Simple graded commutative algebras. (Q961020)

Let \(\Gamma\) be an Abelian group endowed with a biadditive map \(\langle\cdot\mid\cdot\rangle\colon\Gamma\times\Gamma\to\mathbb Z_2\). An algebra \(\mathcal A\) is defined to be \(\Gamma\)-commutative if \(\mathcal A\) is \(\Gamma\)-graded: \(\mathcal A=\bigoplus_{\gamma\in\Gamma}\mathcal A_\gamma\) with \(\mathcal A_\gamma\mathcal A_{\gamma'}\subseteq\mathcal A_{\gamma+\gamma'}\) for any \(\gamma,\gamma'\in\Gamma\), and for any homogeneous elements \(a\in\mathcal A_\gamma\), \(b\in\mathcal A_{\gamma'}\), \(ab=(-1)^{\langle\gamma\mid\gamma'\rangle}ba\). The first main result asserts that if \(\mathcal A\) is a \(\Gamma\)-commutative algebra and \(\Gamma\) is finitely generated, then \(\Gamma\) and \(\langle\cdot\mid\cdot\rangle\) can be substituted by \(\mathbb Z_2^n\), for some \(n\) and the standard ``inner product''. Thus, for instance, the classical real division algebra of quaternions \(\mathbb H\) is \(\mathbb Z_2^3\)-commutative with \(\deg(i)=(\overline 1,\overline 0,\overline 1)\), \(\deg(j)=(\overline 0,\overline 1,\overline 1)\) and \(\deg(k)=(\overline 1,\overline 1,\overline 0)\). (Note that \(\mathbb H\) is naturally \(\mathbb Z_2^2\)-graded, but then the inner product on \(\mathbb Z_2^2\) does not work.) The second main result asserts that over \(\mathbb R\) or \(\mathbb C\), the only simple finite-dimensional associative and graded-commutative algebras are the Clifford algebras. Some nonassociative extensions are considered, too, where the so called \textit{tiny Kaplansky superalgebra} plays a key role. This is a nonunital three-dimensional simple Jordan superalgebra.