Cyclic homology and the determinant of the Cartan matrix
Let \(\Lambda = \bigoplus^ \infty_{i = 0} \Lambda_ i\) be a graded algebra over the field \(K\) and let \(R\) denote the ideal \(\bigoplus^ \infty_{i = 1} \Lambda_ i\). Each \(\Lambda_ i\) should be finite dimensional over \(K\), and \(D = \Lambda_ 0 = D_ 1 \times \cdots \times D_ m\) should be a product of finitely many finite-dimensional separable division algebras over \(K\). (\(\Lambda\) may even be \(\mathbb{N}^ n\) graded.) The \(\mathbb{N}\) graded Cartan matrix \(C_ \Lambda (x)\) of \(\Lambda\) is the \(m \times m\) matrix with entries \(C_{ij}(x) \in \mathbb{Z} x\) given by \(C_{ij} (x) = \sum_ k \dim_{D_ j} (D_ i \Lambda_ k D_ j)x^ k\). The author defines the cyclic homology \(HC_ * (\Lambda, R)\) as the homology of a suitable bicomplex. The gradings on \(\Lambda\) and \(R\) are then shown to induce a grading on cyclic homology: \[ HC_ * (\Lambda, R) = \sum_ k HC^ k_ * (\Lambda, R) \] which is then used (when \(K\) has characteristic 0) to define the graded Euler characteristic of \(HC_ * (\Lambda, R)\) as the power series in \(\mathbb{Z} x\) defined by \[ \chi (HC_ * (\Lambda, R)) (x) = \sum_ k \sum_ i (-1)^ i \dim_ K HC^ k_ i (\Lambda, R) x^ k. \] The main theorem of the paper calculates the Euler characteristic in terms of the logarithm of the determinant of the Cartan matrix (and vice versa): the author shows that \[ \chi (HC_ * (\Lambda, R)) (x) = \sum^ \infty_{k = 1} \log \text{det} C_ \Lambda (x^ k) \sum_{d\mid k} (\mu(d)/d) \] where \(\mu\) is of course the Möbius function.
finite-dimensional separable division algebras
scientific article; zbMATH DE number 584670