Lie algebras were introduced toward the end of nineteenth century in order to study some problems arising from geometry.
In the interest of classifying these objects, the subcategory of
semisimple Lie algebras has been studied.
This is the class of algebra in which I am interested in. The easiest example of such algebra is
almost given by 𝖌=𝖌l_{n},
the set of matrices of size n equiped with the Lie bracket [A,B]=A.B-B.A.
Such an algebra is in fact
called reductive
since only [𝖌,𝖌] is semisimple.

Symmetric Lie algebras

There exists a well known classification of complex semisimple Lie algebras (A_{n}, B_{n}, C_{n}, D_{n}, E_{6}, E_{7}, E_{8}, F_{4} et G_{2}).
When the groundfield is real, the classification is heavily based upon the complex one.
Concretely, let 𝖌_{ℝ} be a real semisimple Lie algebra and let θ_{ℝ} be its Cartan involution.
Then, decomposing 𝖌_{ℝ} into θ_{ℝ}-eigenspaces respectively associated to +1 and -1, one gets 𝖌_{ℝ}=𝖐_{ℝ}+𝖕_{ℝ}.
This decomposition is linked with the decomposition of any real Lie group into compact and non-compact part.
For instance the Polar decomposition is such a decomposition when 𝖌=𝖌l_{n}.

When one complexifies these notions, one gets a pair (𝖌_{ℂ},θ_{ℂ}) associated to a decomposition 𝖌_{ℂ}=𝖐_{ℂ}+𝖕_{ℂ}.
Then 𝖐_{ℂ} is a (reductive) Lie subalgebra of 𝖌_{ℂ} and 𝖕_{ℂ} is an ad 𝖐_{ℂ}-module where ad(x)=[x,.] is the Lie algebra adjoint action.
A pair (𝖌,θ) is called a symmetric Lie algebra when 𝖌 is a Lie algebra and θ is an involution of 𝖌.

The above described method yields to a one to one correspondance between real semisimple Lie subalgebras and complex symmetric Lie subalgebras.
This correspondance is made deeper by the so-called Kostant Sekiguchi correspondance.
The latter establishes a bijection between nilpotent orbits of 𝖌_{ℝ} and nilpotent orbits of 𝖕_{ℂ}.

A particular case of symmetric Lie algebra is when 𝖌 is isomorphic to the direct sum of two copies 𝖌_{0} and 𝖌_{1} of a Lie algebra 𝖌',
and when θ(𝖌_{i})=𝖌_{1-i}, i=0,1.
Then, the Lie algebra 𝖐={x+θ(x)| x∈ 𝖌_{1}} is isomorphic to 𝖌'
while the ad 𝖐-module 𝖕={x-θ(x)| x∈ 𝖌_{1}} is isomorphic to the ad 𝖌'-module 𝖌'.
That's why symmetric Lie are often considered as a generalization of Lie algebras.
Furthermore, lots of notions arising from Lie theory
(e.g. Cartan subalgebras, root systems, Dynkin diagrams, 𝖘l_{2}-triples,...)
have a symmetric generalization (e.g. Cartan subspaces, restricted root systems, Satake diagrams, normal 𝖘l_{2}-triples,...).

Geometry in Lie algebras

There is an another level structure on (semisimple complex) Lie algebras.
Denoting by G the algebraic adjoint group of 𝖌, we can consider
𝖌 as a G-variety under the adjoint action Ad.
We can then study some properties in the framework of algebraic geometry.

For instance, the notion of semisimple or nilpotent element of 𝖌 can be defined geometrically.
Semisimple elements are elements whose (G-)orbit is closed,
while nilpotents elements are elements whose G-orbit contains 0 in its closure.
Again, this structure has a symmetric analogue.
Indeed, one can define the connected subgroup K of G having 𝖐 as Lie algebra.
Then, 𝖕 is an Ad K-variety.

One can then study various G-varieties arising from this setting.
From a global perspective, I try to generalize or understand some properties of analogue varieties in symmetric Lie algebras

Example 1: The commuting variety

As a first exemple, we can talk about the commuting variety 𝕮(𝖌):={(x,y)∈𝖌×𝖌| [x,y]=0}.
This variety is irreducible (Richardson 1979).
Similarily, it has been recently shown (Premet 2003) that the nilpotent commuting variety
𝕮^{nil}(𝖌)=𝕮(𝖌)∩𝓝×𝓝 is equidimensional
and that its irreducible components are indexed by so-called distinguished nilpotent orbits.

From a symmetric point of view, one should speak about the study of the symmetric commuting variety 𝕮(𝖕)=𝕮 (𝖌)∩(𝖕×𝖕). In the general case, It is not irreducible nor equidimensionnal.
Personnally, I have been interested in the study of the symmetric nilpotent commuting variety
𝕮^{nil}(𝖕)=𝕮^{nil}(𝖌)∩(𝖕×𝖕).
In 15 out of 20 cases, I could prove the following: 𝕮^{nil}(𝖕) is equidimensional.

Example 2: Sheets

Sheets (or G-sheets) are some other examples of varieties already studied.
These are irreducible components of subsets of the form
𝖌^{(m)}:={x∈𝖌| dim G.x=m}; m∈ ℕ.
The study of these varieties is linked with various geometric problems arising in Lie theory.
For instance, the proof by Richardson of the irreducibility of the commuting variety is based upon some results about sheets.

A part of the study of sheets has been made through an object called Slodowy Slice.
When (e,h,f) is an 𝖘l_{2}-triple such that e belong to a sheet S, a Slodowy slice of S is e+X:=S∩(e+𝖌^{f}).
In particular, one can show that S=G.(e+X) and that a geometric quotient of S can be expressed as a quotient of X by a finite group.

The K-sheets are a good symmetric analogue to sheets.
They are defined as irreducible components of subsets of the form
𝖕^{(m)}:={x∈ 𝖕| dim K.x=m}, m∈ℕ.
In my study
of these objects, I could show that the parametrization through
the Slodowy slice still had some good properties
when 𝖌=𝖌l_{n}.
In particular, it has been possible to prove that K-sheets are the regular elements of the closure of sets of the form
K.(e+X∩𝖕)
when (e,h,f) is a normal 𝖘l_{2}-triple.

- Cool non? -
Michaël Bulois Last update: 17 December 2011 - nice website, isn't it?-