781 research outputs found
Filtrations and Distortion in Infinite-Dimensional Algebras
A tame filtration of an algebra is defined by the growth of its terms, which
has to be majorated by an exponential function. A particular case is the degree
filtration used in the definition of the growth of finitely generated algebras.
The notion of tame filtration is useful in the study of possible distortion of
degrees of elements when one algebra is embedded as a subalgebra in another. A
geometric analogue is the distortion of the (Riemannian) metric of a (Lie)
subgroup when compared to the metric induced from the ambient (Lie) group. The
distortion of a subalgebra in an algebra also reflects the degree of complexity
of the membership problem for the elements of this algebra in this subalgebra.
One of our goals here is to investigate, mostly in the case of associative or
Lie algebras, if a tame filtration of an algebra can be induced from the degree
filtration of a larger algebra
Gaudin subalgebras and wonderful models
Gaudin hamiltonians form families of r-dimensional abelian Lie subalgebras of
the holonomy Lie algebra of the arrangement of reflection hyperplanes of a
Coxeter group of rank r. We consider the set of principal Gaudin subalgebras,
which is the closure in the appropriate Grassmannian of the set of spans of
Gaudin hamiltonians. We show that principal Gaudin subalgebras form a smooth
projective variety isomorphic to the De Concini-Procesi compactification of the
projectivized complement of the arrangement of reflection hyperplanes.Comment: 13 pages, 2 figures; added detailed description of the B_2 and B_3
cases in the new versio
Relation algebras from cylindric algebras, I
Accepted versio
The complexity of the list homomorphism problem for graphs
We completely classify the computational complexity of the list H-colouring
problem for graphs (with possible loops) in combinatorial and algebraic terms:
for every graph H the problem is either NP-complete, NL-complete, L-complete or
is first-order definable; descriptive complexity equivalents are given as well
via Datalog and its fragments. Our algebraic characterisations match important
conjectures in the study of constraint satisfaction problems.Comment: 12 pages, STACS 201
Conservative constraint satisfaction re-revisited
Conservative constraint satisfaction problems (CSPs) constitute an important
particular case of the general CSP, in which the allowed values of each
variable can be restricted in an arbitrary way. Problems of this type are well
studied for graph homomorphisms. A dichotomy theorem characterizing
conservative CSPs solvable in polynomial time and proving that the remaining
ones are NP-complete was proved by Bulatov in 2003. Its proof, however, is
quite long and technical. A shorter proof of this result based on the absorbing
subuniverses technique was suggested by Barto in 2011. In this paper we give a
short elementary prove of the dichotomy theorem for the conservative CSP
Hedgehog Bases for A_n Cluster Polylogarithms and An Application to Six-Point Amplitudes
Multi-loop scattering amplitudes in N=4 Yang-Mills theory possess cluster
algebra structure. In order to develop a computational framework which exploits
this connection, we show how to construct bases of Goncharov polylogarithm
functions, at any weight, whose symbol alphabet consists of cluster coordinates
on the cluster algebra. Using such a basis we present a new expression
for the 2-loop 6-particle NMHV amplitude which makes some of its cluster
structure manifest.Comment: 32 pages; v2: minor corrections and clarification
Linear representations of regular rings and complemented modular lattices with involution
Faithful representations of regular -rings and modular complemented
lattices with involution within orthosymmetric sesquilinear spaces are studied
within the framework of Universal Algebra. In particular, the correspondence
between classes of spaces and classes of representables is analyzed; for a
class of spaces which is closed under ultraproducts and non-degenerate finite
dimensional subspaces, the latter are shown to be closed under complemented
[regular] subalgebras, homomorphic images, and ultraproducts and being
generated by those members which are associated with finite dimensional spaces.
Under natural restrictions, this is refined to a --correspondence between
the two types of classes
- …