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Clifford algebra non-zero – MathOverflow

To display that an algebra constructed as a quotient of the tensor algebra of a vector house is nonzero, one of many predominant methods to refer is to assemble representations. We can do that for the Clifford algebra as follows.

Let $V$ breathe a vector house over a bailiwick $okay$ and $(,):Vtimes V to okay$ a symmetric bilinear figure on $V$. The Clifford algebra (for this figure) is given by

$$Cl(V)= T(V)/langle v otimes v – (v,v)rangle.$$

We will assemble a illustration of the Clifford algebra on the outside algebra $bigwedge (V)$.

For $v in V$, outline two $okay$-endomorphisms of $bigwedge(V)$ by

$$ l_v(x) = v wedge x$$

and

$$ delta_v(x) = sum_{j=1}^okay (-1)^{j-1}(v,x_j) x_1 wedge dots wedge widehat{x_j} wedge dots wedge x_k$$

if $x = x_1 wedge dots wedge x_k$.

Then bridle that $l_v^2 = delta_v^2 = 0$, and furthermore that $l_v delta_v + delta_v l_v = (v,v) cdot mathrm{id}$.

Extend the map linear $v mapsto l_v + delta_v$ to an algebra homomorphism from the tensor algebra $T(V)$ to $mathrm{End}_k(bigwedge(V))$. By the earlier comment, this descends to a map, let’s convene it $phi$, from the Clifford algebra to $mathrm{End}_k(bigwedge(V))$.

In explicit, $phi(v)1 = v$, so $V$ injects into the Clifford algebra.

Edit: I consider too that the map

$$ x mapsto phi(x)1$$

offers a linear isomorphism of the Clifford algebra with the outside algebra.

A noble reference for these things is Chevalley’s monograph, The Algebraic Theory of Clifford Algebras and Spinors.

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