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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 6th 2016
   • (edited Sep 6th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAt _[[sesquilinear form]]_ I have added the following general definition: let $A$ be a not-necessarily commutative star-algebra. Let $V$ be a left $A$-module with $A$-linear dual denoted $V^\ast$. Then a sesquilinear form on $V$ is simply an element in the tensor product $$ V^\ast \otimes_A V^\ast \,, $$ where we use the only possible way to regard the left $V$-module as a right $V$-module: by the star-involution. I am wondering if there is anywhere some discussion as to how far one may push dg-algebra over $A$ this way, specifically for the cases where $A$ is a normed division algebra such that the quaternions or the octonions. For instance which structure do we need on $A$ to make sense of the Grassmann algebra $\wedge^\bullet_A V^\ast$ of $V^\ast$ this way? This is motivated by the following: At _[[spin representation]]_ I once put a remark that one may obtain the $N = 1$ super-translation Lie algebra simply by starting with the super-point $\mathbb{R}^{0\vert 2}$, regarded as an abelian super Lie algebra, and then forming the central extension by $\mathbb{R}^3$ which is classified by the cocycle $d \theta_i \wedge d \theta_j \in \wedge^2 (\mathbb{R}^2)^\ast$ $(1 \leq i \leq j \leq 2)$, with $\theta_i$ and $\theta_j$ the two canonical odd-graded coordinates on $\mathbb{R}^{0\vert 2}$. Yesterday with John Huerta we were brainstorming about how to best formulate this such that the statement goes through verbatim for the other real normed division algebras to yield the super-translation Lie algebra alsoin dimensions 4,6 and 10. With sesquilinear forms as above it is obvious: Let $\mathbb{K}$ any of the four real normed division algebras, consider the superpoint $\mathbb{K}^{0\vert 2}$ and then form the central extension of super Lie algebras classified by the sesquilinear forms $d \theta_i \otimes_{\mathbb{K}} d \theta_j$ ($1 \leq i \leq j \leq 2$). These forms being sesquilinear expresses nothing but the spinor pairing of the susy algebra that Baez-Huerta (as reviewed [here](https://ncatlab.org/nlab/show/spin%20representation#ExpressionInTermsOfNormedDivisionAlgebras) ) write as $(\psi,\phi)\mapsto \psi \phi^\dagger + \phi \psi^\dagger$.

   At sesquilinear form I have added the following general definition:

   let $A$ be a not-necessarily commutative star-algebra. Let $V$ be a left $A$-module with $A$-linear dual denoted $V^\ast$. Then a sesquilinear form on $V$ is simply an element in the tensor product

   $$V^\ast \otimes_A V^\ast \,,$$

   where we use the only possible way to regard the left $V$-module as a right $V$-module: by the star-involution.

   I am wondering if there is anywhere some discussion as to how far one may push dg-algebra over $A$ this way, specifically for the cases where $A$ is a normed division algebra such that the quaternions or the octonions.

   For instance which structure do we need on $A$ to make sense of the Grassmann algebra $\wedge^\bullet_A V^\ast$ of $V^\ast$ this way?

   This is motivated by the following:

   At spin representation I once put a remark that one may obtain the $N = 1$ super-translation Lie algebra simply by starting with the super-point $\mathbb{R}^{0\vert 2}$, regarded as an abelian super Lie algebra, and then forming the central extension by $\mathbb{R}^3$ which is classified by the cocycle $d \theta_i \wedge d \theta_j \in \wedge^2 (\mathbb{R}^2)^\ast$ $(1 \leq i \leq j \leq 2)$, with $\theta_i$ and $\theta_j$ the two canonical odd-graded coordinates on $\mathbb{R}^{0\vert 2}$.

   Yesterday with John Huerta we were brainstorming about how to best formulate this such that the statement goes through verbatim for the other real normed division algebras to yield the super-translation Lie algebra alsoin dimensions 4,6 and 10.

   With sesquilinear forms as above it is obvious: Let $\mathbb{K}$ any of the four real normed division algebras, consider the superpoint $\mathbb{K}^{0\vert 2}$ and then form the central extension of super Lie algebras classified by the sesquilinear forms $d \theta_i \otimes_{\mathbb{K}} d \theta_j$ ($1 \leq i \leq j \leq 2$).

   These forms being sesquilinear expresses nothing but the spinor pairing of the susy algebra that Baez-Huerta (as reviewed here ) write as $(\psi,\phi)\mapsto \psi \phi^\dagger + \phi \psi^\dagger$.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 3rd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the definition of positive/negative definity, for completeness ([here](https://ncatlab.org/nlab/show/sesquilinear+form#Definity)) <a href="https://ncatlab.org/nlab/revision/diff/sesquilinear+form/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/sesquilinear+form/3">v3</a>, <a href="https://ncatlab.org/nlab/show/sesquilinear+form">current</a>

   added the definition of positive/negative definity, for completeness (here)

   diff, v3, current