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43

This is not really an answer, but rather a meta-answer as to why there exist many conventions in the first place. The symmetric monoidal category $\mathit{sVect}$ of super-vector spaces has a non-trivial involution $J$. The symmetric monoidal functor $J:\mathit{sVect}\to \mathit{sVect}$ is the identity at the level of objects and at the level of morphisms. ...

22

As Igor mentions in the comments, this is really a question about deformations of the multiplication map of the exterior algebra in the space of associative multiplications. Since this is a pretty general story, let me try to sketch how it works in this case. Let's fix some field $k$ and a $k$-algebra $(A,\mu)$. We want to understand deformations of $(A,\mu)... 20 Here is one conceptual description of the relationship. (I wouldn't call it an explanation; I don't really know why it's true, except that it's because Bott periodicity is true.) KO-theory is the first Weiss-derivative of the$K$-theory of Clifford algebras. More precisely: given a real inner product space$V$, we get a category$M(V)=\mathrm{Mod}(Cl(V))$... 18 The$p$-completed algebraic$K$-theory of the algebraic closure of$\mathbb{Q}_p$, i.e.,$K(\bar{\mathbb{Q}}_p; \mathbb{Z}_p)$, is equivalent to its second loop space, up to an issue about path components. This is due to Suslin. The descent to$\mathbb{Q}_p$is more subtle than the descent from$\mathbb{C}$to$\mathbb{R}$, because the absolute Galois group ... 16 To my mind, it has to do with p.d.e.'s and to your background, geometry or physics. Let me explain.$\newcommand{\pa}{\partial}$Fix$\epsilon=\pm 1$. Define the Clifford algebra using the$\epsilon$rule $$c_jc_k+c_kc_j=2\epsilon\delta_{jk}$$ and we form the Dirac operator $$D=\sum_k c_k\pa_{x^k}$$ Its square is$$D^2=\epsilon\sum_k \pa_{... 14 In addition to the answer of Bertram Arnold, let me point out that there is a very explicit formula for "fermionic" Weyl-Moyal product. Let us assume that your vector space$V$(or module) is defined over a ring containing the rationals. This is sort of crucial in the following and in prime characteristic the story is slightly different. Part of the sport ... 11 Here is another proposal. The key will be to enlarge the category of quadratic vector spaces fairly substantially. Here are three hints leading towards the proposal: Clifford algebras can and should be thought of as$\mathbb{Z}_2$-graded; for example, Clifford algebras over$\mathbb{R}$give every element of the$\mathbb{Z}_2$-graded Brauer group / Brauer-... 11 It may be worthwhile to vary the question ever so slightly to read Question: Is there a conceptual explanation for the tight relationship (or some aspect thereof) between K-theory and superalgebra? This seems to have a deep answer as follows: The$\mathbb{Z}_2$-grading in superalgebra/supergeometry is usefully identified as the low degree pieces of$\...

11

Here's a standard explicit formula: Let $\mathbb{O}\simeq\mathbb{R}^8$ denote the algebra of octonions, and for $x\in\mathbb{O}$, let $L_x$ (respectively $R_x)$ denote the linear map from $\mathbb{O}$ to itself generated by left (respectively, right) multiplication by $x$ and let $C:\mathbb{O}\to\mathbb{O}$ be conjugation in the octonions. Now define $\rho(... 11 Let$(M,g)$be an orientable pseudo-riemannian manifold. Each tangent space$T_xM$is a pseudo-euclidean space and hence has an associated Clifford algebra$CL(T_xM)$, which is the fibre at$x\in M$of the Clifford bundle$Cl(TM)$. If the manifold is spin (a topological condition which says that the oriented orthonormal frame bundle lifts to a spin bundle) ... 10$C_{n,0}$is either a full matrix algebra over$\mathbb{R}$,$\mathbb{C}$, or$\mathbb{H}$, or the direct sum of two such algebras that are isomorphic. The exact description depends on the residue class of$n$(mod$8$) and can be found in textbooks or on wikipedia http://en.wikipedia.org/wiki/Classification_of_Clifford_algebras$\mathrm{Mat}_{k \times k}(\...

10

I would think that these these notes by Akhil Mathew provide the "exact definition" you are asking for: In response to the follow-up question "which is the first Clifford module used in the physics context": Two different representations (modules) of the Clifford algebra were studied in early work on the Dirac equation. Paul Dirac himself used a ...

10

$\def\RR{\mathbb{R}}$This is an answer to flesh out what I wrote in comments. Let $V$ be a real vector space with positive definite inner product. For a unit vector $\vec{u} \in V$, let $s_{\vec{u}}$ be the reflection over $\vec{v}$. Here is something we might hope for: Vague hope: Can we build a ring $R$ whose additive structure comes from the vector ...

9

For any finite-dimensional associative unital $\mathbb{R}$-algebra $A$, the set $S$ of noninvertible elements is the zero-set of a polynomial. Namely, let $f:A \to Hom (A,A)$ be the map $a \mapsto ( x \mapsto ax)$ to the linear endomorphisms of $a$. $f$ is an injective algebra homomorphism, since $f(a)=0$ means $ax =0$ for all $x \in A$, in particular $a= a1 ... 7 A very good place to read about this is the 3. chapter of the book "Heat kernels and Dirac operators" by Nicole, Getzler & Vergne. Up to the wrong sign in your second definition, 1. and 2. are equivalent, as the symbol just collects the highest order terms. Also you obtain from the basic properties of the symbol map that the bundle which admits a Dirac ... 7 To put the problem to a rest, I add my comment as an answer which is, in a nutshell,$Spin(6,2)\cong Spin(4,\mathbb H)$. The existence of this isomorphism follows from the isomorphism of the Satake diagrams and simple connectedness. The tricky thing is the definition of a spin group over the quaternions which is explained in much greater generality in ... 6 I am not aware of a single formula, only recursive formulas involving the "shuffle basis" for Cayley-Dickson spaces. The shuffle basis is not the one commonly used by most researchers. For the shuffle basis,$e_0=1$and$e_{2n}=(e_n,0)$and$e_{2n+1}=(0,e_n)$. Furthermore, the product$e_pe_q=\pm e_{pq}$where$pq$is defined as the 'exclusive or' of the ... 6 Keep in mind that any finite-dimensional representation of a Lie group determines a finite-dimensional representation of its Lie algebra, and for a connected Lie group the induced Lie algebra representation determines the Lie group representation. However, every finite-dimensional representation of$\operatorname{Lie}(\mathrm{Mp}(2,\mathbb R)) = \mathfrak{...

6

I don't think that, strictly speaking, the question by the OP has been properly addressed. Although it is true that real Clifford algebras are isomorphic (through a non-canonical isomorphism of unital associative algebras) to some matrix algebra or direct sum thereof, Clifford algebras are more than just matrix algebras. Let $(V,g)$ be a real regular ...

6

To answer this question, we have to make a few rather natural assumptions, which I give in a slightly different notation. Let the spinor bundle $S$ be associated to a Euclidean vector bundle $E$. Assume that both bundles are locally trivialised over some open coordinate patch $U$ by orthonormal / unitary frames $e_1,\dots, e_n$, $\psi_1,\dots,\psi_N$, such ...

5

You can find all what you want to know in the chapter 1 of the book X.Ma and G. Marinescu:Holomorphic Morse Inequalities and Bergman Kernels. In Kahler case, all the metric connection is the chern connection. And for the general case(hermitian manifold), the connection is called Bismut connection (see section 1.2.3). And the Weitzenböck formula is in ...

5

Update: "The eight Cayley-Dickson doubling products", Adv. Appl. Clifford Algebras 26 (2016) pp 529–551, doi:10.1007/s00006-015-0638-6, arXiv:1707.07318 I now find that 4 of those 8 should also be discarded: each allows zero divisors at the eight dimensional stage. The 4 remaining products are denoted in the paper as $P_0$, $P_3$ (the standard doubling ...

5

Consider the matrix $\sum_i a_iR_i$. One can show that it sends every vector of length 1 to a vector of length $\sqrt{\sum_i a_i^2}$. It follows that if the norm of $a=(a_i)$ is 1, then the matrix $\sum_i a_iR_i$ is orthogonal. In particular, all the matrices $R_i$ are orthogonal. By multiplication from the left or from the right with an orthogonal matrix, ...

5

The essence of "geometric algebra" (better known as Kahler-Atiyah algebra) is the classical Chevalley-Riesz isomorphism, which presents the Clifford algebra of a quadratic space $(V,h)$ as a deformation quantization of the exterior algebra of $V$. The systematic use of this presentation allows for the automatic translation of various computations ...

4

I think that you want to look at the relevant passages in Spin Geometry by Lawson and Michelsohn, particularly Chapter IV, Sections 9 and 10, where they explain in general how the square of a spinor can be written as a sum of differential forms. This gives a general way to see how having a parallel spinor (in any dimension) defines a set of parallel forms ...

4

The answer for both questions 1, 2 is yes, the metric connection in Proposition 9.1.27 is the Levi-Civita induced connection if the background manifold is Kahler. When $X$ is only a Hermitian manifold, the result is not true. One such example is discussed in great detail in Bismut's paper A local index theorem for non-Kahler manifolds, Math. ...

4

A little bit of what you want can be found in Chapter 5 of Gracia-Bondia, Varilly, and Figueroa's book Elements of Noncommutative Geometry. They don't say much about subalgebras, I think, but they do prove (Lemma 5.7, page 182) the result that bivectors in the Clifford algebra $Cl(V)$ are closed under taking commutators, and that the adjoint action of ...

4

About 15 years ago I used Karoubi's description of $K$-theory to solve prove some cutting and pasting formula for the index families of elliptic problems. To do so I needed rephrase Karoubi's theory into something more flexible and more computable. Some of the interpretations I found might be relevant to your question. I will briefly describe ...

4

One more example: for $d=8$ based on octonions. And this are all possible dimensions. $R_1$ takes vectors of length one to vectors of length one hence it is an isometry. By multiplying from the left by $R_1^{-1}$ we may assume that $R_1=\mathop{\rm Id}$. Then for every $v\in S^{d-1}$, the unit sphere in $\mathbb{R}^d$, the map $v\mapsto(R_2v,R_3v,\ldots,R_dv)... 4 This is an alternative way to see this question. By argument given by the current answer given by Ehud Meir we may assume that$R_1$is the identity matrix. Then the hypotheses imply that for$i=2,\cdots,d$the maps$v\rightarrow R_i v$are$d-1$independent vector fields on the unit sphere$S^{d-1}\$. But it is known that the maximal number of independent ...

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