17,849 research outputs found
Moufang sets and structurable division algebras
A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group.
It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. We extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, we show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field k of characteristic different from 2 and 3 arises from a structurable division algebra.
We also obtain explicit formulas for the root groups, the tau-map and the Hua maps of these Moufang sets. This is particularly useful for the Moufang sets arising from exceptional linear algebraic groups
Singular vectors for the algebras
In this paper, we use free field realisations of the A-type principal, or
Casimir, algebras to derive explicit formulae for singular vectors in
Fock modules. These singular vectors are constructed by applying screening
operators to Fock module highest weight vectors. The action of the screening
operators is then explicitly evaluated in terms of Jack symmetric functions and
their skew analogues. The resulting formulae depend on sequences of pairs of
integers that completely determine the Fock module as well as the Jack
symmetric functions.Comment: 18 page
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
We present new deterministic algorithms for several cases of the maximum rank
matrix completion problem (for short matrix completion), i.e. the problem of
assigning values to the variables in a given symbolic matrix as to maximize the
resulting matrix rank. Matrix completion belongs to the fundamental problems in
computational complexity with numerous important algorithmic applications,
among others, in computing dynamic transitive closures or multicast network
codings (Harvey et al SODA 2005, Harvey et al SODA 2006).
We design efficient deterministic algorithms for common generalizations of
the results of Lovasz and Geelen on this problem by allowing linear functions
in the entries of the input matrix such that the submatrices corresponding to
each variable have rank one. We present also a deterministic polynomial time
algorithm for finding the minimal number of generators of a given module
structure given by matrices. We establish further several hardness results
related to matrix algebras and modules. As a result we connect the classical
problem of polynomial identity testing with checking surjectivity (or
injectivity) between two given modules. One of the elements of our algorithm is
a construction of a greedy algorithm for finding a maximum rank element in the
more general setting of the problem. The proof methods used in this paper could
be also of independent interest.Comment: 14 pages, preliminar
Rings of small rank over a Dedekind domain and their ideals
In 2001, M. Bhargava stunned the mathematical world by extending Gauss's
200-year-old group law on integral binary quadratic forms, now familiar as the
ideal class group of a quadratic ring, to yield group laws on a vast assortment
of analogous objects. His method yields parametrizations of rings of degree up
to 5 over the integers, as well as aspects of their ideal structure, and can be
employed to yield statistical information about such rings and the associated
number fields.
In this paper, we extend a selection of Bhargava's most striking
parametrizations to cases where the base ring is not Z but an arbitrary
Dedekind domain R. We find that, once the ideal classes of R are properly
included, we readily get bijections parametrizing quadratic, cubic, and quartic
rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss
composition for which Bhargava is famous. We expect that our results will shed
light on the analytic distribution of extensions of degree up to 4 of a fixed
number field and their ideal structure.Comment: 39 pages, 1 figure. Harvard College senior thesis, edite
An analogue of the Magnus problem for associative algebras
We prove an analogue of the Magnus theorem for associative algebras without
unity over arbitrary fields. Namely, if an algebra is given by n+k generators
and k relations and has an n-element system of generators, then this algebra is
a free algebra of rank n
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