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Konstantinos Kanakoglou
Konstantinos Kanakoglou
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The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.

Their classification, is in general an open project:

  • In the cocommutative case, over a field of $chark=0$, they are all universal enveloping algebras of lie algebras. If $k=\mathbb{C}$, this is by an old result of Milnor and Moore. You can find the statement for any field of char zero at Montgomery's book.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA]arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA]arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.

The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.

Their classification, is in general an open project:

  • In the cocommutative case, over a field of $chark=0$, they are all universal enveloping algebras of lie algebras. If $k=\mathbb{C}$, this is by an old result of Milnor and Moore. You can find the statement for any field of char zero at Montgomery's book.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.

The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.

Their classification, is in general an open project:

  • In the cocommutative case, over a field of $chark=0$, they are all universal enveloping algebras of lie algebras. If $k=\mathbb{C}$, this is by an old result of Milnor and Moore. You can find the statement for any field of char zero at Montgomery's book.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.
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Konstantinos Kanakoglou
Konstantinos Kanakoglou
  • 7.7k
  • 2
  • 23
  • 64

The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.

Their classification, is in general an open project:

  • In the cocommutative case, over the complex numbersa field of $chark=0$, they are all universal enveloping algebras of lie algebras. ThisIf $k=\mathbb{C}$, this is by an old result of Milnor and Moore. You can find the statement for any field of char zero at Montgomery's book.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.

The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.

Their classification, is in general an open project:

  • In the cocommutative case, over the complex numbers, they are all universal enveloping algebras of lie algebras. This is by an old result of Milnor and Moore.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.

The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.

Their classification, is in general an open project:

  • In the cocommutative case, over a field of $chark=0$, they are all universal enveloping algebras of lie algebras. If $k=\mathbb{C}$, this is by an old result of Milnor and Moore. You can find the statement for any field of char zero at Montgomery's book.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.
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Konstantinos Kanakoglou
Konstantinos Kanakoglou
  • 7.7k
  • 2
  • 23
  • 64

The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: https://mathoverflow.net/a/400761/85967Name for a Hopf algebra whose only grouplike element is the identity?.

Their classification, is in general an open project:

  • In the cocommutative case, over the complex numbers, they are all universal enveloping algebras of lie algebras. This is by an old result of Milnor and Moore.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.

The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: https://mathoverflow.net/a/400761/85967

Their classification, is in general an open project:

  • In the cocommutative case, over the complex numbers, they are all universal enveloping algebras of lie algebras. This is by an old result of Milnor and Moore.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.

The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.

Their classification, is in general an open project:

  • In the cocommutative case, over the complex numbers, they are all universal enveloping algebras of lie algebras. This is by an old result of Milnor and Moore.
  • In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.
Source Link
Konstantinos Kanakoglou
Konstantinos Kanakoglou
  • 7.7k
  • 2
  • 23
  • 64