Non-finitary Generalizations of Nil-triangular Subalgebras
of Chevalley Algebras
Let NФ(K) be a niltriangular subalgebra of Chevalley algebra over a field or ring K associated with root system Ф of classical type. For type An−1 it is associated to algebra NT(n, K) of (lower) nil-triangular n×n-matrices over K. The algebra R = NT(Г, K) of all nil-triangular Г-matrices α = || αij ||i,j∈Г over K with indices from chain Г of natural numbers gives its non-finitary generalization. It is proved, (together with radicalness of ring R) that if K is a ring without zero divizors, then ideals Ti,i−1 of all Г-matrices with zeros above i-th row and in columns with numbers > i exhausts all maximal commutative ideals of the ring R and associated Lie rings R(−), and also maximal normal subgroups of adjoint group (it is isomorphic to the generalize unitriangular group UT(Г, K)). As corollary we obtain that the automorphism groups Aut R and Aut R(−) coincide. Partially automorphisms studied earlier, in particulary, for UT(Г, K) when K is a field.
Well-known (1990) special matrix representation of Lie algebras NФ(K) allows to construct non-finitary generalizations NG(K) of type G = BГ, CГ and DГ. Be research automorphisms by transfer to factors of Lie ring NG(K) which is isomorphic to NT(Г, K).
On the other hand, for any chain Г finitary nil-triangular Г-matrices forms finitary Lie algebra FNG(Г, K) of type G = AГ ( i.e., FNT(Г, K)), BГ, CГ and DГ. Earlier automorphisms was studied (V. M. Levchuk and G. S. Sulejmanova, 1987 and 2009) for Lie ring FNT(Г, K) over ring K without zero divizors and, also, for finitary generalizations of unipotent subgroups of Chevalley group of classical type over the field (including twisted types).
Julianna Bekker, Postgraduate Student, Siberian Federal University, 79, Pr. Svobodniy., Krasnoyarsk, 660041, Russian Federation, tel.: 8 923 377 76 30, e-mail: email@example.com
Vladimir Levchuk, Head of Department, Professor, Siberian Federal, University, 79, Pr. Svobodniy., Krasnoyarsk, 660041, Russian Federation, tel.: 89504360807, e-mail: firstname.lastname@example.org
Elena Sotnikova, Master Student, Siberian Federal University, 79, Pr. Svobodniy., Krasnoyarsk, 660041, Russian Federation, tel.: 89130384800, e-mail: email@example.com
Bekker J.V., Levchuk V.M., Sotnikova E.A. Non-finitary Generalizations of Nil-triangular Subalgebras of Chevalley Algebras. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 29, pp. 39-51. (in Russian) https://doi.org/10.26516/1997-7670.2019.29.39
- Levchuk V.M. Automorphisms of unipotent subgroups of chevalley groups. Algebra and Logic, 1990, vol. 29, is. 3, pp. 211-224. https://doi.org/10.1007/BF01979936.
- Levchuk V.M., Litavrin A.V., Hodyunya N.D., Cygankov V.V. Nil’treugol’nye podalgebry algebry Chevalley [Nilthriangular Subalgebras Chevalley Algebra]. Vladikav. mat. zhurnal [Vladikav. mat. magazine], 2015, vol. 17, iss. 2, pp. 37–46.
- Levchuk V.M. Some locally nilpotent rings and their adjoined groups. Matematicheskie Zametki, 1987, vol. 42, no. 5, pp. 631–641. https://doi.org/10.1007/BF01137426.
- Levchuk V.M. Connections between a unitriangular group and certain rings. Chap. 2: Groups of automorphisms. Siberian Mathematical Journal, 1983, vol. 24, is. 4, pp. 543–557. https://doi.org/10.1007/BF00969552.
- Levchuk V.M., Sulejmanova G.S. Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2009, vol. 15, no. 2, pp. 118–127. https://doi.org/10.1134/S0081543809070128
- Merzlyakov YU.I. Equisubgroups of unitriangular groups: a criterion of self normalizability. Russian Ac. Sci. Dokl. Math., 1995, vol. 50, no. 3, pp. 507–511.
- Holubovski V. Algebraicheskie svojstva grupp beskonechnyh matric [Algebraic properties of groups of infinite matrices]. Gliwice, Wydawnictwo Politechniki Slaska, 2017, 140 p.
- Carter R. Simple Groups of Lie type. Wiley and Sons, New York, 1972.
- Jacobson N. Lie Algebras. Int. Publ., New York, 1962.
- Levchuk V.M. Niltriangular subalgebra of Chevalley algebra: enveloping algebra, ideals and automorphisms. Dokl. Math., 2018, vol. 478, no. 1, pp. 23–27.
- Levchuk V.M., Radchenko O.V. Derivations of the locally nilpotent matrix rings. Journal of Algebra and Its Applications, 2010, vol. 9, no. 5, pp. 717–724.
- Slovik R. Bijective maps of infinite triangular and unitriangular matrices preserving commutators. Linear and Multilinear Algebra, 2013, vol. 61.8, pp. 1028–1040.