×

The discriminant controls automorphism groups of noncommutative algebras. (English) Zbl 1337.16032

Let \(A\) be an associative filtered algebra over a field \(k\) such that \(\text{gr\,}A\) is a domain, and \(A\) is affine over its center \(R\). Under some assumption on discriminant of the extension \(A/R\) it is shown that \(A_1\) is stable under every automorphism of \(A\). Any automorphism of the polynomial algebra \(A[t]\) has the form \(h(t)=ct+r\), \(c\in k^*\), \(r\in R\), and \(g(A)=A\). If the base field \(k\) has characteristic zero, then every locally nilpotent derivation of \(A\) is zero and the automorphism group is an extension of \((k^*)\) by a finite group.
In particular let \(W_n\) be the algebra generated by \(x_1,\ldots,x_n\) with defining relations \(x_ix_j+x_jx_i=1\) for all \(i\neq j\). Then the automorphism group of \(W_n\) is a direct product of the symmetric group \(S_n\) and the group \(\{\pm 1\}\).

MSC:

16W20 Automorphisms and endomorphisms
16W70 Filtered associative rings; filtrational and graded techniques
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Alaca, S.; Williams, K. S., Introductory algebraic number theory, (2004), Cambridge Univ. Press Cambridge · Zbl 1035.11001
[2] Alev, J.; Chamarie, M., Dérivations et automorphismes de quelques algébres quantiques, Comm. Algebra, 20, 6, 1787-1802, (1992) · Zbl 0760.17003
[3] Alev, J.; Dumas, F., Rigidité des plongements des quotients primitifs minimaux de \(U_q(\mathit{sl}(2))\) dans l’algébre quantique de Weyl-hayashi, Nagoya Math. J., 143, 119-146, (1996) · Zbl 0862.16019
[4] Andruskiewitsch, N.; Dumas, F., On the automorphisms of \(U_q^+(g)\), (Quantum Groups, IRMA Lect. Math. Theor. Phys., vol. 12, (2008), Eur. Math. Soc. Zürich), 107-133 · Zbl 1182.17003
[5] Artin, M.; Schelter, W., Graded algebras of global dimension 3, Adv. Math., 66, 171-216, (1987) · Zbl 0633.16001
[6] Bavula, V. V.; Jordan, D. A., Isomorphism problems and groups of automorphisms for generalized Weyl algebras, Trans. Amer. Math. Soc., 353, 2, 769-794, (2001) · Zbl 0961.16016
[7] Ceken, S.; Palmieri, J.; Wang, Y.-H.; Zhang, J. J., Discriminant criterion and the automorphism group of quantized algebras, (2014), preprint
[8] S. Ceken, J. Palmieri, Y.-H. Wang, J.J. Zhang, Invariant theory for quantum Weyl algebras under finite group action, in preparation. · Zbl 1372.16044
[9] Chan, K.; Walton, C.; Wang, Y.-H.; Zhang, J. J., Hopf actions on filtered regular algebras, J. Algebra, 397, 68-90, (2014) · Zbl 1306.16026
[10] Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V., Discriminants, resultants and multidimensional determinants, Mod. Birkhäuser Class., (2008), Birkhäuser Boston · Zbl 1138.14001
[11] Gómez-Torrecillas, J.; El Kaoutit, L., The group of automorphisms of the coordinate ring of quantum symplectic space, Beiträge Algebra Geom., 43, 2, 597-601, (2002) · Zbl 1016.16033
[12] Goodearl, K. R.; Yakimov, M. T., Unipotent and Nakayama automorphisms of quantum nilpotent algebras, (2013), preprint
[13] Humphreys, J. E., Linear algebraic groups, Grad. Texts in Math., vol. 21, (1975), Springer-Verlag New York, Heidelberg · Zbl 0325.20039
[14] Kirkman, E.; Kuzmanovich, J.; Zhang, J. J., Invariants of \((- 1)\)-skew polynomial rings under permutation representations, (Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, Contemp. Math., vol. 623, (2014), Amer. Math. Soc. Providence, RI), 155-192 · Zbl 1335.16030
[15] McConnell, J. C.; Robson, J. C., Noncommutative Noetherian rings, (2001), Amer. Math. Soc. Providence, RI, with the cooperation of L.W. Small, revised edition · Zbl 0980.16019
[16] Reiner, I., Maximal orders, London Math. Soc. Monogr. New Ser., vol. 28, (2003), The Clarendon Press, Oxford University Press Oxford · Zbl 1024.16008
[17] Shestakov, I.; Umirbaev, U., The tame and the wild automorphisms of polynomial rings in three variables, J. Amer. Math. Soc., 17, 1, 197-227, (2004) · Zbl 1056.14085
[18] Stein, W. A., Algebraic number theory: a computational approach, preprint
[19] Suárez-Alvarez, M.; Vivas, Q., Automorphisms and isomorphism of quantum generalized Weyl algebras, (2012), preprint
[20] Tornheim, L., The Sylvester-franke theorem, Amer. Math. Monthly, 59, 6, 389-391, (1952) · Zbl 0046.01004
[21] Yakimov, M., The launois-lenagan conjecture, J. Algebra, 392, 1-9, (2013) · Zbl 1293.16035
[22] Yakimov, M., Rigidity of quantum tori and the andruskiewitsch-dumas conjecture, Selecta Math., 20, 2, 421-464, (2014) · Zbl 1300.17014
[23] Yekutieli, A.; Zhang, J. J., Dualizing complexes and perverse modules over differential algebras, Compos. Math., 141, 3, 620-654, (2005) · Zbl 1123.16033
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.