# zbMATH — the first resource for mathematics

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 $$\text{degree\,}A_n^{q,\Lambda}=m^n$$.

##### MSC:
 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:
##### References:
 [1] Awami M., Bull. Soc. Math. Belg. Sér A 40 pp 175– (1988) [2] De Concini and Procesi, C. 1993.Quantum Groups, Lecture Notes in Mathematics Vol. 1565, 31–140. Springer Verlag. · Zbl 0795.17005 [3] Fukada N., Bull. Austral. Math. Soc. 57 pp 403– (1998) · Zbl 0923.16005 [4] Goodearl K. R., J. Algebra 150 pp 324– (1992) · Zbl 0779.16010 [5] Goodearl K. R., 16. London Math. Soc. Student Text Series 16, in: An Introduction to Noncommutative Noetherian Rings (1989) · Zbl 0679.16001 [6] Jakobsen H. P., J. Algebra 196 pp 458– (1997) · Zbl 0886.17015 [7] Jakobsen H. P., J. Algebra 246 pp 70– (2001) · Zbl 1077.16044 [8] Jordan D., J. Algebra 174 pp 267– (1995) · Zbl 0833.16025 [9] Jordan D., J. Pure and Applied Algebra 98 pp 45– (1995) [10] Jøndrup S., Proc. Amer. Math. Soc. 128 pp 1301– (2000) · Zbl 0949.16023 [11] Jøndrup S., Noncommutative Plane Geometry [12] McConnell J. C., Noncommutative Noetherian Rings (1987) · Zbl 0644.16008 [13] Morikawa H., Nagoya Math. J. 113 pp 153– (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.