Mikhail Kochetov

Gradings on simple Lie algebras

Introduction Gradings on algebras Associative algebras Classical Lie algebras Composition algebras and type

Classification of group gradings on simple Lie algebras of types A, B, C and D

G_2

Gradings on the exceptional Lie algebras

Jordan algebras and type

F_4

F_4

and

Other simple Lie algebras in characteristic zero Lie algebras of Cartan type in prime characteristic Affine group schemes Irreducible root systems Bibliography Index of Notation Index

G_2

Abstract 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.

revisited

All gradings by abelian groups are classified on the following al- gebras over an algebraically closed field F: the simple Lie algebra of type G2 (charF 6 2,3), the exceptional simple Jordan algebra (charF 6 2), and the simple Lie algebra of type F4 (charF 6 2).

Group gradings on simple Lie algebras in positive characteristic

In this paper we describe all gradings by a finite abelian group G on the following Lie algebras over an algebraically closed field F of characteristic p ≠ 2: sl n (F) (n not divisible by p), so n (F) (n > 5, n ≠ 8) and sp n (F) (n > 6, n even).

Graded modules over classical simple Lie algebras with a grading

Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D4, where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.

Estimating effective elasticity tensors from Christoffel equations

We consider the problem of obtaining the orientation and elasticity parameters of an effective tensor of particular symmetry that corresponds to measurable traveltime and polarization quantities. These quantities — the wavefront-slowness and polarization vectors — are used in the Christoffel equation, a characteristic equation of the elastodynamic equation that brings seismic concepts to our formulation and relates experimental data to the elasticity tensor. To obtain an effective tensor of particular symmetry, we do not assume its orientation; thus, the regression using the residuals of the Christoffel equation results in a nonlinear optimization problem. We find the absolute extremum and, to avoid numerical instability of a global search, obtain an accurate initial guess using the tensor of given symmetry closest to the generally anisotropic tensor obtained from data by linear regression. The issue is twofold. First, finding the closest tensor of particular symmetry without assuming its orientation is challenging. Second, the closest tensor is not the effective tensor in the sense of regression because the process of finding it carries neither seismic concepts nor statistical information; rather, it relies on an abstract norm in the space of elasticity tensors. To include seismic concepts and statistical information, we distinguish between the closest tensor of particular symmetry and the effective one; the former is the initial guess to search for the latter.