Mixing
Subgroups of $textSpin(7)$.
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I am interested in subgroups of $textSpin(7)$ and identifying certain elementary properties that they have. Specifically relating to commutativity. Let me apologise at this point for my ignorance and sloppy presentation below. I know little about this subject, which has recently come up in my work.
I write down the extended Dynkin diagram $widetilde B_3$ corresponding to the Lie algebra of $textSpin(7)$
with appended long root $alpha_0$, and short root $alpha_3$.
Removing the short node $alpha_3$ I get $widetilde A_2$ coresponding to the $SU(3)cong textSpin(6)$ subgroup. Cutting out $alpha_1$ leaves something with a fold symmetry that gives the $G_2leq textSpin(7)$ subgroup inclusion.
On the other hand, if we take out $alpha_2$ then we are left with three $A_1$'s. It's pretty clear that the pair given by $alpha_0$, $alpha_1$ are the two commuting $textSpin(3)$-subgroups that together constitute the canonical $textSpin(4)leq textSpin(7)$. The final $A_1$ corresponds to the short root $alpha_3$, and is what is confusing me. My intuition is telling me that it is a third copy of $textSpin(3)$, covering the $SO(3)leq SO(7)$ which is complementary to the $SO(4)$ subgroup.
As such this third $A_1$ should be another $textSpin(3)$ which commutes with the $textSpin(4)$ identified previously. However, my understanding runs out here, and the presence of the half-length root indicates to me that what I have actually identified is in fact an $(S^3times S^3times S^3)/mathbbZ_2$ subgroup containing a $textSpin(4)$ subgroup which does not commute with the other factor.
Have I identified a commuting $textSpin(3)$ and $textSpin(4)$ subgroups, or is it rather the second option $(S^3times S^3times S^3)/mathbbZ_2$?
Moreover, the diagram indicates to me the presence of three distinguished homotopy classes of maps $S^3rightarrow textSpin(7)$. The first two come from $textSpin(4)$, with $alpha_0$ corresponding to the canonical generator $i_0:S^3cong textSpin(3)hookrightarrow textSpin(7)$, and $alpha_1$ its negative, $i_1=-i_0$. Since $frac=2$, the corresponding homotopy class of $alpha_3$ should be $i_3=2cdot i_0$, which would follow from elementary properties of the Dynkin index. I'm not entirely sure this is what I want to see.
Am I correct about the identifications here?
group-theory lie-groups lie-algebras
asked Aug 1 at 7:23
Tyrone
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I am interested in subgroups of $textSpin(7)$ and identifying certain elementary properties that they have. Specifically relating to commutativity. Let me apologise at this point for my ignorance and sloppy presentation below. I know little about this subject, which has recently come up in my work.
I write down the extended Dynkin diagram $widetilde B_3$ corresponding to the Lie algebra of $textSpin(7)$
with appended long root $alpha_0$, and short root $alpha_3$.
Removing the short node $alpha_3$ I get $widetilde A_2$ coresponding to the $SU(3)cong textSpin(6)$ subgroup. Cutting out $alpha_1$ leaves something with a fold symmetry that gives the $G_2leq textSpin(7)$ subgroup inclusion.
On the other hand, if we take out $alpha_2$ then we are left with three $A_1$'s. It's pretty clear that the pair given by $alpha_0$, $alpha_1$ are the two commuting $textSpin(3)$-subgroups that together constitute the canonical $textSpin(4)leq textSpin(7)$. The final $A_1$ corresponds to the short root $alpha_3$, and is what is confusing me. My intuition is telling me that it is a third copy of $textSpin(3)$, covering the $SO(3)leq SO(7)$ which is complementary to the $SO(4)$ subgroup.
As such this third $A_1$ should be another $textSpin(3)$ which commutes with the $textSpin(4)$ identified previously. However, my understanding runs out here, and the presence of the half-length root indicates to me that what I have actually identified is in fact an $(S^3times S^3times S^3)/mathbbZ_2$ subgroup containing a $textSpin(4)$ subgroup which does not commute with the other factor.
Have I identified a commuting $textSpin(3)$ and $textSpin(4)$ subgroups, or is it rather the second option $(S^3times S^3times S^3)/mathbbZ_2$?
Moreover, the diagram indicates to me the presence of three distinguished homotopy classes of maps $S^3rightarrow textSpin(7)$. The first two come from $textSpin(4)$, with $alpha_0$ corresponding to the canonical generator $i_0:S^3cong textSpin(3)hookrightarrow textSpin(7)$, and $alpha_1$ its negative, $i_1=-i_0$. Since $frac=2$, the corresponding homotopy class of $alpha_3$ should be $i_3=2cdot i_0$, which would follow from elementary properties of the Dynkin index. I'm not entirely sure this is what I want to see.
Am I correct about the identifications here?
group-theory lie-groups lie-algebras
asked Aug 1 at 7:23
Tyrone
3,1311925
This question has an open bounty worth +50
reputation from Tyrone ending ending at 2018-08-12 18:20:53Z">in 7 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I am interested in subgroups of $textSpin(7)$ and identifying certain elementary properties that they have. Specifically relating to commutativity. Let me apologise at this point for my ignorance and sloppy presentation below. I know little about this subject, which has recently come up in my work.
I write down the extended Dynkin diagram $widetilde B_3$ corresponding to the Lie algebra of $textSpin(7)$
with appended long root $alpha_0$, and short root $alpha_3$.
Removing the short node $alpha_3$ I get $widetilde A_2$ coresponding to the $SU(3)cong textSpin(6)$ subgroup. Cutting out $alpha_1$ leaves something with a fold symmetry that gives the $G_2leq textSpin(7)$ subgroup inclusion.
On the other hand, if we take out $alpha_2$ then we are left with three $A_1$'s. It's pretty clear that the pair given by $alpha_0$, $alpha_1$ are the two commuting $textSpin(3)$-subgroups that together constitute the canonical $textSpin(4)leq textSpin(7)$. The final $A_1$ corresponds to the short root $alpha_3$, and is what is confusing me. My intuition is telling me that it is a third copy of $textSpin(3)$, covering the $SO(3)leq SO(7)$ which is complementary to the $SO(4)$ subgroup.
As such this third $A_1$ should be another $textSpin(3)$ which commutes with the $textSpin(4)$ identified previously. However, my understanding runs out here, and the presence of the half-length root indicates to me that what I have actually identified is in fact an $(S^3times S^3times S^3)/mathbbZ_2$ subgroup containing a $textSpin(4)$ subgroup which does not commute with the other factor.
Have I identified a commuting $textSpin(3)$ and $textSpin(4)$ subgroups, or is it rather the second option $(S^3times S^3times S^3)/mathbbZ_2$?
Moreover, the diagram indicates to me the presence of three distinguished homotopy classes of maps $S^3rightarrow textSpin(7)$. The first two come from $textSpin(4)$, with $alpha_0$ corresponding to the canonical generator $i_0:S^3cong textSpin(3)hookrightarrow textSpin(7)$, and $alpha_1$ its negative, $i_1=-i_0$. Since $frac=2$, the corresponding homotopy class of $alpha_3$ should be $i_3=2cdot i_0$, which would follow from elementary properties of the Dynkin index. I'm not entirely sure this is what I want to see.
Am I correct about the identifications here?
group-theory lie-groups lie-algebras
asked Aug 1 at 7:23
Tyrone
3,1311925
I am interested in subgroups of $textSpin(7)$ and identifying certain elementary properties that they have. Specifically relating to commutativity. Let me apologise at this point for my ignorance and sloppy presentation below. I know little about this subject, which has recently come up in my work.
I write down the extended Dynkin diagram $widetilde B_3$ corresponding to the Lie algebra of $textSpin(7)$
with appended long root $alpha_0$, and short root $alpha_3$.
Removing the short node $alpha_3$ I get $widetilde A_2$ coresponding to the $SU(3)cong textSpin(6)$ subgroup. Cutting out $alpha_1$ leaves something with a fold symmetry that gives the $G_2leq textSpin(7)$ subgroup inclusion.
On the other hand, if we take out $alpha_2$ then we are left with three $A_1$'s. It's pretty clear that the pair given by $alpha_0$, $alpha_1$ are the two commuting $textSpin(3)$-subgroups that together constitute the canonical $textSpin(4)leq textSpin(7)$. The final $A_1$ corresponds to the short root $alpha_3$, and is what is confusing me. My intuition is telling me that it is a third copy of $textSpin(3)$, covering the $SO(3)leq SO(7)$ which is complementary to the $SO(4)$ subgroup.
As such this third $A_1$ should be another $textSpin(3)$ which commutes with the $textSpin(4)$ identified previously. However, my understanding runs out here, and the presence of the half-length root indicates to me that what I have actually identified is in fact an $(S^3times S^3times S^3)/mathbbZ_2$ subgroup containing a $textSpin(4)$ subgroup which does not commute with the other factor.
Have I identified a commuting $textSpin(3)$ and $textSpin(4)$ subgroups, or is it rather the second option $(S^3times S^3times S^3)/mathbbZ_2$?
Moreover, the diagram indicates to me the presence of three distinguished homotopy classes of maps $S^3rightarrow textSpin(7)$. The first two come from $textSpin(4)$, with $alpha_0$ corresponding to the canonical generator $i_0:S^3cong textSpin(3)hookrightarrow textSpin(7)$, and $alpha_1$ its negative, $i_1=-i_0$. Since $frac=2$, the corresponding homotopy class of $alpha_3$ should be $i_3=2cdot i_0$, which would follow from elementary properties of the Dynkin index. I'm not entirely sure this is what I want to see.
Am I correct about the identifications here?
group-theory lie-groups lie-algebras
asked Aug 1 at 7:23
Tyrone
3,1311925
edited 6 hours ago
asked Aug 1 at 7:23
Tyrone
3,1311925
asked Aug 1 at 7:23
Tyrone
3,1311925
asked Aug 1 at 7:23
Tyrone
3,1311925
3,1311925
This question has an open bounty worth +50
reputation from Tyrone ending ending at 2018-08-12 18:20:53Z">in 7 days.
Looking for an answer drawing from credible and/or official sources.
This question has an open bounty worth +50
reputation from Tyrone ending ending at 2018-08-12 18:20:53Z">in 7 days.
Looking for an answer drawing from credible and/or official sources.
add a comment |Â
add a comment |Â
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