Quantum Matrix Spaces. II. Coordinate Approach

  • Yuri I. ManinEmail author
Part of the CRM Short Courses book series (CRMSC)


Consider a general quadratic algebra
$$ A=\mathbb {K}\langle \tilde{x}_1,\dots ,\tilde{x}_n\rangle /(r_\alpha ), $$
where \(\mathbb {K}\langle \tilde{x}\rangle \) means a free associative algebra generated by the \(\tilde{x}_j\), and
$$\begin{aligned} r_\alpha =r_\alpha (\tilde{x})=\sum _{i, j} c_\alpha ^{ij}\tilde{x}_i\tilde{x}_j \quad \text {for }\alpha =1,\dots , m\;, \end{aligned}$$
are linearly independent elements of \(\mathbb {K}\langle \tilde{x}_1,\dots ,\tilde{x}_n\rangle _2\). We define \(R :=(r_\alpha )\) and \(x_i :=\tilde{x}_i\bmod R\); we also denote by R the set of relations in any algebra to appear later.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Max Planck Institute for MathematicsBonnGermany

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