240 research outputs found
Classification of group gradings on simple Lie algebras of types A, B, C and D
For a given abelian group G, we classify the isomorphism classes of
G-gradings on the simple Lie algebras of types A_n (n >= 1), B_n (n >= 2), C_n
(n >= 3) and D_n (n > 4), in terms of numerical and group-theoretical
invariants. The ground field is assumed to be algebraically closed of
characteristic different from 2.Comment: 20 pages, no figure
Large Restricted Lie Algebras
We establish some results about large restricted Lie algebras similar to
those known in the Group Theory. As an application we use this group-theoretic
approach to produce some examples of restricted as well as ordinary Lie
algebras which can serve as counterexamples for various Burnside-type
questions
Actions of Maximal Growth
We study acts and modules of maximal growth over finitely generated free
monoids and free associative algebras as well as free groups and free group
algebras. The maximality of the growth implies some other specific properties
of these acts and modules that makes them close to the free ones; at the same
time, we show that being a strong "infiniteness" condition, the maximality of
the growth can still be combined with various finiteness conditions, which
would normally make finitely generated acts finite and finitely generated
modules finite-dimensional
Involutions on graded matrix algebras
In this paper we describe graded automorphisms and antiautomorphisms of
finite order on matrix algebras endowed with a group gradings by a finite
abelian group over an arbitrary algebraically closed field of charcteristic
different from 2
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
Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra
Given a grading on a nonassociative algebra
by an abelian group , we have two subgroups of the group of
automorphisms of : the automorphisms that stabilize each homogeneous
component (as a subspace) and the automorphisms that permute the
components. By the Weyl group of we mean the quotient of the latter
subgroup by the former. In the case of a Cartan decomposition of a semisimple
complex Lie algebra, this is the automorphism group of the root system, i.e.,
the so-called extended Weyl group. A grading is called fine if it cannot be
refined. We compute the Weyl groups of all fine gradings on matrix algebras,
octonions and the Albert algebra over an algebraically closed field (of
characteristic different from 2 in the case of the Albert algebra).Comment: 23 pages. References update
- …