Add perwass2009geometric, jadczyk2019notes, garling2011clifford

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}

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}

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}

trees/spin-0001.tree

@@ -15,3 +15,10 @@
 \transclude{spin-0004}
 
 \transclude{spin-0005}
+
+\transclude{spin-0006}
+
+\transclude{spin-0007}
+
+\transclude{spin-0008}
+

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 $\mathfrak{S}_{p, q}$.
+}

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:
+$$
+\begin{aligned}
+& \operatorname{Pin}(q):=\left\{s \in \Gamma(q)^{+} \cup \Gamma(q)^{-}: N(s)= \pm 1\right\} \\
+& \operatorname{Spin}(q):=\left\{s \in \Gamma(q)^{+}: N(s)= \pm 1\right\} \\
+& \operatorname{Spin}^{+}(q):=\left\{s \in \Gamma(q)^{+}: N(s)=+1\right\} .
+\end{aligned}
+$$
+}

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.
+}