Modular Lie algebra
Encyclopedia
In mathematics
, a modular Lie algebra is a Lie algebra
over a field
of positive characteristic.
The theory of modular Lie algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential map to establish a tight connection between properties of a modular Lie algebra and the corresponding algebraic group
.
Although serious study of modular Lie algebras was initiated by Nathan Jacobson
in 1950s, their representation theory in the semisimple case was advanced only recently due to the influential Lusztig conjectures, which have been partially proved.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a modular Lie algebra is a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of positive characteristic.
The theory of modular Lie algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential map to establish a tight connection between properties of a modular Lie algebra and the corresponding algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
.
Although serious study of modular Lie algebras was initiated by Nathan Jacobson
Nathan Jacobson
Nathan Jacobson was an American mathematician....
in 1950s, their representation theory in the semisimple case was advanced only recently due to the influential Lusztig conjectures, which have been partially proved.