the Peter-Weyl theorem for
Constructing Extremal Compatible Quantum Observables by Means of Two Mutually Unbiased Bases
We describe a particular class of pairs of quantum observables which are extremal in the convex set of all pairs of compatible quantum observables. The pairs in this class are constructed as uniformly noisy versions of two mutually unbiased bases (MUB) with possibly different noise intensities affecting each basis. We show that not all pairs of MUB can be used in this construction, and we provide a criterion for determining those MUB that actually do yield extremal compatible observables. We apply our criterion to all pairs of Fourier conjugate MUB, and we prove that in this case extremality is achieved if and only if the quantum system Hilbert space is odd-dimensional. Remarkably, this fact is no longer true for general non-Fourier conjugate MUB, as we show in an example. Therefore, the presence or the absence of extremality is a concrete geometric manifestation of MUB inequivalence, that already materializes by comparing sets of no more than two bases at a time
HIGHEST WEIGHT HARISH-CHANDRA SUPERMODULES AND THEIR GEOMETRIC REALIZATIONS
In this paper we discuss the highest weight kr-finite representations of the pair (r, kr) consisting of r, a real form of a complex basic Lie superalgebra of classical type ( 60 A(n, n)), and the maximal compact subalgebra kr of r,0, together with their geometric global realizations. These representations occur, as in the ordinary setting, in the superspaces of sections of holomorphic super vector bundles on the associated Hermitian superspaces Gr/Kr
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