5,026 research outputs found
Whose Grass Is Greener? Green Marketing: Toward a Uniform Approach for Responsible Environmental Advertising
An axial algebra is a commutative non-associative algebra generated by
primitive idempotents, called axes, whose adjoint action on is semisimple
and multiplication of eigenvectors is controlled by a certain fusion law.
Different fusion laws define different classes of axial algebras.
Axial algebras are inherently related to groups. Namely, when the fusion law
is graded by an abelian group , every axis leads to a subgroup of
automorphisms of . The group generated by all is called the
Miyamoto group of the algebra. We describe a new algorithm for constructing
axial algebras with a given Miyamoto group. A key feature of the algorithm is
the expansion step, which allows us to overcome the -closeness restriction
of Seress's algorithm computing Majorana algebras.
At the end we provide a list of examples for the Monster fusion law, computed
using a MAGMA implementation of our algorithm.Comment: 31 page
Computing âSmallâ 1âHomological Models for Commutative Differential Graded Algebras
We use homological perturbation machinery specific for the algebra category
[13] to give an algorithm for computing the differential structure of a small 1â
homological model for commutative differential graded algebras (briefly, CDGAs).
The complexity of the procedure is studied and a computer package in Mathematica
is described for determining such models.Ministerio de EducaciĂłn y Ciencia PB98â1621âC02â02Junta de AndalucĂa FQMâ014
Testing isomorphism of graded algebras
We present a new algorithm to decide isomorphism between finite graded
algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it
runs in time polynomial in the order of the input algebras. We introduce
heuristics that often dramatically improve the performance of the algorithm and
report on an implementation in Magma
Regular subalgebras and nilpotent orbits of real graded Lie algebras
For a semisimple Lie algebra over the complex numbers, Dynkin (1952)
developed an algorithm to classify the regular semisimple subalgebras, up to
conjugacy by the inner automorphism group. For a graded semisimple Lie algebra
over the complex numbers, Vinberg (1979) showed that a classification of a
certain type of regular subalgebras (called carrier algebras) yields a
classification of the nilpotent orbits in a homogeneous component of that Lie
algebra. Here we consider these problems for (graded) semisimple Lie algebras
over the real numbers. First, we describe an algorithm to classify the regular
semisimple subalgebras of a real semisimple Lie algebra. This also yields an
algorithm for listing, up to conjugacy, the carrier algebras in a real graded
semisimple real algebra. We then discuss what needs to be done to obtain a
classification of the nilpotent orbits from that; such classifications have
applications in differential geometry and theoretical physics. Our algorithms
are implemented in the language of the computer algebra system GAP, using our
package CoReLG; we report on example computations
Computation of Minimal Homogeneous Generating Sets and Minimal Standard Bases for Ideals of Free Algebras
Let \KX =K\langle X_1,\ldots ,X_n\rangle be the free algebra generated by
over a field . It is shown that with respect to any
weighted -gradation attached to \KX, minimal homogeneous
generating sets for finitely generated graded (two-sided) ideals of \KX can
be algorithmically computed, and that if an ungraded (two-sided) ideal of
\KX has a finite Gr\"obner basis \G with respect to a graded monomial
ordering on \KX, then a minimal standard basis for can be computed via
computing a minimal homogeneous generating set of the associated graded ideal
\langle\LH (I)\rangle.Comment: 13 pages. Algorithm1, Algorithm 2, and Algorithm 3 are revise
Multigraded Hilbert Series of noncommutative modules
In this paper, we propose methods for computing the Hilbert series of
multigraded right modules over the free associative algebra. In particular, we
compute such series for noncommutative multigraded algebras. Using results from
the theory of regular languages, we provide conditions when the methods are
effective and hence the sum of the Hilbert series is a rational function.
Moreover, a characterization of finite-dimensional algebras is obtained in
terms of the nilpotency of a key matrix involved in the computations. Using
this result, efficient variants of the methods are also developed for the
computation of Hilbert series of truncated infinite-dimensional algebras whose
(non-truncated) Hilbert series may not be rational functions. We consider some
applications of the computation of multigraded Hilbert series to algebras that
are invariant under the action of the general linear group. In fact, in this
case such series are symmetric functions which can be decomposed in terms of
Schur functions. Finally, we present an efficient and complete implementation
of (standard) graded and multigraded Hilbert series that has been developed in
the kernel of the computer algebra system Singular. A large set of tests
provides a comprehensive experimentation for the proposed algorithms and their
implementations.Comment: 28 pages, to appear in Journal of Algebr
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