Timeline for Geometric or conceptual way to understand supersymmetry algebra
Current License: CC BY-SA 3.0
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| Aug 5, 2017 at 17:48 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 5, 2017 at 17:38 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 5, 2017 at 16:04 | comment | added | Konstantinos Kanakoglou | And when (4) is viewed as a subalgebra of (1)-(3) then the anticommutator expressions imply that $Q^2=(Q^+)^2=0$. | |
| Aug 5, 2017 at 16:01 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 5, 2017 at 15:59 | comment | added | Konstantinos Kanakoglou | aahhh, in this sense, you are absolutely right. To be more precise, the squared expressions "live" inside the universal enveloping algebra of the LS (and not in the LS itself). In fact, in the level of the LS these are just the anticommutators $\{Q,Q\}=\{Q^+,Q^+\}=0$. I will edit to make that clear. | |
| Aug 5, 2017 at 15:52 | comment | added | მამუკა ჯიბლაძე | I did just mean that squaring is not part of Lie superalgebra structure. | |
| Aug 5, 2017 at 15:45 | comment | added | Konstantinos Kanakoglou | Also, i did not understand the last sentence of your comment: "... But then the last equalities would not be there..". Which equalities do you refer to ? | |
| Aug 5, 2017 at 15:45 | comment | added | Konstantinos Kanakoglou | @ მამუკა ჯიბლაძე, yes the algebra defined in terms of generators $Q, Q^+, H$ and relations (4) is a Lie superalgebra. It is -maybe the simplest- susy toy model and it is well known in the relevant literature. See for ex aip.scitation.org/doi/citedby/10.1063/1.528170 or bolvan.ph.utexas.edu/~vadim/classes/13f/Witten1982.pdf but there are plenty of other references. Regarding your question on whether it is a known Lie superalgebra, i am not sure right now as to which Lie superalgebra it corresponds to (in the sense of the Kac classification/terminology of Lie superalgebras). | |
| Aug 5, 2017 at 14:29 | comment | added | მამუკა ჯიბლაძე | It is an exciting explanation but as such it stirs my appetite :D Could you please also explain what kind of mathematical object is your (4)? Is it some known Lie superalgebra? But then the last equalities would not be there... | |
| Aug 5, 2017 at 14:21 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 4, 2017 at 17:50 | comment | added | Konstantinos Kanakoglou | The above example can be generalized so that any Lie superalgebra can be derived in a similar fashion (given that we know a fd representation of the LS at hand). This is actually the inverse of a method which i had used in the past motivated by attempts to construct new representations of Lie superalgebras, through their "realizations" via quotients of the UEA of the HW Lie superalgebra. Such ideas are investigated at arxiv.org/abs/0912.1070v1 and arxiv.org/abs/1104.0696 | |
| Aug 4, 2017 at 17:49 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 4, 2017 at 17:17 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 4, 2017 at 17:07 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 4, 2017 at 16:57 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 4, 2017 at 15:58 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 4, 2017 at 15:45 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 2, 2017 at 3:13 | history | edited | Konstantinos Kanakoglou | CC BY-SA 3.0 |
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| Aug 2, 2017 at 2:54 | history | answered | Konstantinos Kanakoglou | CC BY-SA 3.0 |