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nLab > Latest Changes: Clifford algebra

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJan 27th 2017
- (edited Jan 27th 2017)
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Author: Urs
Format: MarkdownItexThe entry _[[Clifford algebra]]_ used to state the classification and Bott periodicty over the complex numbers, but not over the real numbers. I have added in now the relevant statements, straight from Lawson-Michelson: * _[Classification over the real numbers](https://ncatlab.org/nlab/show/Clifford+algebra#ClassificationOverTheRealNumbers)_. Just the bare statements so far.
The entry Clifford algebra used to state the classification and Bott periodicty over the complex numbers, but not over the real numbers. I have added in now the relevant statements, straight from Lawson-Michelson:
- Classification over the real numbers.
Just the bare statements so far.
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeApr 26th 2020
- (edited Apr 26th 2020)
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Author: Urs
Format: MarkdownItexadded some more publication data to the references, added pointer to one more brief introduction, and reorganized the items slightly <a href="https://ncatlab.org/nlab/revision/diff/Clifford+algebra/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/Clifford+algebra/23">v23</a>, <a href="https://ncatlab.org/nlab/show/Clifford+algebra">current</a>

added some more publication data to the references, added pointer to one more brief introduction, and reorganized the items slightly

diff, v23, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeMay 5th 2020
- PermaLink
Author: Urs
Format: MarkdownItexadded ([here](https://ncatlab.org/nlab/show/Clifford+algebra#RelationToOrthogonalLieAlgebras)) the statement and its proof (i.e. the elementary computation) of the Clifford representation of the orthognal Lie algebras <a href="https://ncatlab.org/nlab/revision/diff/Clifford+algebra/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/Clifford+algebra/25">v25</a>, <a href="https://ncatlab.org/nlab/show/Clifford+algebra">current</a>

added (here) the statement and its proof (i.e. the elementary computation) of the Clifford representation of the orthognal Lie algebras

diff, v25, current
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeSep 4th 2021
- (edited Sep 4th 2021)
- PermaLink
Author: Urs
Format: MarkdownItexhave added these two pointers: * [[Robert A. Wilson]], *A group-theorist's perspective on symmetry groups in physics* ([arXiv:2009.14613](https://arxiv.org/abs/2009.14613)) * [[Robert A. Wilson]], *On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics*, Adv. Appl. Clifford Algebras 31, 59 (2021) ([doi:10.1007/s00006-021-01160-5]( https://doi.org/10.1007/s00006-021-01160-5)) Will also add these to *[[standard model of particle physics]]*. <a href="https://ncatlab.org/nlab/revision/diff/Clifford+algebra/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/Clifford+algebra/27">v27</a>, <a href="https://ncatlab.org/nlab/show/Clifford+algebra">current</a>
have added these two pointers:
- Robert A. Wilson, A group-theorist’s perspective on symmetry groups in physics (arXiv:2009.14613)
- Robert A. Wilson, On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics, Adv. Appl. Clifford Algebras 31, 59 (2021) (doi:10.1007/s00006-021-01160-5)
Will also add these to standard model of particle physics.

diff, v27, current
- CommentRowNumber5.
- CommentAuthornLab edit announcer
- CommentTimeOct 26th 2022
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Author: nLab edit announcer
Format: MarkdownItexIn quaternions i, j, and k square to -1. Therefore it should be $Cl_{3,0} \simeq \mathbb{H}$, instead of $Cl_{2,0} \simeq \mathbb{H}$, right? Anonymous <a href="https://ncatlab.org/nlab/revision/diff/Clifford+algebra/29">diff</a>, <a href="https://ncatlab.org/nlab/revision/Clifford+algebra/29">v29</a>, <a href="https://ncatlab.org/nlab/show/Clifford+algebra">current</a>

In quaternions i, j, and k square to -1. Therefore it should be $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Cl</mi> <mrow><mn>3</mn><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">Cl_{3,0} \simeq \mathbb{H}</annotation></semantics></math>$ , instead of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Cl</mi> <mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">Cl_{2,0} \simeq \mathbb{H}</annotation></semantics></math>$ , right?

Anonymous

diff, v29, current
- CommentRowNumber6.
- CommentAuthorDavid_Corfield
- CommentTimeOct 26th 2022
- (edited Oct 26th 2022)
- PermaLink
Author: David_Corfield
Format: MarkdownItexIsn't it that from the two basis elements of $\mathbb{R}^{(2,0)}$, $e_1$ and $e_2$ squaring to $-1$, the Clifford algebra is generated by $\{1, e_1, e_2, e_1 e_2\}$, the latter three generators squaring to $-1$?

Isn’t it that from the two basis elements of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{(2,0)}</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>e</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">e_1</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>e</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">e_2</annotation></semantics></math>$ squaring to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math>$ , the Clifford algebra is generated by $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><msub><mi>e</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>1</mn></msub><msub><mi>e</mi> <mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1, e_1, e_2, e_1 e_2\}</annotation></semantics></math>$ , the latter three generators squaring to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math>$ ?
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeOct 26th 2022
- PermaLink
Author: Urs
Format: MarkdownItexYes. Ideally the entry would explain how the identification works. (Myself, i have no time right now, maybe later...)

Yes. Ideally the entry would explain how the identification works. (Myself, i have no time right now, maybe later…)
- CommentRowNumber8.
- CommentAuthorDavid_Corfield
- CommentTimeOct 26th 2022
- PermaLink
Author: David_Corfield
Format: MarkdownItexI've rolled back to version #28.

I’ve rolled back to version #28.
- CommentRowNumber9.
- CommentAuthorDavid_Corfield
- CommentTimeOct 26th 2022
- PermaLink
Author: David_Corfield
Format: MarkdownItexAdded a small clarification regarding #6. <a href="https://ncatlab.org/nlab/revision/diff/Clifford+algebra/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/Clifford+algebra/31">v31</a>, <a href="https://ncatlab.org/nlab/show/Clifford+algebra">current</a>

Added a small clarification regarding #6.

diff, v31, current
- CommentRowNumber10.
- CommentAuthornLab edit announcer
- CommentTimeMar 28th 2023
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Author: nLab edit announcer
Format: MarkdownItexAdded a small clarification regarding #6. Aurelius <a href="https://ncatlab.org/nlab/revision/diff/Clifford+algebra/32">diff</a>, <a href="https://ncatlab.org/nlab/revision/Clifford+algebra/32">v32</a>, <a href="https://ncatlab.org/nlab/show/Clifford+algebra">current</a>

Added a small clarification regarding #6.

Aurelius

diff, v32, current
- CommentRowNumber11.
- CommentAuthornLab edit announcer
- CommentTimeAug 19th 2024
- PermaLink
Author: nLab edit announcer
Format: MarkdownItexAdded [[algebra - contents]] to sidebar Anonymouse <a href="https://ncatlab.org/nlab/revision/diff/Clifford+algebra/36">diff</a>, <a href="https://ncatlab.org/nlab/revision/Clifford+algebra/36">v36</a>, <a href="https://ncatlab.org/nlab/show/Clifford+algebra">current</a>

Added algebra - contents to sidebar

Anonymouse

diff, v36, current

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nLab > Latest Changes: Clifford algebra