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edited Jul 25, 2018 at 14:14

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $k$$\mathbb{C}[[h]]$-algebras. Why is this true? The reason they cite is that $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$. Now, I know how to compute Hochschild cohomology of simpler algebras, but several problems arise for me in this setup:

I am unsure how to proceed in the computation of $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))$ (or more generally, replacing $U(\mathfrak{g})$ with $T(V)$, the tensor algebra of a vector space). How can I conclude that this is $0$?
Why does the "trivial" deformation look like a power series ring? They say that the multiplication in $U_h(\mathfrak{g})$ looks like "multiplication modulo $h$" in $U(\mathfrak{g})$; what does this mean precisely?
The authors instead deform $U(\mathfrak{g})$ as a Hopf algebra, but don't elaborate on what such a deformation should look like, what cohomology theory classifies such deformations, etc. Where can I find a resource that concretely answers these questions? An English translation of Drinfel'd's paper "Quasi-Hopf algebras" would be a good starting point, but I haven't been able to find one.

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $k$-algebras. Why is this true? The reason they cite is that $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$. Now, I know how to compute Hochschild cohomology of simpler algebras, but several problems arise for me in this setup:

I am unsure how to proceed in the computation of $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))$ (or more generally, replacing $U(\mathfrak{g})$ with $T(V)$, the tensor algebra of a vector space). How can I conclude that this is $0$?
Why does the "trivial" deformation look like a power series ring? They say that the multiplication in $U_h(\mathfrak{g})$ looks like "multiplication modulo $h$" in $U(\mathfrak{g})$; what does this mean precisely?
The authors instead deform $U(\mathfrak{g})$ as a Hopf algebra, but don't elaborate on what such a deformation should look like, what cohomology theory classifies such deformations, etc. Where can I find a resource that concretely answers these questions? An English translation of Drinfel'd's paper "Quasi-Hopf algebras" would be a good starting point, but I haven't been able to find one.

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $\mathbb{C}[[h]]$-algebras. Why is this true? The reason they cite is that $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$. Now, I know how to compute Hochschild cohomology of simpler algebras, but several problems arise for me in this setup:

I am unsure how to proceed in the computation of $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))$ (or more generally, replacing $U(\mathfrak{g})$ with $T(V)$, the tensor algebra of a vector space). How can I conclude that this is $0$?
Why does the "trivial" deformation look like a power series ring? They say that the multiplication in $U_h(\mathfrak{g})$ looks like "multiplication modulo $h$" in $U(\mathfrak{g})$; what does this mean precisely?
The authors instead deform $U(\mathfrak{g})$ as a Hopf algebra, but don't elaborate on what such a deformation should look like, what cohomology theory classifies such deformations, etc. Where can I find a resource that concretely answers these questions? An English translation of Drinfel'd's paper "Quasi-Hopf algebras" would be a good starting point, but I haven't been able to find one.

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asked Jul 23, 2018 at 19:10

cantwellnc

Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?

I have read in these lecture notes that every deformation $U_h(\mathfrak{g})$ of $U(\mathfrak{g})$ is trivial, i.e. isomorphic to $U(\mathfrak{g})[[h]]$ as associative $k$-algebras. Why is this true? The reason they cite is that $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$. Now, I know how to compute Hochschild cohomology of simpler algebras, but several problems arise for me in this setup:

I am unsure how to proceed in the computation of $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))$ (or more generally, replacing $U(\mathfrak{g})$ with $T(V)$, the tensor algebra of a vector space). How can I conclude that this is $0$?
Why does the "trivial" deformation look like a power series ring? They say that the multiplication in $U_h(\mathfrak{g})$ looks like "multiplication modulo $h$" in $U(\mathfrak{g})$; what does this mean precisely?
The authors instead deform $U(\mathfrak{g})$ as a Hopf algebra, but don't elaborate on what such a deformation should look like, what cohomology theory classifies such deformations, etc. Where can I find a resource that concretely answers these questions? An English translation of Drinfel'd's paper "Quasi-Hopf algebras" would be a good starting point, but I haven't been able to find one.