39,777 research outputs found
Resolutions for unit groups of orders
We present a general algorithm for constructing a free resolution for unit
groups of orders in semisimple rational algebras. The approach is based on
computing a contractible -complex employing the theory of minimal classes of
quadratic forms and Opgenorth's theory of dual cones. The information from the
complex is then used together with Wall's perturbation lemma to obtain the
resolution
Partially-massless higher-spin algebras and their finite-dimensional truncations
The global symmetry algebras of partially-massless (PM) higher-spin (HS)
fields in (A)dS are studied. The algebras involving PM generators up to
depth are defined as the maximal symmetries of free conformal
scalar field with order wave equation in dimensions. We review
the construction of these algebras by quotienting certain ideals in the
universal enveloping algebra of isometries. We discuss another
description in terms of Howe duality and derive the formula for computing trace
in these algebras. This enables us to explicitly calculate the bilinear form
for this one-parameter family of algebras. In particular, the bilinear form
shows the appearance of additional ideal for any non-negative integer values of
, which coincides with the annihilator of the one-row -box
Young diagram representation of . Hence, the
corresponding finite-dimensional coset algebra spanned by massless and PM
generators is equivalent to the symmetries of this representation.Comment: 22 pages, references added, revised version, accepted to JHE
Computing with Free Algebras, in
Abstract We describe arithmetic computations in terms of operations on some well known free algebras (S1S, S2S and ordered rooted binary trees) while emphasizing the common structure present in all of them when seen as isomorphic with the set of natural numbers. Constructors and deconstructors seen through an initial algebra semantics are generalized to recursively defined functions obeying similar laws. Implementations using GHC's view construct are discussed, based on the free algebra of rooted ordered binary trees. Categories and Subject Descriptors D.3.3 [PROGRAMMING LANGUAGES]: Language Constructs and Features-Data types and structures General Terms Algorithms, Languages, Theory Keywords arithmetic computations with free algebras, generalized constructors, declarative modeling of computational phenomena, bijective Gödel numberings and algebraic datatypes
Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gr{\"o}bner bases
This paper is a sequel to "Computing diagonal form and Jacobson normal form
of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We
present a new fraction-free algorithm for the computation of a diagonal form of
a matrix over a certain non-commutative Euclidean domain over a computable
field with the help of Gr\"obner bases. This algorithm is formulated in a
general constructive framework of non-commutative Ore localizations of
-algebras (OLGAs). We split the computation of a normal form of a matrix
into the diagonalization and the normalization processes. Both of them can be
made fraction-free. For a matrix over an OLGA we provide a diagonalization
algorithm to compute and with fraction-free entries such that
holds and is diagonal. The fraction-free approach gives us more information
on the system of linear functional equations and its solutions, than the
classical setup of an operator algebra with rational functions coefficients. In
particular, one can handle distributional solutions together with, say,
meromorphic ones. We investigate Ore localizations of common operator algebras
over and use them in the unimodularity analysis of transformation
matrices . In turn, this allows to lift the isomorphism of modules over an
OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of
this lifting with the solutions of the original system of equations. Moreover,
we prove some new results concerning normal forms of matrices over non-simple
domains. Our implementation in the computer algebra system {\sc
Singular:Plural} follows the fraction-free strategy and shows impressive
performance, compared with methods which directly use fractions. Since we
experience moderate swell of coefficients and obtain simple transformation
matrices, the method we propose is well suited for solving nontrivial practical
problems.Comment: 25 pages, to appear in Journal of Symbolic Computatio
Affine Lie Algebras in Massive Field Theory and Form-Factors from Vertex Operators
We present a new application of affine Lie algebras to massive quantum field
theory in 2 dimensions, by investigating the limit of the q-deformed
affine symmetry of the sine-Gordon theory, this limit occurring
at the free fermion point. Working in radial quantization leads to a
quasi-chiral factorization of the space of fields. The conserved charges which
generate the affine Lie algebra split into two independent affine algebras on
this factorized space, each with level 1 in the anti-periodic sector, and level
in the periodic sector. The space of fields in the anti-periodic sector can
be organized using level- highest weight representations, if one supplements
the \slh algebra with the usual local integrals of motion. Introducing a
particle-field duality leads to a new way of computing form-factors in radial
quantization. Using the integrals of motion, a momentum space bosonization
involving vertex operators is formulated. Form-factors are computed as vacuum
expectation values in momentum space. (Based on talks given at the Berkeley
Strings 93 conference, May 1993, and the III International Conference on
Mathematical Physics, String Theory, and Quantum Gravity, Alushta, Ukraine,
June 1993.)Comment: 13 pages, CLNS 93/125
- …