The Jacobian module of a Lie algebra
HTML articles powered by AMS MathViewer
- by J. P. Brennan, M. V. Pinto and W. V. Vasconcelos PDF
- Trans. Amer. Math. Soc. 321 (1990), 183-196 Request permission
Abstract:
There is a natural way to associate to the commuting variety $C(A)$ of an algebra $A$ a module over a polynomial ring. It serves as a vehicle to study the arithmetical properties of $C(A)$, particularly Cohen-Macaulayness. The focus here is on Lie algebras and some of their representations.References
- Luchezar L. Avramov, Complete intersections and symmetric algebras, J. Algebra 73 (1981), no. 1, 248–263. MR 641643, DOI 10.1016/0021-8693(81)90357-4
- N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Hermann, Paris, 1975 (French). Actualités Sci. Indust., No. 1364. MR 0453824
- David A. Buchsbaum and David Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259–268. MR 314819, DOI 10.1016/0021-8693(73)90044-6
- J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204. MR 142592, DOI 10.1098/rspa.1962.0170
- F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbücher für Mathematik, Band 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
- Murray Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961), 324–348. MR 132079, DOI 10.2307/1970336
- Robert M. Guralnick, A note on pairs of matrices with rank one commutator, Linear and Multilinear Algebra 8 (1979/80), no. 2, 97–99. MR 552353, DOI 10.1080/03081087908817305
- Klaus Hulek, A remark on certain matrix varieties, Linear and Multilinear Algebra 10 (1981), no. 3, 169–172. MR 630145, DOI 10.1080/03081088108817409
- J. Herzog, A. Simis, and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), no. 2, 661–683. MR 737891, DOI 10.1090/S0002-9947-1984-0737891-6
- Craig Huneke, On the symmetric algebra of a module, J. Algebra 69 (1981), no. 1, 113–119. MR 613861, DOI 10.1016/0021-8693(81)90131-9
- C. Huneke and M. Rossi, The dimension and components of symmetric algebras, J. Algebra 98 (1986), no. 1, 200–210. MR 825143, DOI 10.1016/0021-8693(86)90023-2
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- Tadeusz Józefiak, Ideals generated by minors of a symmetric matrix, Comment. Math. Helv. 53 (1978), no. 4, 595–607. MR 511849, DOI 10.1007/BF02566100
- H. Kleppe and D. Laksov, The algebraic structure and deformation of Pfaffian schemes, J. Algebra 64 (1980), no. 1, 167–189. MR 575789, DOI 10.1016/0021-8693(80)90140-4
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- T. S. Motzkin and Olga Taussky, Pairs of matrices with property $L$. II, Trans. Amer. Math. Soc. 80 (1955), 387–401. MR 86781, DOI 10.1090/S0002-9947-1955-0086781-5
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- Michael G. Neubauer, The variety of pairs of matrices with rank$(AB-BA)\leq 1$, Proc. Amer. Math. Soc. 105 (1989), no. 4, 787–792. MR 931743, DOI 10.1090/S0002-9939-1989-0931743-4
- R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math. 38 (1979), no. 3, 311–327. MR 535074
- Pierre Samuel, Anneaux gradués factoriels et modules réflexifs, Bull. Soc. Math. France 92 (1964), 237–249 (French). MR 186702
- Aron Simis and Wolmer V. Vasconcelos, On the dimension and integrality of symmetric algebras, Math. Z. 177 (1981), no. 3, 341–358. MR 618200, DOI 10.1007/BF01162067
- Aron Simis and Wolmer V. Vasconcelos, Krull dimension and integrality of symmetric algebras, Manuscripta Math. 61 (1988), no. 1, 63–78. MR 939141, DOI 10.1007/BF01153583
- N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Hermann, Paris, 1975 (French). Actualités Sci. Indust., No. 1364. MR 0453824
- David A. Buchsbaum and David Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259–268. MR 314819, DOI 10.1016/0021-8693(73)90044-6
- J. A. Eagon and D. G. Northcott, Ideals defined by matrices and a certain complex associated with them, Proc. Roy. Soc. London Ser. A 269 (1962), 188–204. MR 142592, DOI 10.1098/rspa.1962.0170
- F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbücher für Mathematik, Band 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
- Murray Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961), 324–348. MR 132079, DOI 10.2307/1970336
- Robert M. Guralnick, A note on pairs of matrices with rank one commutator, Linear and Multilinear Algebra 8 (1979/80), no. 2, 97–99. MR 552353, DOI 10.1080/03081087908817305
- Klaus Hulek, A remark on certain matrix varieties, Linear and Multilinear Algebra 10 (1981), no. 3, 169–172. MR 630145, DOI 10.1080/03081088108817409
- J. Herzog, A. Simis, and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), no. 2, 661–683. MR 737891, DOI 10.1090/S0002-9947-1984-0737891-6
- Craig Huneke, On the symmetric algebra of a module, J. Algebra 69 (1981), no. 1, 113–119. MR 613861, DOI 10.1016/0021-8693(81)90131-9
- C. Huneke and M. Rossi, The dimension and components of symmetric algebras, J. Algebra 98 (1986), no. 1, 200–210. MR 825143, DOI 10.1016/0021-8693(86)90023-2
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- Tadeusz Józefiak, Ideals generated by minors of a symmetric matrix, Comment. Math. Helv. 53 (1978), no. 4, 595–607. MR 511849, DOI 10.1007/BF02566100
- H. Kleppe and D. Laksov, The algebraic structure and deformation of Pfaffian schemes, J. Algebra 64 (1980), no. 1, 167–189. MR 575789, DOI 10.1016/0021-8693(80)90140-4
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- T. S. Motzkin and Olga Taussky, Pairs of matrices with property $L$. II, Trans. Amer. Math. Soc. 80 (1955), 387–401. MR 86781, DOI 10.1090/S0002-9947-1955-0086781-5
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- Michael G. Neubauer, The variety of pairs of matrices with rank$(AB-BA)\leq 1$, Proc. Amer. Math. Soc. 105 (1989), no. 4, 787–792. MR 931743, DOI 10.1090/S0002-9939-1989-0931743-4
- R. W. Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math. 38 (1979), no. 3, 311–327. MR 535074
- Pierre Samuel, Anneaux gradués factoriels et modules réflexifs, Bull. Soc. Math. France 92 (1964), 237–249 (French). MR 186702
- Aron Simis and Wolmer V. Vasconcelos, On the dimension and integrality of symmetric algebras, Math. Z. 177 (1981), no. 3, 341–358. MR 618200, DOI 10.1007/BF01162067
- Aron Simis and Wolmer V. Vasconcelos, Krull dimension and integrality of symmetric algebras, Manuscripta Math. 61 (1988), no. 1, 63–78. MR 939141, DOI 10.1007/BF01153583
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 321 (1990), 183-196
- MSC: Primary 13H10; Secondary 13C05, 13C15
- DOI: https://doi.org/10.1090/S0002-9947-1990-0958883-0
- MathSciNet review: 958883