 # Clifford algebra non-zero – MathOverflow Answer

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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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