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数学

Discovery

Andrius Kulikauskas

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Lietuvių kalba

Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

Book.LieAlgebrasChoiceTemplates istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2019 balandžio 30 d., 14:23 atliko AndriusKulikauskas -
Pridėtos 46-49 eilutės:

'''Orthogonalities'''

* Real (orthogonal), complex (unitary), quaternion (symplectic)
2019 balandžio 16 d., 15:06 atliko AndriusKulikauskas -
Pakeistos 49-51 eilutės iš
* [[https://ncatlab.org/nlab/show/geometric+shape+for+higher+structures | Geometric shape for higher structures]
į:
* [[https://ncatlab.org/nlab/show/geometric+shape+for+higher+structures | Geometric shape for higher structures]]
Pridėta 54 eilutė:
* Ask: What is a "globe" and a "globular set"?
2019 balandžio 16 d., 14:53 atliko AndriusKulikauskas -
Pakeistos 45-49 eilutės iš
Not satifactory. :(
į:
Not satisfactory. :(

'''Readings'''

* [[https://ncatlab.org/nlab/show/geometric+shape+for+higher+structures | Geometric shape for higher structures]
2018 lapkričio 02 d., 20:27 atliko AndriusKulikauskas -
Pakeista 50 eilutė iš:
* How are the Weyl groups related to the choice templates? Use the relationships between the choice templates and the Weyl groups, and the Weyl groups and the root systems, to relate the root systems and the choice systems.
į:
* How are the Weyl groups related to the choice templates? Use the relationships between the choice templates and the Weyl groups, and the Weyl groups and the root systems, to relate the root systems and the choice systems. Note the problematic identification as I had expected to identify the coordinate systems with {$C_n$} but their symmetry group instead may match that of {$D_n$}.
2018 lapkričio 02 d., 20:20 atliko AndriusKulikauskas -
Ištrinta 49 eilutė:
* What does a reflection actually mean in terms of the root systems?
Ištrinta 50 eilutė:
* Why is the order of the Weyl group {$C_n$} divided by 2?
2018 lapkričio 02 d., 13:34 atliko AndriusKulikauskas -
Pakeistos 31-33 eilutės iš
* {$B_n$}: By way of the hyperplane {$\pm(0-e_1)$}. Swapping {$e_1$} with {$0$} as with the cross polytopes. This takes place in the framework {$e_1 - 0$} so the minus sign is a feature of the pairing.
* {$D_n$}: By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2--e_1)$}. Swapping {$e_1$} with {$e_2$} and then {$e_2$} with {$-e_1$}
* {$C_n$}: By way of the hyperplane {$\pm(e_1--e_1)$}. Swapping {$e_1$} with {$-e_1$} directly. Then there is one minus sign in the pairing, and one carried by {$-e_1$}.
į:
* {$B_n$}: By swapping with 0. By way of the hyperplane {$\pm(0-e_1)$}. Swapping {$e_1$} with {$0$} as with the cross polytopes. This takes place in the framework {$e_1 - 0$} so the minus sign is a feature of the pairing.
* {$D_n$}: By swapping with any root {$e_2$}. By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2--e_1)$}. Swapping {$e_1$} with {$e_2$} and then {$e_2$} with {$-e_1$}
* {$C_n$}: By swapping directly with {$-e_1$}. By way of the hyperplane {$\pm(e_1--e_1)$}. Swapping {$e_1$} with {$-e_1$} directly. Then there is one minus sign in the pairing, and one carried by {$-e_1$}.
2018 lapkričio 02 d., 13:33 atliko AndriusKulikauskas -
Pakeista 32 eilutė iš:
* {$D_n$}: By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2--e_1)$}. Swapping {$e_1$} with {$-e_2$} and then {$-e_2$} with {$-e_1$}
į:
* {$D_n$}: By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2--e_1)$}. Swapping {$e_1$} with {$e_2$} and then {$e_2$} with {$-e_1$}
2018 lapkričio 02 d., 13:32 atliko AndriusKulikauskas -
Pakeista 32 eilutė iš:
* {$D_n$}: By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2+e_1)$}. Swapping {$e_1$} with {$-e_2$} and then {$-e_2$} with {$-e_1$}
į:
* {$D_n$}: By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2--e_1)$}. Swapping {$e_1$} with {$-e_2$} and then {$-e_2$} with {$-e_1$}
2018 lapkričio 02 d., 13:30 atliko AndriusKulikauskas -
Pakeista 30 eilutė iš:
There are three ways of defining the reflections {$e_1 \leftrightarrow -e_1$}:
į:
There are three ways of defining the reflections {$e_1 \Leftrightarrow -e_1$}:
2018 lapkričio 02 d., 13:30 atliko AndriusKulikauskas -
Pakeistos 26-27 eilutės iš
||{$\pm(e_2-e_1)$}||{$e_1 \leftrightarrow e_2, -e_1 \Leftrightarrow -e_2$}||
||{$\pm(e_2+e_1)$}||{$e_1 \leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}||
į:
||{$\pm(e_2-e_1)$}||{$e_1 \Leftrightarrow e_2, -e_1 \Leftrightarrow -e_2$}||
||{$\pm(e_2+e_1)$}||{$e_1 \Leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}||
2018 lapkričio 02 d., 13:29 atliko AndriusKulikauskas -
Pakeistos 26-28 eilutės iš
||{$\pm(e_2-e_1)$}||{$e_1 \leftrightarrow e_2, -e_1 \leftrightarrow -e_2$}||
||{$\pm(e_2+e_1)$}||{$e_1 \leftrightarrow -e_2, -e_1 \leftrightarrow e_2$}||
||{$\pm(e_1)$}||{$e_1 \leftrightarrow -e_1$}||
į:
||{$\pm(e_2-e_1)$}||{$e_1 \leftrightarrow e_2, -e_1 \Leftrightarrow -e_2$}||
||{$\pm(e_2+e_1)$}||{$e_1 \leftrightarrow -e_2, -e_1 \Leftrightarrow e_2$}||
||{$\pm(e_1)$}||{$e_1 \Leftrightarrow -e_1$}||
2018 lapkričio 01 d., 16:07 atliko AndriusKulikauskas -
Pridėta 50 eilutė:
* What does a reflection actually mean in terms of the root systems?
2018 lapkričio 01 d., 16:06 atliko AndriusKulikauskas -
Pridėtos 36-37 eilutės:

The sign can be attributed to the element or to the pairing template. So try to think that through.
2018 lapkričio 01 d., 16:04 atliko AndriusKulikauskas -
Pakeistos 40-42 eilutės iš
* {$B_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X \eq Y$} is ok.
* {$D_n$}: {$X, Y$} are taken from {${\dots -e_2
,-e_1,e_1,e_2 \dots e_n}$} and {$X \neq Y$}.
* {$C_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and ...
į:
* {$B_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X$} and {$Y$} may refer to the same number so long as they have the same sign, in which case they are understood to coincide.
* {$D_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X$} and {$Y$} must refer to different numbers.
* {$C_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X$} and {$Y$} may refer to the same number so long as they have the same sign, in which case they are understood to add.
Not satifactory. :(
2018 lapkričio 01 d., 15:54 atliko AndriusKulikauskas -
Pakeistos 37-41 eilutės iš
Consider {$X-Y$} as a template.
* {$A_n$}: {$X \neq Y$} are taken from
{${e_1,e_2 \dots e_n}$}.
*
{$B_n$}:
*
{$D_n$}:
*
{$C_n$}:
į:
Consider {$X-Y$} as a template where the minus sign imposes an ordering.
*
{$A_n$}: {$X, Y$} are taken from {${e_1,e_2 \dots e_n}$} and {$X \neq Y$} is ruled out by {$X-Y=0$}.
Consider {${X,Y}$} as a template where the set indicates inclusion with no ordering.
* {$B_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X \eq Y$} is ok.
* {$D_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and {$X \neq Y$}.
* {$C_n$}: {$X, Y$} are taken from {${\dots -e_2,-e_1,e_1,e_2 \dots e_n}$} and ...
2018 lapkričio 01 d., 15:29 atliko AndriusKulikauskas -
Pakeistos 37-38 eilutės iš
Consider {$A-B$} as a template.
* {$A_n$}: {$A \neq B$} are taken from {${e_1,e_2 \dots e_n}$}.
į:
Consider {$X-Y$} as a template.
* {$A_n$}: {$X \neq Y$} are taken from {${e_1,e_2 \dots e_n}$}.
* {$B_n$}:
* {$D_n$}:
* {$C_n$}:
2018 lapkričio 01 d., 15:26 atliko AndriusKulikauskas -
Pakeista 38 eilutė iš:
* {$A_n$}: {$A /neq B$} are taken from {${1,2 /dots n}$}.
į:
* {$A_n$}: {$A \neq B$} are taken from {${e_1,e_2 \dots e_n}$}.
2018 lapkričio 01 d., 15:25 atliko AndriusKulikauskas -
Pakeista 38 eilutė iš:
* {$A_n$}: A and B are taken from {${1,2/cdot n}$}.
į:
* {$A_n$}: {$A /neq B$} are taken from {${1,2 /dots n}$}.
2018 lapkričio 01 d., 15:24 atliko AndriusKulikauskas -
Pridėtos 34-38 eilutės:

'''Pairing template'''

Consider {$A-B$} as a template.
* {$A_n$}: A and B are taken from {${1,2/cdot n}$}.
2018 lapkričio 01 d., 15:20 atliko AndriusKulikauskas -
Pridėtos 41-42 eilutės:
* What geometrical information do the lengths of the roots carry, above and beyond the Weyl group?
* What is the geometric relation between {$B_n$} and {$C_n$}?
2018 lapkričio 01 d., 15:13 atliko AndriusKulikauskas -
Pakeistos 31-33 eilutės iš
* {$B_n$}: By way of the hyperplane {$\pm(0-e_1)$}.
* {$D_n$}: By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2+e_1)$}
* {$C_n$}: By way of the hyperplane {$\pm(e_1--e_1)$}.
į:
* {$B_n$}: By way of the hyperplane {$\pm(0-e_1)$}. Swapping {$e_1$} with {$0$} as with the cross polytopes. This takes place in the framework {$e_1 - 0$} so the minus sign is a feature of the pairing.
*
{$D_n$}: By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2+e_1)$}. Swapping {$e_1$} with {$-e_2$} and then {$-e_2$} with {$-e_1$}
* {$C_n$}: By way of the hyperplane {$\pm(e_1--e_1)$}. Swapping {$e_1$} with {$-e_1$} directly. Then there is one minus sign in the pairing, and one carried by {$-e_1
$}.
2018 lapkričio 01 d., 15:06 atliko AndriusKulikauskas -
Pakeista 31 eilutė iš:
* {$B_n$}: By way of the hyperplane {$\pm(e_1)$}.
į:
* {$B_n$}: By way of the hyperplane {$\pm(0-e_1)$}.
Pakeista 33 eilutė iš:
* {$C_n$}: By way of the hyperplane {$\pm(e_1+e_1)$}.
į:
* {$C_n$}: By way of the hyperplane {$\pm(e_1--e_1)$}.
2018 lapkričio 01 d., 15:05 atliko AndriusKulikauskas -
Pakeistos 31-33 eilutės iš
* {$B_n$}By way of the hyperplane {$\pm(e_1)$}.
* {$D_n$}By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2+e_1)$}
* {$C_n$}By way of the hyperplane {$\pm(e_1+e_1)$}.
į:
* {$B_n$}: By way of the hyperplane {$\pm(e_1)$}.
* {$D_n$}: By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2+e_1)$}
* {$C_n$}: By way of the hyperplane {$\pm(e_1+e_1)$}.
2018 lapkričio 01 d., 15:05 atliko AndriusKulikauskas -
Pakeistos 30-35 eilutės iš
į:
There are three ways of defining the reflections {$e_1 \leftrightarrow -e_1$}:
* {$B_n$}By way of the hyperplane {$\pm(e_1)$}.
* {$D_n$}By composing the hyperplanes {$\pm(e_2-e_1)$} and {$\pm(e_2+e_1)$}
* {$C_n$}By way of the hyperplane {$\pm(e_1+e_1)$}.
Pakeista 39 eilutė iš:
* Why is the order of the Weyl group {$D_n$} divided by 2?
į:
* Why is the order of the Weyl group {$C_n$} divided by 2?
2018 lapkričio 01 d., 14:51 atliko AndriusKulikauskas -
Pridėtos 24-28 eilutės:

||'''Hyperplane =''' {$\pm$} '''Root'''||'''Transposition'''||
||{$\pm(e_2-e_1)$}||{$e_1 \leftrightarrow e_2, -e_1 \leftrightarrow -e_2$}||
||{$\pm(e_2+e_1)$}||{$e_1 \leftrightarrow -e_2, -e_1 \leftrightarrow e_2$}||
||{$\pm(e_1)$}||{$e_1 \leftrightarrow -e_1$}||
2018 lapkričio 01 d., 14:44 atliko AndriusKulikauskas -
Pakeista 30 eilutė iš:
* Interpret the three ways of defining the reflection {$e_{i} \leftrightarrow -e_{i}}. How is this related to turning around the counting?
į:
* Interpret the three ways of defining the reflection {$e_{i} \leftrightarrow -e_{i}$}. How is this related to turning around the counting?
2018 lapkričio 01 d., 14:44 atliko AndriusKulikauskas -
Pridėta 30 eilutė:
* Interpret the three ways of defining the reflection {$e_{i} \leftrightarrow -e_{i}}. How is this related to turning around the counting?
2018 lapkričio 01 d., 14:40 atliko AndriusKulikauskas -
Pridėtos 23-25 eilutės:
The Weyl group of a root system is the group generated by the reflections across the hyperplanes of the roots. So the Weyl group is given by the directions of the roots but not their lengths.
Pakeistos 28-29 eilutės iš
* How are the Weyl groups related to the root systems? And how are the Weyl groups related to the choice systems? Use these relationships to relate the root systems and the choice systems.
į:
* How are the Weyl groups related to the choice templates? Use the relationships between the choice templates and the Weyl groups, and the Weyl groups and the root systems, to relate the root systems and the choice systems.
* Why is the order of the Weyl group {$D_n$} divided by 2?
2018 lapkričio 01 d., 13:17 atliko AndriusKulikauskas -
Pakeista 24 eilutė iš:
į:
------------------
Pakeista 26 eilutė iš:
į:
------------------
2018 lapkričio 01 d., 13:14 atliko AndriusKulikauskas -
Pakeistos 20-27 eilutės iš
Attach:AlternateDiagonal.png
į:
Attach:AlternateDiagonal.png


>>bgcolor=#FFFFC0<<

* How are the Weyl groups related to the root systems? And how are the Weyl groups related to the choice systems? Use these relationships to relate the root systems and the choice systems.

>><<
2018 lapkričio 01 d., 13:09 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
The root systems of the classical Lie algebras all include the roots of {$A_{n}$}: {$\pm (x_i-x_j)$}. The other root systems all include the additional roots of {$D_{n}$}: {$\pm (x_i+x_j), i\neq j$}. Additionally, {$B_{n}$} has {$\pm x_i$} and {$C_{n}$} has {$\pm (x_i+x_i)$}.
į:
The root systems of the classical Lie algebras all include the roots of {$A_{n}$}: {$\pm (x_i-x_j)$}. The other root systems all include the additional roots of {$D_{n}$}: {$\pm (x_i+x_j), i\neq j$}. Finally, {$B_{n}$} includes {$\pm x_i$} and {$C_{n}$} includes {$\pm (x_i+x_i)$}.
2018 lapkričio 01 d., 13:08 atliko AndriusKulikauskas -
Pakeista 17 eilutė iš:
The root systems of the classical Lie algebras all include the roots of {$A_{n}$}: {$\pm (x_i-x_j)$}. The other root systems all include the additional roots of {$D_{n}$}: {$\pm (x_i+x_j), i\neq j$}. Additionally, {$B_{n}$} has {$\pm x_i$} and {$C_{n}$} has {$\pm x_i+x_i$}.
į:
The root systems of the classical Lie algebras all include the roots of {$A_{n}$}: {$\pm (x_i-x_j)$}. The other root systems all include the additional roots of {$D_{n}$}: {$\pm (x_i+x_j), i\neq j$}. Additionally, {$B_{n}$} has {$\pm x_i$} and {$C_{n}$} has {$\pm (x_i+x_i)$}.
2018 lapkričio 01 d., 13:08 atliko AndriusKulikauskas -
Pakeistos 17-21 eilutės iš
The classical Lie algebras all include {$\pm (x_i-x_j)$}. Additionally, we have:
*
{$\pm (x_i+x_j), \pm x_i$}
* {$\pm (x_i+x_j), \pm 2x_i$}
* {$\pm (x_i+x_j), i\neq j
$}
į:
The root systems of the classical Lie algebras all include the roots of {$A_{n}$}: {$\pm (x_i-x_j)$}. The other root systems all include the additional roots of {$D_{n}$}: {$\pm (x_i+x_j), i\neq j$}. Additionally, {$B_{n}$} has {$\pm x_i$} and {$C_{n}$} has {$\pm x_i+x_i$}.
2018 lapkričio 01 d., 13:02 atliko AndriusKulikauskas -
Pridėtos 16-20 eilutės:

The classical Lie algebras all include {$\pm (x_i-x_j)$}. Additionally, we have:
* {$\pm (x_i+x_j), \pm x_i$}
* {$\pm (x_i+x_j), \pm 2x_i$}
* {$\pm (x_i+x_j), i\neq j$}
2018 lapkričio 01 d., 12:32 atliko AndriusKulikauskas -
Pridėta 11 eilutė:
2018 lapkričio 01 d., 12:31 atliko AndriusKulikauskas -
Pakeistos 11-14 eilutės iš
* {$ \prod_{i}(\varnothing + \leftrightarrow_{i})n $} Simplexes.
* {$ ( + ( + ))^n $} Cross-polytopes.
* {$ (( + )+ ↔)^n $} Cubes.
* {$ ( + )^n $} Cube slices.
į:
* {$ \prod_{i}(\varnothing + \leftrightarrow_{i}) $} Simplexes.
* {$ \prod_{i}(\varnothing + (\leftarrow_{i} + \rightarrow_{i})) $} Cross-polytopes.
* {$ \prod_{i}((\leftarrow_{i} + \rightarrow_{i})+\leftrightarrow_{i}) $} Cubes.
* {$ \prod_{i}(\leftarrow_{i} + \rightarrow_{i}) $} Cube slices.
2018 lapkričio 01 d., 12:27 atliko AndriusKulikauskas -
Pakeista 11 eilutė iš:
* {$ ( + )^n $} Simplexes.
į:
* {$ \prod_{i}(\varnothing + \leftrightarrow_{i})n $} Simplexes.
2018 lapkričio 01 d., 12:19 atliko AndriusKulikauskas -
Pakeistos 10-14 eilutės iš
į:
There are four choice templates:
* {$ (∅ + ↔)^n $} Simplexes.
* {$ (∅ + (← + →))^n $} Cross-polytopes.
* {$ ((← + →)+ ↔)^n $} Cubes.
* {$ (← + →)^n $} Cube slices.
2018 lapkričio 01 d., 12:13 atliko AndriusKulikauskas -
Pakeistos 8-14 eilutės iš
>><<
į:
>><<





Attach:AlternateDiagonal.png
2018 lapkričio 01 d., 11:59 atliko AndriusKulikauskas -
Pakeista 3 eilutė iš:
See: [[Lie theory]]
į:
See: [[Lie theory]], [[Classical Lie groups]]
2018 lapkričio 01 d., 11:59 atliko AndriusKulikauskas -
Pakeistos 1-8 eilutės iš
See: [[Lie theory]]
į:
>>bgcolor=#E9F5FC<<
-----------

See: [[Lie theory]]

'''Interpret the four classical Lie algebras in terms of the four choice templates.'''

-----------
>><<
2018 lapkričio 01 d., 11:58 atliko AndriusKulikauskas -
Pridėta 1 eilutė:
See: [[Lie theory]]

LieAlgebrasChoiceTemplates


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