Split Lie algebra
Encyclopedia
In the mathematical field of Lie theory
, a split Lie algebra is a pair
where 
is a Lie algebra
and
is a splitting Cartan subalgebra, where "splitting" means that for all 
is triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra. Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center.
Over an algebraically closed field
such as the complex numbers, all semisimple Lie algebras are splittable (indeed, the Cartan subalgebra acts not only by triangularizable matrices but a fortiori by diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields.
Split Lie algebras are of interest both because they formalize the split real form of a complex Lie algebra, and because split semisimple Lie algebras (more generally, split reductive Lie algebras) over any field share many properties with semisimple Lie algebras over algebraically closed fields – having essentially the same representation theory, for instance – the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. This is the approach followed in , for instance.
Every complex semisimple Lie algebra has a unique (up to isomorphism) split real Lie algebra, which is also semisimple, and is simple if and only if the complex Lie algebra is.
For real semisimple Lie algebras, split Lie algebras are opposite to compact Lie algebra
s – the corresponding Lie group is "as far as possible" from being compact.
These are the Lie algebras of the split real groups of the complex Lie groups.
Note that for sl and sp, the real form is the real points of (the Lie algebra of) the same algebraic group
, while for so one must use the split forms (of maximally indefinite index), as SO is compact.
Lie theory
Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....
, a split Lie algebra is a pair
where is a Lie algebraLie 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...
and
is a splitting Cartan subalgebra, where "splitting" means that for all is triangularizable. If a Lie algebra admits a splitting, it is called a splittable Lie algebra. Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center.Over an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
such as the complex numbers, all semisimple Lie algebras are splittable (indeed, the Cartan subalgebra acts not only by triangularizable matrices but a fortiori by diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields.
Split Lie algebras are of interest both because they formalize the split real form of a complex Lie algebra, and because split semisimple Lie algebras (more generally, split reductive Lie algebras) over any field share many properties with semisimple Lie algebras over algebraically closed fields – having essentially the same representation theory, for instance – the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. This is the approach followed in , for instance.
Properties
- Over an algebraically closed field, all Cartan subalgebras are conjugate. Over a non-algebraically closed fields, not all Cartan subalgebras are conjugate in general; however, in a splittable semisimple Lie algebra all splitting Cartan algebras are conjugate.
- Over an algebraically closed field, all semisimple Lie algebras are splittable.
- Over a non-algebraically closed field, there exist non-splittable semisimple Lie algebras.
- In a splittable Lie algebra, there may exist Cartan subalgebras that are not splitting.
- Direct sums of splittable Lie algebras and ideals in splittable Lie algebras are splittable.
Split real Lie algebras
For a real Lie algebra, splittable is equivalent to either of these conditions:- The real rank equals the complex rank.
- The Satake diagramSatake diagramIn the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram whose configurations classify simple Lie algebras over the field of real numbers...
has neither black vertices nor arrows.
Every complex semisimple Lie algebra has a unique (up to isomorphism) split real Lie algebra, which is also semisimple, and is simple if and only if the complex Lie algebra is.
For real semisimple Lie algebras, split Lie algebras are opposite to compact Lie algebra
Compact Lie algebra
In the mathematical field of Lie theory, a Lie algebra is compact if it is the Lie algebra of a compact Lie group. Intrinsically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite, though this definition does not quite agree with the previous...
s – the corresponding Lie group is "as far as possible" from being compact.
Examples
The split real forms for the complex semisimple Lie algebras are:-
-
-
-
- Exceptional Lie algebras:
have split real forms EI, EV, EVIII, FI, G.
These are the Lie algebras of the split real groups of the complex Lie groups.
Note that for sl and sp, the real form is the real points of (the Lie algebra of) the same 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...
, while for so one must use the split forms (of maximally indefinite index), as SO is compact.
See also
- Compact Lie algebraCompact Lie algebraIn the mathematical field of Lie theory, a Lie algebra is compact if it is the Lie algebra of a compact Lie group. Intrinsically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite, though this definition does not quite agree with the previous...
- Real form
- Split-complex numberSplit-complex numberIn abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...
- Split orthogonal group