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Algebraic Supergroups with Lie Superalgebras of Classical Type
No full textWe show that every connected affine algebraic supergroup defined over a field k, with diagonalizable maximal torus and whose tangent Lie superalgebra is a k-form of a complex simple Lie superalgebra of classical type is a Chevalley supergroup, as it is defined and constructed explicitly in [R. Fioresi, F. Gavarini, Chevalley Supergroups, Memoirs of the Amer. Math. Soc. 215 (2012), no. 1014]
Deep Learning and Geometric Deep Learning: an introduction for mathematicians and physicists.
No full textIn this expository paper, we want to give a brief introduction, with few key references for further reading, to the inner functioning of the new and successful algorithms of Deep Learning and Geometric Deep Learning with a focus on Graph Neural Networks. We go over the key ingredients for these algorithms: the score and loss function and we explain the main steps for the training of a model. We do not aim to give a complete and exhaustive treatment, but we isolate few concepts to give a fast introduction to the subject. We provide some appendices to complement our treatment discussing Kullback–Leibler divergence, regression, Multi-layer Perceptrons and the Universal Approximation theorem
Super Distributions, Analytic and Algebraic Super Harish-Chandra pairs
No full textWe extend the theory of super Harish-Chandra pairs, originally developed by Kostant
and Koszul for smooth Lie supergroups, to algebraic supergroups over a field of
characteristic zero. We also review the corresponding complex analytic theory
and we give a characterization of the action of an algebraic (resp. complex
analytic) super Harish-Chandra pair on a supervariety (resp. complex analytic
supermanifold)
Chevalley supergroups
Get PDFIn the framework of algebraic supergeometry, we give a construction
of the scheme-theoretic supergeometric analogue of Chevalley groups, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. This provides a unified approach to most of the algebraic supergroups considered so far in literature, and an effective method to construct new ones. As an intermediate step, we prove an existence theorem for Chevalley bases of simple classical
Lie superalgebras and a PBW-like theorem for their associated
Kostant superalgebras
HIGHEST WEIGHT HARISH-CHANDRA SUPERMODULES AND THEIR GEOMETRIC REALIZATIONS
Get PDFIn 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 ( ≠ 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
Superalgebraic methods in the classical theory of representations. Capelli's identity, the Koszul map and the center of the enveloping algebra U(gl(n))
No full textIn this paper we essentially deal with Capelli’s identity and with the center of the enveloping algebra of the general linear Lie algebra gl(n).
By way of elementary motivation, we show that the proof of Capelli’s identity can be reduced to a straightforward computation just by using a touch of superalgebraic notions.
This point of view is extended to the study of the enveloping algebra U(gl(n)).
The notions of determinantal and permanental Capelli bitableaux provide two relevant classes of bases that arise from Straightening Laws.
In the final section, we submit new results on the center of U(gl(n)). These results - which cannot even be expressed
without appealing to the superalgebraic notation - allows a variety of classical and recent results to be almost trivially
proved and be put under one roof
The ACAT Project
No full textThis is a presentation of the ACAT Project of the European Science Foundation, gathering 13 national teams, active in Applied and Computational Algebraic Topology
The local functors of points of supermanifolds
No full textAbstractWe 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
Principio di Entropia, Sistemi Simmetrici Iperbolici e Termodinamica Estesa
No full textSi presenta la problematica della termodinamica dei processi irreversibili discutendo in particolare gli approcci della Meccanica dei Continui e della Teoria Cinetica. Viene evidenziato il duplice ruolo del principio di entropia come criterio di selezione di equazioni costitutive fisicamente valide per soluzioni classiche e criterio di selezione dei processi ammissibili per soluzioni deboli. Infine viene brevemente pre- sentata la Termodinamica Estesa e le sue intime relazioni con i sistemi simmetrici iperbolici di leggi di bilancio
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