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The Niltriangular Subalgebra of the Chevalley Algebra: the Enveloping Algebra, Ideals, and Automorphisms


The enveloping algebra of the niltriangular subalgebra NΦ(K) of the Chevalley algebra of type An−1 is the algebra of niltriangular n × n matrices over K. The enveloping algebras R of other types constructed so far are nonassociative. For classical types, an explicit description of automorphisms of the rings R over any commutative associative ring with an identity is given; in the case where K is a field, all ideals in R are also listed. The enumeration of ideals in R for K = GF(q) leads to a solution of a combinatorial problem concerning ideals of the algebras NΦ(K).

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  1. H. C. Myung, Trans. Am. Math. Soc. 167, 79–88 (1972).

    Article  Google Scholar 

  2. R. Carter, Simple Groups of Lie Type (Wiley, New York, 1972).

    MATH  Google Scholar 

  3. V. M. Levchuk, Algebra Logika 29 (3), 315–338 (1990).

    MathSciNet  Article  Google Scholar 

  4. V. M. Levchuk and G. S. Suleimanova, J. Algebra 349 (1), 98–116 (2012).

    MathSciNet  Article  Google Scholar 

  5. V. M. Levchuk, Sibirsk. Mat. Zh. 24 (4), 64–80 (1983).

    MathSciNet  Google Scholar 

  6. Y. Cao, D. Jiang, and J. Wang, Int. J. Algebra Comput. 17 (3), 527–555 (2007).

    Article  Google Scholar 

  7. V. M. Levchuk and A. V. Litavrin, Sibirsk. Elektron. Mat. Izv. 13, 467–477 (2016).

    Google Scholar 

  8. V. M. Levchuk, Algebra Logika, 15 (5), 558–579 (1976).

    MathSciNet  Article  Google Scholar 

  9. R. Dubish and S. Perlis, Am. J. Math. 73 (3), 439–452 (1951).

    Article  Google Scholar 

  10. G. P. Egorychev and V. M. Levchuk, ACM SIGSAM Bull. 35 (2), 20–34 (2001).

    Article  Google Scholar 

  11. E. Sommers, Canad. Math. Bull. 48 (3), 460–472 (2005).

    MathSciNet  Article  Google Scholar 

  12. F. Kuzucuoglu and V. M. Levchuk, J. Algebra 243 (2), 473–485 (2001).

    MathSciNet  Article  Google Scholar 

  13. C. A. Athanasiadis and V. Reiner, SIAM J. Discr. Math. 18 (2), 397–417 (2004).

    Article  Google Scholar 

  14. C. A. Athanasiadis, Trans. Am. Math. Soc. 357 (1), 179–197 (2005).

    Article  Google Scholar 

  15. V. P. Krivokolesko and V. M. Levchuk, Tr. Inst.Mat. Mekh. Ural Otd. Ross. Akad. Nauk 21 (1), 166–171 (2015).

    Google Scholar 

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Correspondence to V. M. Levchuk.

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Original Russian Text © V.M. Levchuk, 2018, published in Doklady Akademii Nauk, 2018, Vol. 478, No. 2, pp. 137–140.

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Levchuk, V.M. The Niltriangular Subalgebra of the Chevalley Algebra: the Enveloping Algebra, Ideals, and Automorphisms. Dokl. Math. 97, 23–27 (2018).

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