CKForms - Manual for version 2.x

Notation and convention

This instruction descibes the functionality associated with the following article (Bochenski, Jastrzebski, and Tralle 2021).

We use the notation of the theory of real Lie algebras from CoReLG Package, (Dietrich, Faccin, and Graaf 2014).

Function for real Lie algebras

RealRank(g)

The input is a real Lie algebra \(\mathfrak{g}\) (as a Lie algebra object). The output is the real rank of \(\mathfrak{g}\) (the dimension of the Cartan subalgebra of \(\mathfrak{g}\)).

AHypRank(g)

The input is a real Lie algebra \(\mathfrak{g}\) (as a Lie algebra object). The output is the a-hyperbolic rank of \(\mathfrak{g}\).

Example:

gap> g:=RealFormById("A",5,6);
<Lie algebra of dimension 35 over SqrtField>
gap> NameRealForm(g);
"sl(6,R)"
gap> RealRank(g);
5
gap> AHypRank(g);
3

Main procedure

As in Theorem 6 in (Bochenski, Jastrzebski, and Tralle 2021) we are checking three conditions:

• \(L_0\) - Calabi–Markus phenomenon, \(\operatorname{rank}_{\mathbb{R}} \mathfrak{g}=\operatorname{rank}_{\mathbb{R}} \mathfrak{h}\)
• \(L_1\) - \(\operatorname{rank}_{\textrm{a-hyp}}(\mathfrak{g}) =\operatorname{rank}_{\textrm{a-hyp}}(\mathfrak{h})\)
• \(L_2\) - \(\operatorname{rank}_{\textrm{a-hyp}}(\mathfrak{g}) >\operatorname{rank}_{\mathbb{R}} \mathfrak{h}\)
• \(L_3\) - none of the above conditions is met

gap> CheckRankConditions("A",5,6);
g=sl(6,R) | real rank(g)=5 | a-hyp rank(g)=3
----------------------------
#1: h=sl(3,R)+sl(3,R) + a torus of  1 non-compact dimensions | real rank(h)=5 | ahyp rank(h)=2
 | L0-true | L1-false | L2-false | L3-false
----
#2: h=sl(3,C) + a torus of 1 compact dimensions | real rank(h)=2 | ahyp rank(h)=1
 | L0-false | L1-false | L2-true | L3-false
----
#3: h=sl(2,R)+sl(4,R) + a torus of  1 non-compact dimensions | real rank(h)=5 | ahyp rank(h)=3
 | L0-true | L1-true | L2-false | L3-false
----
#4: h=sl(5,R) + a torus of  1 non-compact dimensions | real rank(h)=5 | ahyp rank(h)=2
 | L0-true | L1-false | L2-false | L3-false
----
#5: h=sl(3,R) | real rank(h)=2 | ahyp rank(h)=1
 | L0-false | L1-false | L2-true | L3-false
----
#6: h=sl(2,R)+sl(3,R) | real rank(h)=3 | ahyp rank(h)=2
 | L0-false | L1-false | L2-false | L3-true
----
#7: h=su(4) | real rank(h)=0 | ahyp rank(h)=0
 | L0-false | L1-false | L2-true | L3-false
----
#8: h=su(2,2) | real rank(h)=2 | ahyp rank(h)=2
 | L0-false | L1-false | L2-true | L3-false
----
#9: h=sl(2,H) | real rank(h)=1 | ahyp rank(h)=1
 | L0-false | L1-false | L2-true | L3-false
----
#10: h=sl(4,R) | real rank(h)=3 | ahyp rank(h)=2
 | L0-false | L1-false | L2-false | L3-true
----
#11: h=sp(3,R) | real rank(h)=3 | ahyp rank(h)=3
 | L0-false | L1-true | L2-false | L3-false
----

Note that in the code above the order of the choice of maximal subalgebras \(\mathfrak{h}\) is consistent with the function MaximalReductiveSubalgebras from (Dietrich, Faccin, and Graaf 2014).

All calculations are being done in the “Database - v2” section.

Next if L3 is equal to true, then we check the condition for orbits as in Theorem 5 in (Bochenski, Jastrzebski, and Tralle 2021).

gap> CheckProperSL2RAction("A",5,6,6);
proper

The last argument of the function CheckProperSL2RAction is an index of the maximal subalgebra (subalgs output MaximalReductiveSubalgebras).

References

Bochenski, Maciej, Piotr Jastrzebski, and Aleksy Tralle. 2021. “Homogeneous Spaces of Real Simple Lie Groups with Proper Actions of Non Virtually Abelian Discrete Subgroups: A Calculational Approach.” http://arxiv.org/abs/2106.05777.

Dietrich, H., P. Faccin, and W. A. de Graaf. 2014. “CoReLG, Computation with Real Lie Groups, Version 1.20.” http://users.monash.edu/~heikod/corelg/.