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Introduction E9F5FC

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Questions FFFFC0

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## Book.IntuitingClassicalRootSystems istorija

2018 gruodžio 30 d., 16:54 atliko AndriusKulikauskas -
Pakeista 25 eilutė iš:
Consider this as a problem of the ways of coding counting. And coding is counted by a minus sign. This codes counting both forwards and backwards. But how might the two directions be related? The second direction can be coded with a plus sign. Then the two can be related in three different ways.
į:
Consider this as a problem of the ways of coding counting. And coding is counted by a minus sign. This codes counting both forwards and backwards. But how might the two directions coincide? There are four possibilities. The second direction can be coded with a plus sign. Then the two can be related in three different ways.
2018 gruodžio 30 d., 16:24 atliko AndriusKulikauskas -
Pridėtos 22-25 eilutės:

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Consider this as a problem of the ways of coding counting. And coding is counted by a minus sign. This codes counting both forwards and backwards. But how might the two directions be related? The second direction can be coded with a plus sign. Then the two can be related in three different ways.
2018 gruodžio 09 d., 08:21 atliko AndriusKulikauskas -
2018 gruodžio 09 d., 08:18 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
{$C_n$} includes the case {$i=j$}. {$B_n$} and Dn do not. But {$B_n$} fakes it by having a zero. This yields the same lattice as {$C_n$}. But the root systems bring out the difference. So {$C_n$} gives the conditions needed for this identity. The root systems are building up the context for such an identity.
į:
{$C_n$} includes the case {$i=j$}. {$B_n$} and {$D_n$} do not. But {$B_n$} fakes it by having a zero. This yields the same lattice as {$C_n$}. But the root systems bring out the difference. So {$C_n$} gives the conditions needed for this identity. The root systems are building up the context for such an identity.
2018 gruodžio 09 d., 08:14 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
{$C_n$} includes the case i=j. {$B_n$} and Dn do not. But {$B_n$} fakes it by having a zero. This yields the same lattice as {$C_n$}. But the root systems bring out the difference. So {$C_n$} gives the conditions needed for this identity. The root systems are building up the context for such an identity.
į:
{$C_n$} includes the case {$i=j$}. {$B_n$} and Dn do not. But {$B_n$} fakes it by having a zero. This yields the same lattice as {$C_n$}. But the root systems bring out the difference. So {$C_n$} gives the conditions needed for this identity. The root systems are building up the context for such an identity.
2018 gruodžio 09 d., 08:14 atliko AndriusKulikauskas -
Pakeistos 17-20 eilutės iš
Cn includes the case i=j. Bn and Dn do not. But Bn fakes it by having a zero. This yields the same lattice as Cn. But the root systems bring out the difference. So Cn gives the conditions needed for this identity. The root systems are building up the context for such an identity.

In particular, Bn, Cn, Dn have two conjugate sets of roots because they are establishing the symmetries underlying a pair of expansions linked by a comparison, an identity. Thus we have a value on the left which is being compared to a value on the right. In the case of Cn we may have identity, an equal sign. In the case of Bn we may have a zero. In the case of Dn we have neither.
į:
{$C_n$} includes the case i=j. {$B_n$} and Dn do not. But {$B_n$} fakes it by having a zero. This yields the same lattice as {$C_n$}. But the root systems bring out the difference. So {$C_n$} gives the conditions needed for this identity. The root systems are building up the context for such an identity.

In particular, {$B_n$}, {$C_n$}, {$D_n$} have two conjugate sets of roots because they are establishing the symmetries underlying a pair of expansions linked by a comparison, an identity. Thus we have a value on the left which is being compared to a value on the right. In the case of {$C_n$} we may have identity, an equal sign. In the case of {$B_n$} we may have a zero. In the case of {$D_n$} we have neither.
2018 lapkričio 20 d., 13:34 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
See: [[Math notebook]], [[Classical Lie groups]], [[Classical Lie Root systems]]
į:
See: [[Math notebook]], [[Classical Lie groups]], [[Classical Lie Root systems]], [[Intuiting exceptional root systems]]
2018 lapkričio 20 d., 13:32 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
See: [[Math notebook]], [[Classical Lie groups]], [[Classical Root systems]]
į:
See: [[Math notebook]], [[Classical Lie groups]], [[Classical Lie Root systems]]
2018 lapkričio 20 d., 13:32 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
See: [[Math notebook]], [[Classical Lie groups]]
į:
See: [[Math notebook]], [[Classical Lie groups]], [[Classical Root systems]]
2018 lapkričio 20 d., 13:30 atliko AndriusKulikauskas -
Pakeista 19 eilutė iš:
In particular, Bn, Cn, Dn have two conjugate sets of roots because they are establishing the symmetries underlying a pair of expansions linked by a comparison, an identity. Thus we have a value on the left which is being compared to a value on the right. In the case of Cn we may have identity, an equal sign. In the case of Bn we may have a zero.
į:
In particular, Bn, Cn, Dn have two conjugate sets of roots because they are establishing the symmetries underlying a pair of expansions linked by a comparison, an identity. Thus we have a value on the left which is being compared to a value on the right. In the case of Cn we may have identity, an equal sign. In the case of Bn we may have a zero. In the case of Dn we have neither.
2018 lapkričio 20 d., 13:29 atliko AndriusKulikauskas -
Pridėtos 18-19 eilutės:

In particular, Bn, Cn, Dn have two conjugate sets of roots because they are establishing the symmetries underlying a pair of expansions linked by a comparison, an identity. Thus we have a value on the left which is being compared to a value on the right. In the case of Cn we may have identity, an equal sign. In the case of Bn we may have a zero.
2018 lapkričio 20 d., 13:09 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
Cn includes the case i=j. Bn and Dn do not. But Bn fakes it by having a zero. This yields the same lattice as Cn. But the root systems bring out the difference.
į:
Cn includes the case i=j. Bn and Dn do not. But Bn fakes it by having a zero. This yields the same lattice as Cn. But the root systems bring out the difference. So Cn gives the conditions needed for this identity. The root systems are building up the context for such an identity.
2018 lapkričio 20 d., 13:09 atliko AndriusKulikauskas -
Pridėtos 16-17 eilutės:

Cn includes the case i=j. Bn and Dn do not. But Bn fakes it by having a zero. This yields the same lattice as Cn. But the root systems bring out the difference.
2018 lapkričio 20 d., 13:04 atliko AndriusKulikauskas -
Pakeistos 15-18 eilutės iš
We thus see that all classical root systems contain the roots {$\pm (x_i-x_j)$}.
į:
We thus see that all classical root systems contain the roots {$\pm (x_i-x_j)$}.

The Dynkin diagram for {$D_2$} is a pair of dots. This brings to mind the two conjugate roots of -1. Thus there is this inherent symmetry within {$D_n$}.
2018 lapkričio 20 d., 12:56 atliko AndriusKulikauskas -
Pakeistos 13-15 eilutės iš
||{$D_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}||
į:
||{$D_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}||

We thus see that all classical root systems contain the roots {$\pm (x_i-x_j)$}.
2018 lapkričio 20 d., 12:55 atliko AndriusKulikauskas -
Pakeista 9 eilutė iš:
The four classical root systems are as follows, where throughout, {$i\neq j$}:
į:
The four classical root systems are as follows, where throughout, {$i>j$}:
2018 lapkričio 20 d., 12:53 atliko AndriusKulikauskas -
Pakeista 9 eilutė iš:
The four classical root systems are as follows, where throughout, {$i\neq j}: į: The four classical root systems are as follows, where throughout, {$i\neq j$}: 2018 lapkričio 20 d., 12:53 atliko AndriusKulikauskas - Pakeistos 11-12 eilutės iš ||{$B_n$}||{$\pm (x_i-x_j), \pm x_i$}|| ||{$C_n$}||{$\pm (x_i-x_j), \pm 2x_i$}|| į: ||{$B_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i$}|| ||{$C_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j), \pm 2x_i$}|| 2018 lapkričio 20 d., 12:53 atliko AndriusKulikauskas - Pakeistos 10-13 eilutės iš ||{$A_n$}}||{$\pm (x_i-x_j)||
||{$B_n$}}||{$\pm (x_i-x_j), \pm x_i$}||
||{$C_n$}}||{$\pm (x_i-x_j), \pm 2x_i$}||
||{$D_n$}}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}||
į:
||{$A_n$}||{$\pm (x_i-x_j)$}||
||{$B_n$}||{$\pm (x_i-x_j), \pm x_i$}||
||{$C_n$}||{$\pm (x_i-x_j), \pm 2x_i$}||
||{$D_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}||
2018 lapkričio 20 d., 12:52 atliko AndriusKulikauskas -
Pakeistos 3-4 eilutės iš
See: [[Math notebook]]
į:
See: [[Math notebook]], [[Classical Lie groups]]
Pakeistos 9-13 eilutės iš
The four classical root systems are as follows:
*
*
*
*
į:
The four classical root systems are as follows, where throughout, {$i\neq j}: ||{$A_n$}}||{$\pm (x_i-x_j)||
||{$B_n$}}||{$\pm (x_i-x_j), \pm x_i$}||
||{$C_n$}}||{$\pm (x_i-x_j), \pm 2x_i$}||
||{$D_n$}}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}||
2018 lapkričio 20 d., 12:09 atliko AndriusKulikauskas -
Pridėtos 1-13 eilutės:
>>bgcolor=#E9F5FC<<
-------------
See: [[Math notebook]]

'''Intuit the four classical root systems.'''
-------------
>><<

The four classical root systems are as follows:
*
*
*
*

#### IntuitingClassicalRootSystems

Naujausi pakeitimai

 Puslapis paskutinį kartą pakeistas 2018 gruodžio 30 d., 16:54