283 research outputs found
Black Box White Arrow
The present paper proposes a new and systematic approach to the so-called
black box group methods in computational group theory. Instead of a single
black box, we consider categories of black boxes and their morphisms. This
makes new classes of black box problems accessible. For example, we can enrich
black box groups by actions of outer automorphisms.
As an example of application of this technique, we construct Frobenius maps
on black box groups of untwisted Lie type in odd characteristic (Section 6) and
inverse-transpose automorphisms on black box groups encrypting .
One of the advantages of our approach is that it allows us to work in black
box groups over finite fields of big characteristic. Another advantage is
explanatory power of our methods; as an example, we explain Kantor's and
Kassabov's construction of an involution in black box groups encrypting .
Due to the nature of our work we also have to discuss a few methodological
issues of the black box group theory.
The paper is further development of our text "Fifty shades of black"
[arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black
box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248
Positive trace polynomials and the universal Procesi-Schacher conjecture
Positivstellensatz is a fundamental result in real algebraic geometry
providing algebraic certificates for positivity of polynomials on semialgebraic
sets. In this article Positivstellens\"atze for trace polynomials positive on
semialgebraic sets of matrices are provided. A Krivine-Stengle-type
Positivstellensatz is proved characterizing trace polynomials nonnegative on a
general semialgebraic set using weighted sums of hermitian squares with
denominators. The weights in these certificates are obtained from generators of
and traces of hermitian squares. For compact semialgebraic sets
Schm\"udgen- and Putinar-type Positivstellens\"atze are obtained: every trace
polynomial positive on has a sum of hermitian squares decomposition with
weights and without denominators. The methods employed are inspired by
invariant theory, classical real algebraic geometry and functional analysis.
Procesi and Schacher in 1976 developed a theory of orderings and positivity
on central simple algebras with involution and posed a Hilbert's 17th problem
for a universal central simple algebra of degree : is every totally positive
element a sum of hermitian squares? They gave an affirmative answer for .
In this paper a negative answer for is presented. Consequently, including
traces of hermitian squares as weights in the Positivstellens\"atze is
indispensable
Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories
Kac-Moody groups over have been conjectured to occur as
symmetry groups of supergravities in dimensions less than 3, and their integer
forms are conjecturally U-duality groups. Mathematical
descriptions of , due to Tits, are functorial and not amenable
to computation or applications. We construct Kac-Moody groups over
and using an analog of Chevalley's constructions in finite
dimensions and Garland's constructions in the affine case. We extend a
construction of Eisenstein series on finite dimensional semisimple algebraic
groups using representation theory, which appeared in the context of
superstring theory, to general Kac-Moody groups. This coincides with a
generalization of Garland's Eisenstein series on affine Kac-Moody groups to
general Kac-Moody groups and includes Eisenstein series on and
. For finite dimensional groups, Eisenstein series encode the quantum
corrections in string theory and supergravity theories. Their Kac-Moody analogs
will likely also play an important part in string theory, though their roles
are not yet understood
Pointed Hopf algebra (co)actions on rational functions
This article studies the construction of Hopf algebras acting on a given
algebra in terms of algebra morphisms . The approach is particularly suited for controlling whether
these actions restrict to a given subalgebra of , whether is
pointed, and whether these actions are compatible with a given -structure on
. The theory is applied to the field of rational functions
containing the coordinate ring of the cusp. An explicit example
is described in detail and shown to define a quantum homogeneous space
structure on the cusp, which, unlike the previously known one, extends from
regular to rational functions.Comment: 34 page
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