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
).