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数学

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Andrius Kulikauskas

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Lietuvių kalba

Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Book.IntuitingExceptionalRootSystems istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2018 lapkričio 20 d., 21:53 atliko AndriusKulikauskas -
Pakeista 31 eilutė iš:
||{$E_7$}|| and ||{$E_6$}|| are subsets of ||{$E_8$}||.
į:
{$E_7$} and {$E_6$} are subsets of {$E_8$}.
2018 lapkričio 20 d., 21:52 atliko AndriusKulikauskas -
Pridėtos 30-31 eilutės:

||{$E_7$}|| and ||{$E_6$}|| are subsets of ||{$E_8$}||.
2018 lapkričio 20 d., 21:50 atliko AndriusKulikauskas -
Pridėtos 28-29 eilutės:

||{$E_8$}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}, and {$\frac{1}{2}(\sum_{i=1}^{8}(-1)^{a_i}x_i)$} where {$\sum_{i=1}^{8}{a_i} \in 2\mathbb{Z}$}.
2018 lapkričio 20 d., 21:44 atliko AndriusKulikauskas -
Pakeista 19 eilutė iš:
Thus {$G_2$} unites two copies of {$A_2$}.
į:
Thus {$G_2$} is the disjoint union of two copies of {$A_2$}.
2018 lapkričio 20 d., 21:43 atliko AndriusKulikauskas -
Pridėtos 26-27 eilutės:

Thus {$F_4$} has a copy of {$B_4$} as a subset.
2018 lapkričio 20 d., 21:40 atliko AndriusKulikauskas -
Pakeistos 25-30 eilutės iš
į:
||{$F_4$}||{$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i, \frac{1}{2}(\pm x_1 \pm x_2 \pm x_3 \pm x_4)$}||


-------------------------------

'''Literature'''
2018 lapkričio 20 d., 21:25 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
||{$G_2$}||{$ \pm (x_i-x_j)$} where {$i \neq j$}, {$\pm (3x_i - (x_1+x_2+x_3))$} for all {$i$}.
į:
||{$G_2$}||{$ \pm (x_i-x_j)$} where {$i \neq j$}, and {$\pm (3x_i - (x_1+x_2+x_3))$} for all {$i$}.
2018 lapkričio 20 d., 21:25 atliko AndriusKulikauskas -
Pakeista 23 eilutė iš:
||{$G_2$}||{$ \pm (x_i-x_j), \pm (3x_i - (x_1+x_2+x_3))$} where x_j) - (x_j - x_k)$} for all distinct {$ i,j,k \in \{1,2,3\}$}.
į:
||{$G_2$}||{$ \pm (x_i-x_j)$} where {$i \neq j$}, {$\pm (3x_i - (x_1+x_2+x_3))$} for all {$i$}.
2018 lapkričio 20 d., 21:22 atliko AndriusKulikauskas -
Pakeista 27 eilutė iš:
Notes by [[http://people.maths.ox.ac.uk/mcgerty/Lie12/Lecture16.pdf | Lecture 16]] on [[Lie Algebras]] by [[http://people.maths.ox.ac.uk/mcgerty/ | Kevin McGerty]].
į:
Notes by [[http://people.maths.ox.ac.uk/mcgerty/Lie12/Lecture16.pdf | Lecture 16]] on [[http://people.maths.ox.ac.uk/mcgerty/LieNotes.pdf | Lie Algebras]] by [[http://people.maths.ox.ac.uk/mcgerty/ | Kevin McGerty]].
2018 lapkričio 20 d., 21:09 atliko AndriusKulikauskas -
Pridėtos 21-27 eilutės:
Alternatively,

||{$G_2$}||{$ \pm (x_i-x_j), \pm (3x_i - (x_1+x_2+x_3))$} where x_j) - (x_j - x_k)$} for all distinct {$ i,j,k \in \{1,2,3\}$}.



Notes by [[http://people.maths.ox.ac.uk/mcgerty/Lie12/Lecture16.pdf | Lecture 16]] on [[Lie Algebras]] by [[http://people.maths.ox.ac.uk/mcgerty/ | Kevin McGerty]].
2018 lapkričio 20 d., 20:39 atliko AndriusKulikauskas -
Pridėtos 18-19 eilutės:

Thus {$G_2$} unites two copies of {$A_2$}.
2018 lapkričio 20 d., 20:38 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
||{$G_2$}||{$\pm (x_i-x_j), (x_i - x_j) - (x_j - x_k)$} where {$ i,j,k \in \{1,2,3\}$} are distinct.
į:
||{$G_2$}||{$ (x_i-x_j), (x_i - x_j) - (x_j - x_k)$} for all distinct {$ i,j,k \in \{1,2,3\}$}.
2018 lapkričio 20 d., 20:36 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
||{$G_2$}||{$\pm (x_i-x_j), (x_i - x_j) - (x_j - x_k)$} where {$i,j,k \in \{1,2,3}$} are distinct.
į:
||{$G_2$}||{$\pm (x_i-x_j), (x_i - x_j) - (x_j - x_k)$} where {$ i,j,k \in \{1,2,3\}$} are distinct.
2018 lapkričio 20 d., 20:36 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
||{$G_2$}||{$y_2-y_1, y_1$} where {$y_1=(x_2-x_1), y_2=(x_3-x_2)=(x_3-2x_2+x_1$}.||
į:
||{$G_2$}||{$\pm (x_i-x_j), (x_i - x_j) - (x_j - x_k)$} where {$i,j,k \in \{1,2,3}$} are distinct.
2018 lapkričio 20 d., 20:28 atliko AndriusKulikauskas -
Pakeistos 15-18 eilutės iš
The five exceptional root systems are described in [[https://en.wikipedia.org/wiki/Root_system | Wikipedia's article on root systems]].
į:
The five exceptional root systems are described in [[https://en.wikipedia.org/wiki/Root_system | Wikipedia's article on root systems]].

||{$G_2$}||{$y_2-y_1, y_1$} where {$y_1=(x_2-x_1), y_2=(x_3-x_2)=(x_3-2x_2+x_1$}.||
2018 lapkričio 20 d., 13:35 atliko AndriusKulikauskas -
Pakeistos 13-15 eilutės iš
||{$D_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}||
į:
||{$D_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}||

The five exceptional root systems are described in [[https://en.wikipedia.org/wiki/Root_system | Wikipedia's article on root systems]].
2018 lapkričio 20 d., 13:34 atliko AndriusKulikauskas -
Pridėtos 1-13 eilutės:
>>bgcolor=#E9F5FC<<
-------------
See: [[Math notebook]], [[Classical Lie groups]], [[Classical Lie Root systems]], [[Intuiting classical root systems]]

'''Intuit the five exceptional root systems.'''
-------------
>><<

The four classical root systems are as follows, where throughout, {$i>j$}:
||{$A_n$}||{$\pm (x_i-x_j)$}||
||{$B_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j), \pm x_i$}||
||{$C_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j), \pm 2x_i$}||
||{$D_n$}||{$\pm (x_i-x_j), \pm (x_i+x_j)$}||

IntuitingExceptionalRootSystems


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