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Representations of some PI algebras. (English) Zbl 1035.16020
For a prime affine PI-algebra \(A\) over an algebraically closed field of characteristic zero \(\text{degree\,}A\) is the maximal degree of an irreducible representation of \(A\). Suppose that \(R\) is an affine prime PI-algebra and \(A=R[x;\alpha,\delta]\) its skew polynomial extension. If \(A\) is also a PI-algebra then \[ \text{degree\,}A=\text{degree\,}R[x;\alpha]=l\cdot\text{degree\,}R, \] where \(l\) is the order of the restriction of \(\alpha\) to the center of \(R\). Let a quantized Weil algebra \(A_n^{q,\Lambda}\) be defined in terms of multiparameters \(\lambda_{ij}\) and \(q_i\). Suppose that each \(a_i\lambda_{ji}^{-1}\) is an \(m\)-th root of unity and every \(q_i\) is a primitive \(m\)-th root of unity. It is shown that \(\text{degree\,}A_n^{q,\Lambda}=m^n\).

16S36 Ordinary and skew polynomial rings and semigroup rings
16R99 Rings with polynomial identity
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
Full Text: DOI
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