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Invariant ideals and polynomial forms. (English) Zbl 0998.16017
Let \(G\) be a quasi-simple group of Lie type defined over an infinite field \(F\) of characterstic \(p>0\) and let \(V\) be a finite dimensional vector space over a field \(E\) of the same characteristic \(p\). Assume that \(G\) acts nontrivially on \(V\) by way of the representation \(\phi\colon G\to\text{GL}(V)\), and that \(V\) contains no proper \(G\)-stable subgroup. If \(K\) is a field of characteristic different from \(p\), then the main result of this paper asserts that the augmentation ideal \(\omega K(V)\) of the group algebra \(K[V]\) is the unique proper \(G\)-stable ideal of \(K[V]\). Thus the author completes the work of D. S. Passman and A. E. Zalesskij [Trans. Am. Math. Soc. 353, No. 7, 2971-2982 (2001; Zbl 0981.16024)] by studying the nonrational representations of \(G\).

MSC:
16S34 Group rings
20F50 Periodic groups; locally finite groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16D25 Ideals in associative algebras
20G05 Representation theory for linear algebraic groups
20E32 Simple groups
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[1] V. V. Belyaev, Locally finite Chevalley groups, Studies in group theory, Akad. Nauk SSSR, Ural. Nauchn. Tsentr, Sverdlovsk, 1984, pp. 39 – 50, 150 (Russian). · Zbl 0587.20019
[2] K. Bonvallet, B. Hartley, D. S. Passman, and M. K. Smith, Group rings with simple augmentation ideals, Proc. Amer. Math. Soc. 56 (1976), 79 – 82. · Zbl 0334.16015
[3] Armand Borel and Jacques Tits, Homomorphismes ”abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499 – 571 (French). · Zbl 0272.14013 · doi:10.2307/1970833 · doi.org
[4] A. V. Borovik, Periodic linear groups of odd characteristic, Dokl. Akad. Nauk SSSR 266 (1982), no. 6, 1289 – 1291 (Russian). · Zbl 0511.20023
[5] C. J. B. Brookes and D. M. Evans, Augmentation modules for affine groups, Math. Proc. Cambridge Philos. Soc. 130 (2001), 287-294. CMP 2001:06 · Zbl 1005.20005
[6] Roger W. Carter, Simple groups of Lie type, John Wiley & Sons, London-New York-Sydney, 1972. Pure and Applied Mathematics, Vol. 28. · Zbl 0248.20015
[7] B. Hartley and G. Shute, Monomorphisms and direct limits of finite groups of Lie type, Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 137, 49 – 71. · Zbl 0547.20024 · doi:10.1093/qmath/35.1.49 · doi.org
[8] Jan Ambrosiewicz, On conjugacy classes in linear groups, Demonstratio Math. 26 (1993), no. 2, 359 – 362. · Zbl 0805.20033
[9] B. Hartley and A. E. Zalesskiĭ, Confined subgroups of simple locally finite groups and ideals of their group rings, J. London Math. Soc. (2) 55 (1997), no. 2, 210 – 230. · Zbl 0866.16016 · doi:10.1112/S0024610796004759 · doi.org
[10] Gerald J. Janusz, Algebraic number fields, 2nd ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, Providence, RI, 1996. · Zbl 0854.11001
[11] F. Leinen and O. Puglisi, Ideals in group algebras of simple locally finite groups of \(1\)-type, to appear. · Zbl 1065.16023
[12] J. M. Osterburg, D. S. Passman, and A. E. Zalesskiĭ\kern.15em, Invariant ideals of abelian group algebras under the multiplicative action of a field, II, Proc. Amer. Math. Soc., to appear. · Zbl 0992.16022
[13] Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. · Zbl 0368.16003
[14] D. S. Passman and A. E. Zalesskiĭ\kern.15em, Invariant ideals of abelian group algebras and representations of groups of Lie type. Trans. Amer. Math. Soc. 353 (2001), 2971-2982. · Zbl 0981.16024
[15] J. E. Roseblade and P. F. Smith, A note on hypercentral group rings, J. London Math. Soc. (2) 13 (1976), no. 1, 183 – 190. · Zbl 0328.16010 · doi:10.1112/jlms/s2-13.1.183 · doi.org
[16] Gary M. Seitz, Abstract homomorphisms of algebraic groups, J. London Math. Soc. (2) 56 (1997), no. 1, 104 – 124. · Zbl 0904.20038 · doi:10.1112/S0024610797005176 · doi.org
[17] Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. · Zbl 1196.22001
[18] Simon Thomas, An identification theorem for the locally finite nontwisted Chevalley groups, Arch. Math. (Basel) 40 (1983), no. 1, 21 – 31. , https://doi.org/10.1007/BF01192748 Simon Thomas, The classification of the simple periodic linear groups, Arch. Math. (Basel) 41 (1983), no. 2, 103 – 116. · Zbl 0518.20039 · doi:10.1007/BF01196865 · doi.org
[19] A. E. Zalesskiĭ\kern.15em, Intersection theorems in group rings (in Russian), No. 395-74, VINITI, Moscow, 1974.
[20] A. E. Zalesskiĭ, Group rings of inductive limits of alternating groups, Algebra i Analiz 2 (1990), no. 6, 132 – 149 (Russian); English transl., Leningrad Math. J. 2 (1991), no. 6, 1287 – 1303. · Zbl 0711.20004
[21] A. E. Zalesskiĭ, A simplicity condition for the fundamental ideal of the modular group algebra of a locally finite group, Ukrain. Mat. Zh. 43 (1991), no. 7-8, 1088 – 1091 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 43 (1991), no. 7-8, 1021 – 1024 (1992). · Zbl 0789.20019 · doi:10.1007/BF01058712 · doi.org
[22] A. E. Zalesskiĭ, Group rings of simple locally finite groups, Finite and locally finite groups (Istanbul, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 471, Kluwer Acad. Publ., Dordrecht, 1995, pp. 219 – 246. · Zbl 0839.16021 · doi:10.1007/978-94-011-0329-9_9 · doi.org
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