Verma modules over  Lie algebras defined by torsion-free
abelian groups

V. Mazorchuk

We study a submodule structure of standard highest weight modules over Lie algebras defined by torsion-free abelian groups. All such modules for higher rank Virasoro algebras and for algebras defined by $(\Bbb Q,+)$ are classified. We also prove Futorny theorem for the~algebras defined by $(\Bbb Q,+)$.

1. B. Feigin, D. Fuchs Representations of the Virasoro algebra// preprint of seminar on supermanifolds 194 5, D. Leites, ed.

2. V.M. Futorny Weight representations of the semisimple, finite-dimensional Lie algebras// Ph.D. Thesis, Kiev - 1986

3. M. Golenischeva-Kutuzova, D. Lebedev $\Bbb Z$-graded trigonometrical Lie subalgebras in $\hat{A}_{\infty}$, $\hat{B}_{\infty}$, $\hat{C}_{\infty}$, $\hat{D}_{\infty}$ and their vertex-operator representations// Func. anal. and appl.-- 1993 V. 27 p.12--24

4. O. Mathieu Classification of Harish-Chandra modules over the Virasoro Lie algebra// Invent. math. – 1992V. 107p.225–234

5. B. Moody, A. Pianzola Lie algebras with triangular decomposition, J. Wiley – 1994

6. J. Patera, H. Zassenhaus The higher rank Virasoro Algebras// Commun. Math. Phys. – 1991V. 136 p.1–14

7. Eswara S. Rao Representations of Witt Algebras// Publ. RIMS, Kyoto Univ -- 1994 V.30 p.191--201

8. M.V. Saveliev, A.M. Vershik Continual analogues of contragradient Lie algebras// Commun. Math. Phys. V.126 № 2 p.367--378

9. A. Vershik Lie algebras generated by dynamical systems// St. Petersburg Math. J.-- 1993 V.4 №6 p.1143-- 1151

10. M.I. Zaytseva The set of orderings on abelian group// Uspechi Math. Nauk-1953 V. VIII № 1 p.135-137 (Russian)

	Verma modules over Lie algebras defined by torsion-free abelian groups
Author	V. Mazorchuk Department of Mechanics and Mathematics,Kyiv Taras Shevchenko University,64, Volodymyrska st., 252033, Kyiv, Ukraine
Abstract	We study a submodule structure of standard highest weight modules over Lie algebras defined by torsion-free abelian groups. All such modules for higher rank Virasoro algebras and for algebras defined by $(\Bbb Q,+)$ are classified. We also prove Futorny theorem for the~algebras defined by $(\Bbb Q,+)$.
Keywords	submodule structure, standard highest weight modules, Lie algebras, torsion-free abelian groups, higher rank Virasoro algebras, Futorny theorem
DOI	doi:10.30970/ms.9.2.120-124
Reference	1. B. Feigin, D. Fuchs Representations of the Virasoro algebra// preprint of seminar on supermanifolds 194 5, D. Leites, ed. 2. V.M. Futorny Weight representations of the semisimple, finite-dimensional Lie algebras// Ph.D. Thesis, Kiev - 1986 3. M. Golenischeva-Kutuzova, D. Lebedev $\Bbb Z$-graded trigonometrical Lie subalgebras in $\hat{A}_{\infty}$, $\hat{B}_{\infty}$, $\hat{C}_{\infty}$, $\hat{D}_{\infty}$ and their vertex-operator representations// Func. anal. and appl.-- 1993 V. 27 p.12--24 4. O. Mathieu Classification of Harish-Chandra modules over the Virasoro Lie algebra// Invent. math. – 1992V. 107p.225–234 5. B. Moody, A. Pianzola Lie algebras with triangular decomposition, J. Wiley – 1994 6. J. Patera, H. Zassenhaus The higher rank Virasoro Algebras// Commun. Math. Phys. – 1991V. 136 p.1–14 7. Eswara S. Rao Representations of Witt Algebras// Publ. RIMS, Kyoto Univ -- 1994 V.30 p.191--201 8. M.V. Saveliev, A.M. Vershik Continual analogues of contragradient Lie algebras// Commun. Math. Phys. V.126 № 2 p.367--378 9. A. Vershik Lie algebras generated by dynamical systems// St. Petersburg Math. J.-- 1993 V.4 №6 p.1143-- 1151 10. M.I. Zaytseva The set of orderings on abelian group// Uspechi Math. Nauk-1953 V. VIII № 1 p.135-137 (Russian)
Pages	120-124
Volume	9
Issue	2
Year	1998
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Verma modules over Lie algebras defined by torsion-free abelian groups