Universal property for finite-dimensional Clifford algebras

Here’s my definition of a Clifford algebra:

Definition: Let $B(cdot,cdot)$ be a symmetric bilinear form on a vector space $V$ over $mathbb{K}$ and $Q$ its associated quadratic form. The Clifford algebra associated to the quadratic space $(V,Q)$ is a pair $mathcal{C}l(V,Q)$ where $mathcal{C}l(V,Q)$ is a $mathbb{K}$ associative algebra with identity $1$ and $varphi: V to mathcal{C}l(V,Q)$ is a Clifford map satisfying the following properties:

(a) $varphi(u)varphi(v)+varphi(v)varphi(u) = 2B(u,v)1$ for every $u,v in V$.

(b) The subspace $text{Im}varphi$ generates the algebra $mathcal{C}l(V,Q)$.

(c) For every Clifford map $phi: V to mathcal{A}$ on $(V,Q)$ there exists a homomorphism of algebras $f: mathcal{C}l(V,Q) to mathcal{A}$ such that $phi = fcirc varphi$.

Suppose $V$ is finite dimensional, with dimension $n$. Then $mathcal{C}l(V,Q)$ has dimension $2^{n}$ and is generated by $1$, $varphi(x_{i})$, $varphi(x_{i})varphi(x_{j})$, ($i<j$),…, $varphi(x_{1})cdots varphi(x_{n})$, where ${x_{1},…,x_{n}}$ is a basis for $V$. Moreover, one can prove that the Clifford map $varphi$ is injective. Thus, we may identify $V$ with the image of $varphi$, so that $V$ can be treated as a subspace of $mathcal{C}l(V,Q)$. Thus, $varphi(x_{i})$ becomes simply $x_{i}$.

In the literature, we often find the definition of a Clifford algebra with dimension $2^{n}$ as an algebra generated by (the generators) $1$, $x_{i}$, $x_{i}x_{j}$ ($i<j$),…,$x_{1}cdots x_{n}$, and these elements satisfy:
$$x_{i}x_{j} + x_{j}x_{i} = 2delta_{ij}$$
Let’s call this definition our second version of a Clifford algebra.

As I stressed before, this second case is a simplified version of the above defition, in which $V$ has dimension $n$. The generators $1$, $x_{i}$, $x_{i}x_{j}$ ($i< j$),…, $x_{1}cdots x_{n}$ are identifications of $1$, $varphi(x_{i})$, $varphi(x_{i})varphi(x_{j})$, ($i<j$),…, $varphi(x_{1})cdots varphi(x_{n})$ and $B$ is a bilinear form in which the basis of $V$ is orthonormal.

In summary, the second version follows from the first. But taking the second version as the definition of a Clifford algebra seems a little odd to me because it does not mention property (c) of the above definition, which plays an important role in the abstract theory of Clifford algebra. So my question is: does property (c) holds trivially in the case of finite dimensional vector spaces $V$, so it does not have to be demanded in the definition of such a Clifford algebra, as the second version seems to imply? If not, why is this property commonly omitted in so many references?

mp.mathematical physics – What is the relationship between the Dirac algebra and the Clifford algebra?

While I’m still trying to understand the issues raised on my previous question, I decided to first address the Clifford algebra used on formulating the famous Dirac equation. In this context, what is found in physics books is the following. There are four $ntimes n$ matrices (let’s take $n=4$ for simplicity) $gamma^{mu}$, $mu = 0,1,2,3$ satisfying:
$${gamma^{mu},gamma^{nu}} := gamma^{mu}gamma^{nu}+gamma^{nu}gamma^{mu} = 2g_{munu}I$$
where $g = mbox{diag}(1,-1,-1,-1)$ is the Minkowski metric. These relations should define a Clifford algebra. Moreover, a copy of $mathbb{C}^{4}$ on which the Dirac matrices $gamma^{mu}$ act is called a spinor space, and its elements are called (Dirac) spinors.

As I mentioned in my previous question, the mathematical definition (I know) of a Clifford algebra is the following.

Definition: Let $V$ be a $mathbb{K}$-vector space, $varphi: V times V to mathbb{K}$ a symmetric bilinear map and $Phi: V to mathbb{K}$ the quadratic form associated to $varphi$. A Clifford algebra $mathcal{Cl}(V, Phi)$ associated to $V$ and $Phi$ is a $mathbb{K}$-associative algebra with unit together with a linear map $i_{Phi}:V to mathcal{Cl}(V,Phi)$ such that:

(a) $(i_{Phi}(v))^{2}=Phi(v)cdot 1$, $forall v in V$,

(b) (Universal Property) For every $mathbb{K}$-algebra $A$ and every linear map $f: V to A$ such that $(f(v))^{2}=Phi(v)cdot 1_{A}$ ($forall v in V$), there exists a unique $mathbb{K}$-homomorphism $bar{f}: mathcal{Cl}(V,Phi)to A$ such that $f = bar{f}circ i_{Phi}$.

Property (a) can be rephrased in an equivalent form:
$$i_{Phi}(v)i_{Phi}(w) + i_{Phi}(w)i_{Phi}(v) = 2varphi(v,w)cdot 1$$

I’m trying to relate the physicist approach to the above definition. When it comes to Clifford algebras, there are a lot of information out in the internet and it is really difficult to focus on what’s important if you have no background on the subject. According to Wikipedia, the Dirac algebra should be $mathcal{Cl}_{4}(mathbb{C})$ or $mathcal{Cl}_{1,3}(mathbb{C})$ which, frankly, I don’t know exactly what it means.

My guess is to take $V = mathbb{R}^{4}$ the Minkowski space and $varphi$ the Minkowski inner product:
$$varphi(x,y) = x_{0}y_{0}-x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}$$
If $gamma^{mu}$, $mu = 0,1,2,3$ are complex $4times 4$ known matrices, we can define $i_{Phi}$ by sending each element $e_{mu}$ of the canonical basis of $mathbb{R}^{4}$ to its associated $gamma^{mu}$ and extend $i_{Phi}$ by linearity. However:

(1) If all this reasoning is correct, I don’t know how to prove that the universal property is satisfied.

(2) Wikipedia says this construction should be $mathcal{Cl}_{4}(mathbb{C})$ and $mathcal{Cl}_{1,3}(mathbb{C})$ and I’m taking $mathbb{R}^{4}$ instead of $mathbb{C}$, so I don’t know what is the connection between these approaches.

Can someone help me with these problems?

Efficient computation of scalar part in Clifford algebra

Problem: Let $Cl(d)$ be the Clifford algebra corresponding to the vector space $mathbb{R}^d$ with the usual inner product. Given $v_1, dots, v_k in mathbb{R}^d$, compute
the scalar part of the product $v_1 dots v_k in Cl(d)$ in an efficient way.

One (inefficient) approach would be to first write the $v_j$ in some (ordered) orthonormal basis ${e_i}_{1 leq i leq d}$, then expand and simplify the product using that $e_i e_j = -e_j e_i$ when $i > j$ and that $e_i^2 = -1$, and finally take the scalar part (constant term) of the result. The problem with this approach is that, when expanding the product, one gets a number of monomials that is exponential in $d$, since $Cl(d)$ is of real dimension $2^d$ with basis given by ordered products of distinct $e_i$. (Note that in my case $k$ is allowed to be of the order of $d$.)

Since I am not interested in the full product of the $v_j$ but just on the scalar part of it, I wonder if there is an efficient algorithm for its computation. So far, I have tried using some of the well-known identities that hold in $Cl(d)$, and taking a look at some of the software for doing calculations in Clifford algebras but I couldn’t find this exact functionality.

differential equations – Control and observability of Clifford algebra and quaternion valued systems?

Good evening everybody, I’m asking you if there is a mathematical theory about control and observability of differential systems within the quaternionic and Clifford (geometric) algebras. And if so, please provide me some literature. Thank you.
H. Banouh.

The inner product of a Clifford Algebra

Any Clifford algebra $Cl(k, p)$ carries an induced inner product, which is the “trace” on its 0-blade: $<A>_0$ for a given element $A$ of the algebra.

This inner product, when restricted to the generating vector space $V$, gives back the inner product on $V$ (this fact, by the way, suggests that Cliff can be seen as a functor from the category of finite dimensional vectors spaces armed with an inner product to itself).

Now, my question (maybe entirely trivial, but I could not find it in the standard literature, for instance in Lounesto’s great book):

what is the interplay of the Clifford product and its induced inner product? Are there any formal laws?

Any clue?

PS I have done some little googling, and I came up to some refs on inner product associative algebras, but unfortunately there seem to be a certain latitude as far as their definition.

List Manipulation – Grassman and Clifford algebras as quotient of tensor algebra

I worked with tensors in Mathematica. It's great to have TensorProduct with linearity and everything. Perhaps you can use these structures to work with Grassman algebras or Clifford algebras, both of which are quotient of tensor algebra: T / v ^ 2 is the Grassman algebra, T / (v ^ 2 + norm (v)) the Clifford algebra algebra.

After thinking about it for three days, I still do not know exactly how to switch from tensors to Grassman / Clifford.

ct.category theory – What minimal structure is required to define Clifford modules as abstractly as possible?

Start with a square shape $ q $ on a vector space $ V $, A module $ M $ above the corresponding Clifford algebra is determined by a map $ cdot: V otimes M to M $ satisfying $ v cdot (v cdot m) = – q (v) m $,

Now try to abstract this as follows. The bilinear form $ B (x, y) = q (x) + q (y) -q (x + y) $ determines a natural transformation $ varepsilon: TT to text {identity} $, from where $ T $ is the endofunctor $ T = V otimes $ on vector spaces. A clifford module structure $ M $ In this regard, it is a morphism $ mu: TM to M $and – here begins my question – a certain relationship between the composite material $ mu circ T mu: TTM to TM to M $, and $ varepsilon_M: TTM to M $,

The question is what minimal structure is needed to capture this relationship. Seemingly either a kind of nonadditive transformation $ delta: T to TT $ is necessary to express $ v mapsto v otimes v $or a kind of self-distributing law $ text {switch}: TT to TT $, In the latter case, however, the additive structure seems to be needed to express it $ x cdot (y cdot m) + y cdot (x cdot m) = B (x, y) m $,

Has any of this been done anywhere? Is it possible to avoid the additive structure, at least with some restrictions? For example when $ B $ is not degenerate, the endofunctor $ T $ becomes independent, maybe you can use that somehow, I do not know how.