The characterizations of the quantum Witt algebra at roots of unity (Q1321037)

The two axiomatic characterizations of the quantum Witt algebra (\(q\)- deformation of a Witt algebra) at roots of unity are given. The quantum Witt algebra is basically considered as \(\mathbb{Z}\)-graded algebra with an action of the deformed Euler field \(\sigma f(z)= {1\over 2} (f(qz)+ f(q^{-1} z))\). It is stated that the quantum Witt algebra is a unique (graded) quantum Lie algebra, i.e., skew-symmetric with \(\sigma\)-skew Jacobi identity, with at most one-dimensional graded components. Secondly, it is characterized in terms of the flexible condition \(xy \sigma(z)= \sigma(x) yz\). For generic \(q\) the result was obtained earlier.