On generalized Witt algebras in one variable

Ki-bong NAM; Jonathan Pakianathan

Article Details

On generalized Witt algebras in one variable

Authors:

Ki-bong NAM | Jonathan Pakianathan

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2011 Vol.35 Issue 3, pp.405-436

Document Type:

Article

Publication Date:

2/2011

Document Language:

Turkish

Document ID (Org No.):

2000747938

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Keywords:

Abstract:

 

We study a class of infinite dimensional Lie algebras called  generalized Witt algebras (in one variable).  These include the  classical Witt algebra and the centerless Virasoro algebra as  important examples.  We show that any such generalized Witt algebra is a semisimple,  indecomposable Lie algebra which does not contain any abelian Lie  subalgebras of dimension greater than one.  We develop an invariant of these generalized Witt algebras called the  spectrum, and use it to show that there exist infinite families of  nonisomorphic, simple, generalized Witt algebras and infinite families  of nonisomorphic, nonsimple, generalized Witt algebras.  We develop a machinery that can be used to study the endomorphisms of  a generalized Witt algebra in the case that the spectrum is  discrete.''  We use this to show that, among other things, every  nonzero Lie algebra endomorphism of the classical Witt algebra is an  automorphism and every endomorphism of the centerless Virasoro algebra  fixes a canonical element up to scalar multiplication.  However, not every injective Lie algebra endomorphism of the  centerless Virasoro algebra is an automorphism.

 

We study a class of infinite dimensional Lie algebras called  generalized Witt algebras (in one variable).  These include the  classical Witt algebra and the centerless Virasoro algebra as  important examples.  We show that any such generalized Witt algebra is a semisimple,  indecomposable Lie algebra which does not contain any abelian Lie  subalgebras of dimension greater than one.  We develop an invariant of these generalized Witt algebras called the  spectrum, and use it to show that there exist infinite families of  nonisomorphic, simple, generalized Witt algebras and infinite families  of nonisomorphic, nonsimple, generalized Witt algebras.  We develop a machinery that can be used to study the endomorphisms of  a generalized Witt algebra in the case that the spectrum is  discrete.''  We use this to show that, among other things, every  nonzero Lie algebra endomorphism of the classical Witt algebra is an  automorphism and every endomorphism of the centerless Virasoro algebra  fixes a canonical element up to scalar multiplication.  However, not every injective Lie algebra endomorphism of the  centerless Virasoro algebra is an automorphism.

Article Details

On generalized Witt algebras in one variable

Authors:

Ki-bong NAM | Jonathan Pakianathan

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2011 Vol.35 Issue 3, pp.405-436

Document Type:

Article

Publication Date:

2/2011

Document Language:

Turkish

Document ID (Org No.):

2000747938

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Infinite dimensional Lie algebra

Virasoro algebra

Issues

Article Details

On generalized Witt algebras in one variable

Authors: Ki-bong NAM | Jonathan Pakianathan

Publisher: TUBITAK

Journal: Turkish Journal of Mathematics 2011 Vol.35 Issue 3, pp.405-436

Document Type: Article

Publication Date: 2/2011

Document Language: Turkish

Document ID (Org No.): 2000747938

Frequency: Bimonthly

eISSN: 1303-6149

pISSN: 1300-0098

Infinite dimensional Lie algebra

Virasoro algebra

Issues

Authors:

Ki-bong NAM | Jonathan Pakianathan

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2011 Vol.35 Issue 3, pp.405-436

Document Type:

Article

Publication Date:

2/2011

Document Language:

Turkish

Document ID (Org No.):

2000747938

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098