Geodesic
On algebras of skew polynomials generated by quadratic homogeneous relations
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part IX, Tome 301 (2003), pp. 144-171 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider algebras, with two generators $a$ and $b$, generated by the quadratic relations $ba=\alpha a^2+\beta ab+\gamma b^2$, where the coefficients $\alpha$, $\beta$, and $\gamma$ belong to an arbitrary field $F$ of characteristics $0$. We find conditions for the algebra to be expressed as a skew polynomial algebra with generator $b$ over the polynomial ring $F[a]$. These conditions are equivalent to the existence of the Poincaré–Birkhoff–Witt basis, i.e., basis of the form $\{a^m,b^n\}$. 
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A. V. Golovashkin; V. M. Maximov. On algebras of skew polynomials generated by quadratic homogeneous relations. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part IX, Tome 301 (2003), pp. 144-171. http://geodesic.mathdoc.fr/item/ZNSL_2003_301_a3/

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