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D.15.1 fpadim_lib
- Library:
- fpadim.lib
- Purpose:
- Algorithms for quotient algebras in the letterplace case
- Authors:
- Grischa Studzinski, grischa.studzinski@rwth-aachen.de
Support: Joint projects LE 2697/2-1 and KR 1907/3-1 of the Priority Programme SPP 1489:
'Algorithmische und Experimentelle Methoden in Algebra, Geometrie und Zahlentheorie'
of the German DFG
- Overview:
- Given the free algebra A = K<x_1,...,x_n> and a (finite) Groebner basis
GB = {g_1,..,g_w}, one is interested in the K-dimension and in the
explicit K-basis of A/<GB>.
Therefore one is interested in the following data:
- the Ufnarovskij graph induced by GB
- the mistletoes of A/<GB>
- the K-dimension of A/<GB>
- the Hilbert series of A/<GB>
The Ufnarovskij graph is used to determine whether A/<GB> has finite
K-dimension. One has to check if the graph contains cycles.
For the whole theory we refer to [ufna]. Given a
reduced set of monomials GB one can define the basis tree, whose vertex
set V consists of all normal monomials w.r.t. GB. For every two
monomials m_1, m_2 in V there is a direct edge from m_1 to m_2, if and
only if there exists x_k in {x_1,..,x_n}, such that m_1*x_k = m_2. The
set M = {m in V | there is no edge from m to another monomial in V} is
called the set of mistletoes. As one can easily see it consists of
the endpoints of the graph. Since there is a unique path to every
monomial in V the whole graph can be described only from the knowledge
of the mistletoes. Note that V corresponds to a basis of A/<GB>, so
knowing the mistletoes we know a K-basis. For more details see
[studzins]. This package uses the Letterplace format introduced by
[lls]. The algebra can either be represented as a Letterplace ring or
via integer vectors: Every variable will only be represented by its
number, so variable one is represented as 1, variable two as 2 and so
on. The monomial x_1*x_3*x_2 for example will be stored as (1,3,2).
Multiplication is concatenation. Note that there is no algorithm for
computing the normal form yet, but for our case it is not needed.
References:
[ufna] Ufnarovskij: Combinatorical and asymptotic methods in algebra, 1990
[lls] Levandovskyy, La Scala: Letterplace ideals and non-commutative
Groebner bases, 2009
[studzins] Studzinski: Dimension computations in non-commutative,
associative algebras, Diploma thesis, RWTH Aachen, 2010
Assumptions:
- basering is always a Letterplace ring
- all intvecs correspond to Letterplace monomials
- if you specify a different degree bound d,
d <= attrib(basering,uptodeg) holds
In the procedures below, 'iv' stands for intvec representation
and 'lp' for the letterplace representation of monomials
Procedures:
See also:
freegb_lib.
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