From 554ea5761b40f8fe9cfca46af150b75d169fdc6f Mon Sep 17 00:00:00 2001 From: utensil Date: Sun, 28 Apr 2024 15:48:21 +0800 Subject: [PATCH] Add perwass2009geometric, jadczyk2019notes, garling2011clifford --- trees/refs/garling2011clifford.tree | 12 ++++++++ trees/refs/jadczyk2019notes.tree | 10 +++++++ trees/refs/perwass2009geometric.tree | 12 ++++++++ trees/spin-0001.tree | 7 +++++ trees/spin-0006.tree | 15 ++++++++++ trees/spin-0007.tree | 21 +++++++++++++ trees/spin-0008.tree | 44 ++++++++++++++++++++++++++++ 7 files changed, 121 insertions(+) create mode 100644 trees/refs/garling2011clifford.tree create mode 100644 trees/refs/jadczyk2019notes.tree create mode 100644 trees/refs/perwass2009geometric.tree create mode 100644 trees/spin-0006.tree create mode 100644 trees/spin-0007.tree create mode 100644 trees/spin-0008.tree diff --git a/trees/refs/garling2011clifford.tree b/trees/refs/garling2011clifford.tree new file mode 100644 index 0000000..c0882eb --- /dev/null +++ b/trees/refs/garling2011clifford.tree @@ -0,0 +1,12 @@ +\title{Clifford algebras: an introduction} +\taxon{reference} + +\meta{bibtex}{\startverb +@book{garling2011clifford, + title={Clifford algebras: an introduction}, + author={Garling, David JH}, + volume={78}, + year={2011}, + publisher={Cambridge University Press} +} +\stopverb} diff --git a/trees/refs/jadczyk2019notes.tree b/trees/refs/jadczyk2019notes.tree new file mode 100644 index 0000000..da76c86 --- /dev/null +++ b/trees/refs/jadczyk2019notes.tree @@ -0,0 +1,10 @@ +\title{Jadczyk's Notes on Clifford Algebras} +\taxon{reference} + +\meta{bibtex}{\startverb +@article{jadczyk2019notes, + title={Notes on Clifford Algebras}, + author={Jadczyk, Arkadiusz}, + year={2019} +} +\stopverb} \ No newline at end of file diff --git a/trees/refs/perwass2009geometric.tree b/trees/refs/perwass2009geometric.tree new file mode 100644 index 0000000..708aa97 --- /dev/null +++ b/trees/refs/perwass2009geometric.tree @@ -0,0 +1,12 @@ +\title{Geometric algebra with applications in engineering} +\taxon{reference} + +\meta{bibtex}{\startverb +@book{perwass2009geometric, + title={Geometric algebra with applications in engineering}, + author={Perwass, Christian and Edelsbrunner, Herbert and Kobbelt, Leif and Polthier, Konrad}, + volume={4}, + year={2009}, + publisher={Springer} +} +\stopverb} \ No newline at end of file diff --git a/trees/spin-0001.tree b/trees/spin-0001.tree index fbc7dd6..72404cb 100644 --- a/trees/spin-0001.tree +++ b/trees/spin-0001.tree @@ -15,3 +15,10 @@ \transclude{spin-0004} \transclude{spin-0005} + +\transclude{spin-0006} + +\transclude{spin-0007} + +\transclude{spin-0008} + diff --git a/trees/spin-0006.tree b/trees/spin-0006.tree new file mode 100644 index 0000000..da988fa --- /dev/null +++ b/trees/spin-0006.tree @@ -0,0 +1,15 @@ +\import{base-macros} + +\texdef{Spin group}{perwass2009geometric}{ +A versor is a multivector that can be expressed as the geometric product of a number of non-null 1-vectors. That is, a versor $\boldsymbol{V}$ can be written as $\boldsymbol{V}=\prod_{i=1}^k \boldsymbol{n}_i$, where $\left\{\boldsymbol{n}_1, \ldots, \boldsymbol{n}_k\right\} \subset \mathbb{G}_{p, q}^{\varnothing 1}$ with $k \in \mathbb{N}^{+}$, is a set of not necessarily linearly independent vectors. + +The subset of versors of $\mathbb{G}_{p, q}$ together with the geometric product, forms a group, the Clifford group, denoted by $\mathfrak{G}_{p, q}$. + +A versor $\boldsymbol{V} \in \mathfrak{G}_{p, q}$ is called unitary if $\boldsymbol{V}^{-1}=\tilde{\boldsymbol{V}}$, i.e. $\boldsymbol{V} \widetilde{\boldsymbol{V}}=+1$. + +The set of unitary versors of $\mathfrak{G}_{p, q}$ forms a subgroup $\mathfrak{P}_{p, q}$ of the Clifford group $\mathfrak{G}_{p, q}$, called the pin group. + +A versor $\boldsymbol{V} \in \mathfrak{G}_{p, q}$ is called a spinor if it is unitary $(\boldsymbol{V} \tilde{\boldsymbol{V}}=1)$ and can be expressed as the geometric product of an even number of 1-vectors. This implies that a spinor is a linear combination of blades of even grade. + +The set of spinors of $\mathfrak{G}_{p, q}$ forms a subgroup of the pin group $\mathfrak{P}_{p, q}$, called the spin group, which is denoted by $\mathfrak{S}_{p, q}$. +} \ No newline at end of file diff --git a/trees/spin-0007.tree b/trees/spin-0007.tree new file mode 100644 index 0000000..cf713b6 --- /dev/null +++ b/trees/spin-0007.tree @@ -0,0 +1,21 @@ +\import{base-macros} + +\texdef{Spin group}{jadczyk2019notes}{ + +We define the Clifford group $\Gamma=\Gamma(q)$ to be the group of all invertible elements $u \in \mathrm{Cl}(q)$ which have the property that uyu ${ }^{-1}$ is in $M$ whenever $y$ is in $M$. We define $\Gamma(q)^{ \pm}$as the intersection of $\Gamma(q)$ and $\mathrm{Cl}(q)_{ \pm}$. + +For every element $u \in \Gamma(q)$ we define the spinor norm $N(u)$ by the formula +$$ +N(u)=\tau(u) u, +$$ +where $\tau$ is the main involution of the Clifford algebra $\mathrm{Cl}(q)$. + +The following groups are called spin groups: +$$ +\begin{aligned} +& \operatorname{Pin}(q):=\left\{s \in \Gamma(q)^{+} \cup \Gamma(q)^{-}: N(s)= \pm 1\right\} \\ +& \operatorname{Spin}(q):=\left\{s \in \Gamma(q)^{+}: N(s)= \pm 1\right\} \\ +& \operatorname{Spin}^{+}(q):=\left\{s \in \Gamma(q)^{+}: N(s)=+1\right\} . +\end{aligned} +$$ +} \ No newline at end of file diff --git a/trees/spin-0008.tree b/trees/spin-0008.tree new file mode 100644 index 0000000..5bb9fcf --- /dev/null +++ b/trees/spin-0008.tree @@ -0,0 +1,44 @@ +\import{base-macros} + +\texdef{Spin group}{garling2011clifford}{ +Suppose that $(E, q)$ is a regular quadratic space. We consider the action of $\mathcal{G}(E, q)$ on $\mathcal{A}(E, q)$ by adjoint conjugation. We set +$$ +A d_g^{\prime}(a)=g a g^{-1}, +$$ +for $g \in \mathcal{G}(E, q)$ and $a \in \mathcal{A}(E, q)$. + +We restrict attention to those elements of $\mathcal{G}(E, q)$ which stabilize $E$. The Clifford group $\Gamma=\Gamma(E, q)$ is defined as +$$ +\left\{g \in \mathcal{G}(E, q): A d_g^{\prime}(x) \in E \text { for } x \in E\right\} . +$$ + +If $g \in \Gamma(E, q)$, we set $\alpha(g)(x)=A d_g^{\prime}(x)$. Then $\alpha(g) \in G L(E)$, and $\alpha$ is a homomorphism of $\Gamma$ into $G L(E) . \alpha$ is called the graded vector representation of $\Gamma$. + +It is customary to scale the elements of $\Gamma(E, q)$; we set +$$ +\begin{aligned} +\operatorname{Pin}_{\infty}(E, q) & =\{g \in \Gamma(E, q): \Delta(g)= \pm 1\}, \\ +\Gamma_1(E, q) & =\{g \in \Gamma(E, q): \Delta(g)=1\} . +\end{aligned} +$$ + +If $(E, q)$ is a Euclidean space, then $\operatorname{Pin}(E, q)=\Gamma_1(E, q)$; otherwise, $\Gamma_1(E, q)$ is a subgroup of $\operatorname{Pin}(E, q)$ of index 2. +We have a short exact sequence +$$ +1 \longrightarrow D_2 \xrightarrow{\subseteq} \operatorname{Pin}(E, q) \xrightarrow{\alpha} O(E, q) \longrightarrow 1 ; +$$ +$\operatorname{Pin}_{\infty}(E, q)$ is a double cover of $O(E, q)$. + +In fact there is more interest in the subgroup $\operatorname{Spin}(E, q)$ of $\operatorname{Pin}(E, q)$ consisting of products of an even number of unit vectors in $E$. Thus $\operatorname{Spin}(E, q)=\operatorname{Pin}(E, q) \cap \mathcal{A}^{+}(E, q)$ and +$$ +\operatorname{Spin}(E, q)=\left\{g \in \mathcal{A}^{+}(E, q): g E=E g \text { and } \Delta(g)= \pm 1\right\} . +$$ + +If $x, y$ are unit vectors in $E$ then $\alpha(x y)=\alpha(x) \alpha(y) \in S O(E, q)$, so that $\alpha(\operatorname{Spin}(E, q)) \subseteq S O(E, q)$. Conversely, every element of $S O(E, q)$ is the product of an even number of simple reflections, and so $S O(E, q) \subseteq \alpha\left(\operatorname{Spin}(E, q)\right)$. Thus $\alpha\left(\operatorname{Spin}(E, q)\right)=S O(E, q)$, and we have a short exact sequence. +$$ +1 \longrightarrow D_2 \xrightarrow{\subseteq} \operatorname{Spin}(E, q) \xrightarrow{\alpha} S O(E, q) \longrightarrow 1 ; +$$ +$\operatorname{Spin}(E, q)$ is a double cover of $S O(E, q)$. + +Note also that if $a \in \operatorname{Spin}(E, q)$ and $x \in E$ then $\alpha(a)(x)=a x a^{-1}$; conjugation and adjoint conjugation by elements of $\operatorname{Spin}(E, q)$ are the same. +} \ No newline at end of file