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I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie algebras which are defined on field with characteristic zero which its commutation relations are:

[J_m,J_n]=(m-n)J_m+n,

[J_m,P_n]=(m-n)P_m+n,

[P_m,P_n]=0.

In this connection, I infinitesimally deform the last commutator(the ideal of algebra) by adding some terms in its RHS and reach to an infinite dimensional rigid Lie algebra.The first commutator is a subalgebra which is known as Witt algebra and will remain rigid by deformation procedure. But I am not sure about the second commutator which can be deformed or not. This problem has connection with this question: Is the rigid Lie algebra I mentioned above "unique" and independent of how I deform the initial Lie algebra? In fact, if I started with other commutation relations would I reach to another rigid Lie algebra? Is there any theorem about "uniqueness" of rigid Lie algebra which is derived in deformation procedure?

I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie algebras which are defined on field with characteristic zero which its commutation relations are:

$$[J_m,J_n]=(m-n)J_{m+n},$$

$$[J_m,P_n]=(m-n)P_{m+n},$$

$$[P_m,P_n]=0.$$

In this connection, I infinitesimally deform the last commutator(the ideal of algebra) by adding some terms in its RHS and reach to an infinite dimensional rigid Lie algebra.The first commutator is a subalgebra which is known as Witt algebra and will remain rigid by deformation procedure. But I am not sure about the second commutator which can be deformed or not. This problem has connection with this question: Is the rigid Lie algebra I mentioned above "unique" and independent of how I deform the initial Lie algebra? In fact, if I started with other commutation relations would I reach to another rigid Lie algebra? Is there any theorem about "uniqueness" of rigid Lie algebra which is derived in deformation procedure?

I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie algebras which are defined on field with characteristic zero. In this connection, I infinitesimally deform one of commutation relations (Lie brackets) by adding some terms in its RHS and reach to an infinite dimensional rigid Lie algebra (Witt Lie algebra).The question is: Is this rigid Lie algebra unique and independent of how I deform the initial Lie algebra? In fact, if I started with other commutation relations would I reach to another rigid Lie algebra? Is there any theorem about "uniqueness" of rigid Lie algebra which is derived in deformation procedure?

I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie algebras which are defined on field with characteristic zero which its commutation relations are:

[J_m,J_n]=(m-n)J_m+n,

[J_m,P_n]=(m-n)P_m+n,

[P_m,P_n]=0. 

In this connection, I infinitesimally deform the last commutator(the ideal of algebra) by adding some terms in its RHS and reach to an infinite dimensional rigid Lie algebra.The first commutator is a subalgebra which is known as Witt algebra and will remain rigid by deformation procedure. But I am not sure about the second commutator which can be deformed or not. This problem has connection with this question: Is the rigid Lie algebra I mentioned above "unique" and independent of how I deform the initial Lie algebra? In fact, if I started with other commutation relations would I reach to another rigid Lie algebra? Is there any theorem about "uniqueness" of rigid Lie algebra which is derived in deformation procedure?

I am studying deformation and rigidity of some infinite dimensional Lie algebras. In this connection, I deform one of commutation relations (Lie brackets) by adding some terms and reach to an infinite dimensional rigid Lie algebra.The question is: This rigid Lie algebra is unique and independent of how I deform the initial Lie algebra? In fact, if I started with other commutation relations would I reach to another rigid Lie algebra? Is there any theorem about "uniqueness" of rigid Lie algebra which is derived in deformation procedure?

I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie algebras which are defined on field with characteristic zero. In this connection, I infinitesimally deform one of commutation relations (Lie brackets) by adding some terms in its RHS and reach to an infinite dimensional rigid Lie algebra (Witt Lie algebra).The question is: Is this rigid Lie algebra unique and independent of how I deform the initial Lie algebra? In fact, if I started with other commutation relations would I reach to another rigid Lie algebra? Is there any theorem about "uniqueness" of rigid Lie algebra which is derived in deformation procedure?