Reference Request: Structure constants for G2
$begingroup$
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $Phi^+$ of $T$ in $U$, and for each root subgroup $U_{alpha}$ of $U$, let $u_{alpha}: mathbb R rightarrow U_{alpha}$ be an isomorphism.
For all $alpha, beta in Phi^+$, there exist unique real numbers $C_{alpha,beta,i,j}$ (depending on the $u_{alpha}$ and the ordering) such that for all $x, y in mathbb R$,
$$u_{alpha}(x) u_{beta}(y) u_{alpha}(x)^{-1} = u_{beta}(y) prodlimits_{substack{i,j>0\ ialpha + j beta in Phi^+}} u_{ialpha+jbeta}(C_{alpha,beta,i,j}x^iy^j)$$
I want to work out some examples with unipotent groups of exceptional semisimple groups, and am looking for table of structure constants for the root system G2. Does anyone know a reference where an ordering on the roots is chosen and these constants are written down?
reference-request rt.representation-theory lie-groups algebraic-groups lie-algebras
asked Feb 10 at 2:21
D_SD_S
1,801619
$endgroup$
add a comment |
$begingroup$
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $Phi^+$ of $T$ in $U$, and for each root subgroup $U_{alpha}$ of $U$, let $u_{alpha}: mathbb R rightarrow U_{alpha}$ be an isomorphism.
For all $alpha, beta in Phi^+$, there exist unique real numbers $C_{alpha,beta,i,j}$ (depending on the $u_{alpha}$ and the ordering) such that for all $x, y in mathbb R$,
$$u_{alpha}(x) u_{beta}(y) u_{alpha}(x)^{-1} = u_{beta}(y) prodlimits_{substack{i,j>0\ ialpha + j beta in Phi^+}} u_{ialpha+jbeta}(C_{alpha,beta,i,j}x^iy^j)$$
I want to work out some examples with unipotent groups of exceptional semisimple groups, and am looking for table of structure constants for the root system G2. Does anyone know a reference where an ordering on the roots is chosen and these constants are written down?
reference-request rt.representation-theory lie-groups algebraic-groups lie-algebras
asked Feb 10 at 2:21
D_SD_S
1,801619
$endgroup$
3
$begingroup$
If you differentiate and set $x,y$ to zero, surely these expressions will be more familiar, as Lie brackets of root vectors. Then they are known in the literature, using Chevalley bases, or at least their is an algorithm to uncover them.
$endgroup$
– Ben McKay
Feb 10 at 7:02
1
$begingroup$
Some time ago I wrote a memo for myself on split-octonions and $G_2$, madore.org/~david/.misc/20140711-split-octonions.pdf — what you are looking for is the second table on page 2, right? It's easy to compute, but I don't know where you could find it in the published literature.
$endgroup$
– Gro-Tsen
Feb 10 at 9:18
$begingroup$
Correction: $U$ is the unipotent radical of $B$ in this formulation.
$endgroup$
– Jim Humphreys
Feb 10 at 15:16
add a comment |
$begingroup$
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $Phi^+$ of $T$ in $U$, and for each root subgroup $U_{alpha}$ of $U$, let $u_{alpha}: mathbb R rightarrow U_{alpha}$ be an isomorphism.
For all $alpha, beta in Phi^+$, there exist unique real numbers $C_{alpha,beta,i,j}$ (depending on the $u_{alpha}$ and the ordering) such that for all $x, y in mathbb R$,
$$u_{alpha}(x) u_{beta}(y) u_{alpha}(x)^{-1} = u_{beta}(y) prodlimits_{substack{i,j>0\ ialpha + j beta in Phi^+}} u_{ialpha+jbeta}(C_{alpha,beta,i,j}x^iy^j)$$
I want to work out some examples with unipotent groups of exceptional semisimple groups, and am looking for table of structure constants for the root system G2. Does anyone know a reference where an ordering on the roots is chosen and these constants are written down?
reference-request rt.representation-theory lie-groups algebraic-groups lie-algebras
asked Feb 10 at 2:21
D_SD_S
1,801619
$endgroup$
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $Phi^+$ of $T$ in $U$, and for each root subgroup $U_{alpha}$ of $U$, let $u_{alpha}: mathbb R rightarrow U_{alpha}$ be an isomorphism.
For all $alpha, beta in Phi^+$, there exist unique real numbers $C_{alpha,beta,i,j}$ (depending on the $u_{alpha}$ and the ordering) such that for all $x, y in mathbb R$,
$$u_{alpha}(x) u_{beta}(y) u_{alpha}(x)^{-1} = u_{beta}(y) prodlimits_{substack{i,j>0\ ialpha + j beta in Phi^+}} u_{ialpha+jbeta}(C_{alpha,beta,i,j}x^iy^j)$$
I want to work out some examples with unipotent groups of exceptional semisimple groups, and am looking for table of structure constants for the root system G2. Does anyone know a reference where an ordering on the roots is chosen and these constants are written down?
reference-request rt.representation-theory lie-groups algebraic-groups lie-algebras
reference-request rt.representation-theory lie-groups algebraic-groups lie-algebras
asked Feb 10 at 2:21
D_SD_S
1,801619
asked Feb 10 at 2:21
D_SD_S
1,801619
edited Feb 10 at 15:41
D_S
asked Feb 10 at 2:21
D_SD_S
1,801619
asked Feb 10 at 2:21
D_SD_S
1,801619
asked Feb 10 at 2:21
D_SD_S
1,801619
1,801619
3
$begingroup$
If you differentiate and set $x,y$ to zero, surely these expressions will be more familiar, as Lie brackets of root vectors. Then they are known in the literature, using Chevalley bases, or at least their is an algorithm to uncover them.
$endgroup$
– Ben McKay
Feb 10 at 7:02
1
$begingroup$
Some time ago I wrote a memo for myself on split-octonions and $G_2$, madore.org/~david/.misc/20140711-split-octonions.pdf — what you are looking for is the second table on page 2, right? It's easy to compute, but I don't know where you could find it in the published literature.
$endgroup$
– Gro-Tsen
Feb 10 at 9:18
$begingroup$
Correction: $U$ is the unipotent radical of $B$ in this formulation.
$endgroup$
– Jim Humphreys
Feb 10 at 15:16
add a comment |
3
$begingroup$
If you differentiate and set $x,y$ to zero, surely these expressions will be more familiar, as Lie brackets of root vectors. Then they are known in the literature, using Chevalley bases, or at least their is an algorithm to uncover them.
$endgroup$
– Ben McKay
Feb 10 at 7:02
1
$begingroup$
Some time ago I wrote a memo for myself on split-octonions and $G_2$, madore.org/~david/.misc/20140711-split-octonions.pdf — what you are looking for is the second table on page 2, right? It's easy to compute, but I don't know where you could find it in the published literature.
$endgroup$
– Gro-Tsen
Feb 10 at 9:18
$begingroup$
Correction: $U$ is the unipotent radical of $B$ in this formulation.
$endgroup$
– Jim Humphreys
Feb 10 at 15:16
3
3
$begingroup$
If you differentiate and set $x,y$ to zero, surely these expressions will be more familiar, as Lie brackets of root vectors. Then they are known in the literature, using Chevalley bases, or at least their is an algorithm to uncover them.
$endgroup$
– Ben McKay
Feb 10 at 7:02
$begingroup$
If you differentiate and set $x,y$ to zero, surely these expressions will be more familiar, as Lie brackets of root vectors. Then they are known in the literature, using Chevalley bases, or at least their is an algorithm to uncover them.
$endgroup$
– Ben McKay
Feb 10 at 7:02
1
1
$begingroup$
Some time ago I wrote a memo for myself on split-octonions and $G_2$, madore.org/~david/.misc/20140711-split-octonions.pdf — what you are looking for is the second table on page 2, right? It's easy to compute, but I don't know where you could find it in the published literature.
$endgroup$
– Gro-Tsen
Feb 10 at 9:18
$begingroup$
Some time ago I wrote a memo for myself on split-octonions and $G_2$, madore.org/~david/.misc/20140711-split-octonions.pdf — what you are looking for is the second table on page 2, right? It's easy to compute, but I don't know where you could find it in the published literature.
$endgroup$
– Gro-Tsen
Feb 10 at 9:18
$begingroup$
Correction: $U$ is the unipotent radical of $B$ in this formulation.
$endgroup$
– Jim Humphreys
Feb 10 at 15:16
$begingroup$
Correction: $U$ is the unipotent radical of $B$ in this formulation.
$endgroup$
– Jim Humphreys
Feb 10 at 15:16
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
"Simple groups of Lie type" by R. W. Carter, a table after Section 12.4. But there only the values of $C_{alphabeta11}$ are listed. An explicit form of commutator formulas inside $U^+$ is given in Table IV of "Chevalley groups over commutative rings: I. Elementary calculations" by N. Vavilov and E. Plotkin, see the picture below.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 10:39
Andrei SmolenskyAndrei Smolensky
1,2681122
$endgroup$
$begingroup$
Thanks! So just to be clear $ij$ means $ialpha + j beta$, where $alpha$ is the short root, right?
$endgroup$
– D_S
Feb 10 at 15:58
$begingroup$
@D_S Indeed, as indicated by the presence of $3alpha+2beta$.
$endgroup$
– Andrei Smolensky
Feb 10 at 16:17
add a comment |
$begingroup$
SGA III, Expose XXIII, Section 3.4.
answered Feb 10 at 9:54
Peter McNamaraPeter McNamara
5,5912754
$endgroup$
2
$begingroup$
You can find the Réédition de SGA3 at webusers.imj-prg.fr/~patrick.polo/SGA3
$endgroup$
– David Roberts
Feb 10 at 10:41
add a comment |
$begingroup$
Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No special assumption about the characteristic of the field is needed.) My own later treatment of the classification of simple algebraic groups followed the same method in GTM 21 (Linear Algebraic Groups, Springer, 1975, 33.5). A similar approach was taken in SGA3, as indicated by Peter McNamara. (The later more elegant approach to the classification due to Takeuchi was worked out in Jantzen's book as well as Springer's textbook.)
Note too that the Lie algebra calculations were done in my earlier book GTM 9, first in the characteristic 0 setting: see 19.3.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 15:12
Jim HumphreysJim Humphreys
41.8k494190
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
"Simple groups of Lie type" by R. W. Carter, a table after Section 12.4. But there only the values of $C_{alphabeta11}$ are listed. An explicit form of commutator formulas inside $U^+$ is given in Table IV of "Chevalley groups over commutative rings: I. Elementary calculations" by N. Vavilov and E. Plotkin, see the picture below.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 10:39
Andrei SmolenskyAndrei Smolensky
1,2681122
$endgroup$
$begingroup$
Thanks! So just to be clear $ij$ means $ialpha + j beta$, where $alpha$ is the short root, right?
$endgroup$
– D_S
Feb 10 at 15:58
$begingroup$
@D_S Indeed, as indicated by the presence of $3alpha+2beta$.
$endgroup$
– Andrei Smolensky
Feb 10 at 16:17
add a comment |
$begingroup$
"Simple groups of Lie type" by R. W. Carter, a table after Section 12.4. But there only the values of $C_{alphabeta11}$ are listed. An explicit form of commutator formulas inside $U^+$ is given in Table IV of "Chevalley groups over commutative rings: I. Elementary calculations" by N. Vavilov and E. Plotkin, see the picture below.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 10:39
Andrei SmolenskyAndrei Smolensky
1,2681122
$endgroup$
$begingroup$
Thanks! So just to be clear $ij$ means $ialpha + j beta$, where $alpha$ is the short root, right?
$endgroup$
– D_S
Feb 10 at 15:58
$begingroup$
@D_S Indeed, as indicated by the presence of $3alpha+2beta$.
$endgroup$
– Andrei Smolensky
Feb 10 at 16:17
add a comment |
$begingroup$
"Simple groups of Lie type" by R. W. Carter, a table after Section 12.4. But there only the values of $C_{alphabeta11}$ are listed. An explicit form of commutator formulas inside $U^+$ is given in Table IV of "Chevalley groups over commutative rings: I. Elementary calculations" by N. Vavilov and E. Plotkin, see the picture below.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 10:39
Andrei SmolenskyAndrei Smolensky
1,2681122
$endgroup$
"Simple groups of Lie type" by R. W. Carter, a table after Section 12.4. But there only the values of $C_{alphabeta11}$ are listed. An explicit form of commutator formulas inside $U^+$ is given in Table IV of "Chevalley groups over commutative rings: I. Elementary calculations" by N. Vavilov and E. Plotkin, see the picture below.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 10:39
Andrei SmolenskyAndrei Smolensky
1,2681122
edited Feb 10 at 15:16
YCor
28.2k483136
edited Feb 10 at 15:16
YCor
28.2k483136
edited Feb 10 at 15:16
YCor
28.2k483136
28.2k483136
answered Feb 10 at 10:39
Andrei SmolenskyAndrei Smolensky
1,2681122
answered Feb 10 at 10:39
Andrei SmolenskyAndrei Smolensky
1,2681122
answered Feb 10 at 10:39
Andrei SmolenskyAndrei Smolensky
1,2681122
1,2681122
$begingroup$
Thanks! So just to be clear $ij$ means $ialpha + j beta$, where $alpha$ is the short root, right?
$endgroup$
– D_S
Feb 10 at 15:58
$begingroup$
@D_S Indeed, as indicated by the presence of $3alpha+2beta$.
$endgroup$
– Andrei Smolensky
Feb 10 at 16:17
add a comment |
$begingroup$
Thanks! So just to be clear $ij$ means $ialpha + j beta$, where $alpha$ is the short root, right?
$endgroup$
– D_S
Feb 10 at 15:58
$begingroup$
@D_S Indeed, as indicated by the presence of $3alpha+2beta$.
$endgroup$
– Andrei Smolensky
Feb 10 at 16:17
$begingroup$
Thanks! So just to be clear $ij$ means $ialpha + j beta$, where $alpha$ is the short root, right?
$endgroup$
– D_S
Feb 10 at 15:58
$begingroup$
Thanks! So just to be clear $ij$ means $ialpha + j beta$, where $alpha$ is the short root, right?
$endgroup$
– D_S
Feb 10 at 15:58
$begingroup$
@D_S Indeed, as indicated by the presence of $3alpha+2beta$.
$endgroup$
– Andrei Smolensky
Feb 10 at 16:17
$begingroup$
@D_S Indeed, as indicated by the presence of $3alpha+2beta$.
$endgroup$
– Andrei Smolensky
Feb 10 at 16:17
add a comment |
$begingroup$
SGA III, Expose XXIII, Section 3.4.
answered Feb 10 at 9:54
Peter McNamaraPeter McNamara
5,5912754
$endgroup$
2
$begingroup$
You can find the Réédition de SGA3 at webusers.imj-prg.fr/~patrick.polo/SGA3
$endgroup$
– David Roberts
Feb 10 at 10:41
add a comment |
$begingroup$
SGA III, Expose XXIII, Section 3.4.
answered Feb 10 at 9:54
Peter McNamaraPeter McNamara
5,5912754
$endgroup$
2
$begingroup$
You can find the Réédition de SGA3 at webusers.imj-prg.fr/~patrick.polo/SGA3
$endgroup$
– David Roberts
Feb 10 at 10:41
add a comment |
$begingroup$
SGA III, Expose XXIII, Section 3.4.
answered Feb 10 at 9:54
Peter McNamaraPeter McNamara
5,5912754
$endgroup$
SGA III, Expose XXIII, Section 3.4.
answered Feb 10 at 9:54
Peter McNamaraPeter McNamara
5,5912754
answered Feb 10 at 9:54
Peter McNamaraPeter McNamara
5,5912754
answered Feb 10 at 9:54
Peter McNamaraPeter McNamara
5,5912754
answered Feb 10 at 9:54
Peter McNamaraPeter McNamara
5,5912754
5,5912754
2
$begingroup$
You can find the Réédition de SGA3 at webusers.imj-prg.fr/~patrick.polo/SGA3
$endgroup$
– David Roberts
Feb 10 at 10:41
add a comment |
2
$begingroup$
You can find the Réédition de SGA3 at webusers.imj-prg.fr/~patrick.polo/SGA3
$endgroup$
– David Roberts
Feb 10 at 10:41
2
2
$begingroup$
You can find the Réédition de SGA3 at webusers.imj-prg.fr/~patrick.polo/SGA3
$endgroup$
– David Roberts
Feb 10 at 10:41
$begingroup$
You can find the Réédition de SGA3 at webusers.imj-prg.fr/~patrick.polo/SGA3
$endgroup$
– David Roberts
Feb 10 at 10:41
add a comment |
$begingroup$
Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No special assumption about the characteristic of the field is needed.) My own later treatment of the classification of simple algebraic groups followed the same method in GTM 21 (Linear Algebraic Groups, Springer, 1975, 33.5). A similar approach was taken in SGA3, as indicated by Peter McNamara. (The later more elegant approach to the classification due to Takeuchi was worked out in Jantzen's book as well as Springer's textbook.)
Note too that the Lie algebra calculations were done in my earlier book GTM 9, first in the characteristic 0 setting: see 19.3.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 15:12
Jim HumphreysJim Humphreys
41.8k494190
$endgroup$
add a comment |
$begingroup$
Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No special assumption about the characteristic of the field is needed.) My own later treatment of the classification of simple algebraic groups followed the same method in GTM 21 (Linear Algebraic Groups, Springer, 1975, 33.5). A similar approach was taken in SGA3, as indicated by Peter McNamara. (The later more elegant approach to the classification due to Takeuchi was worked out in Jantzen's book as well as Springer's textbook.)
Note too that the Lie algebra calculations were done in my earlier book GTM 9, first in the characteristic 0 setting: see 19.3.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 15:12
Jim HumphreysJim Humphreys
41.8k494190
$endgroup$
add a comment |
$begingroup$
Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No special assumption about the characteristic of the field is needed.) My own later treatment of the classification of simple algebraic groups followed the same method in GTM 21 (Linear Algebraic Groups, Springer, 1975, 33.5). A similar approach was taken in SGA3, as indicated by Peter McNamara. (The later more elegant approach to the classification due to Takeuchi was worked out in Jantzen's book as well as Springer's textbook.)
Note too that the Lie algebra calculations were done in my earlier book GTM 9, first in the characteristic 0 setting: see 19.3.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 15:12
Jim HumphreysJim Humphreys
41.8k494190
$endgroup$
Probably the earliest reference is the 1956-58 Chevalley seminar, available online in typed format, which has been republished in 2005 as a typeset book edited by P. Cartier: see Chapter 21. (No special assumption about the characteristic of the field is needed.) My own later treatment of the classification of simple algebraic groups followed the same method in GTM 21 (Linear Algebraic Groups, Springer, 1975, 33.5). A similar approach was taken in SGA3, as indicated by Peter McNamara. (The later more elegant approach to the classification due to Takeuchi was worked out in Jantzen's book as well as Springer's textbook.)
Note too that the Lie algebra calculations were done in my earlier book GTM 9, first in the characteristic 0 setting: see 19.3.
edited Feb 10 at 15:16
YCor
28.2k483136
answered Feb 10 at 15:12
Jim HumphreysJim Humphreys
41.8k494190
edited Feb 10 at 15:16
YCor
28.2k483136
edited Feb 10 at 15:16
YCor
28.2k483136
edited Feb 10 at 15:16
YCor
28.2k483136
28.2k483136
answered Feb 10 at 15:12
Jim HumphreysJim Humphreys
41.8k494190
answered Feb 10 at 15:12
Jim HumphreysJim Humphreys
41.8k494190
answered Feb 10 at 15:12
Jim HumphreysJim Humphreys
41.8k494190
41.8k494190
add a comment |
add a comment |
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3
$begingroup$
If you differentiate and set $x,y$ to zero, surely these expressions will be more familiar, as Lie brackets of root vectors. Then they are known in the literature, using Chevalley bases, or at least their is an algorithm to uncover them.
$endgroup$
– Ben McKay
Feb 10 at 7:02
1
$begingroup$
Some time ago I wrote a memo for myself on split-octonions and $G_2$, madore.org/~david/.misc/20140711-split-octonions.pdf — what you are looking for is the second table on page 2, right? It's easy to compute, but I don't know where you could find it in the published literature.
$endgroup$
– Gro-Tsen
Feb 10 at 9:18
$begingroup$
Correction: $U$ is the unipotent radical of $B$ in this formulation.
$endgroup$
– Jim Humphreys
Feb 10 at 15:16