Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.

Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.

1. Are the two $G^{∨}$ and $G^{'}$ defined in different settings? Or are their definitions related? What are the intuitions behind their definitions?

2. Are the two $G^{∨}$ and $G^{'}$ exactly the same?

3. What are the constraints on $G$ to give such groups: $G^{∨}$ and $G^{'}$? (eg, simple, semi-simple, connected, simply-connected, Lie group or not, topological group, etc.)

A Reference on Goddard-Nyuts-Olive dual group: P. Goddard(Cambridge U.), J. Nuyts(UMH, Mons), David I. Olive(CERN and Bohr Inst.) Dec 1, 1976, Published in: Nucl.Phys.B 125 (1977) 1-28 DOI: 10.1016/0550-3213(77)90221-8 https://www.sciencedirect.com/science/article/pii/0550321377902218?via%3Dihub

Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.

Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.

1. Are the two $G^{∨}$ and $G^{'}$ defined in different settings? Or are their definitions related? What are the intuitions and motivations behind their definitions?

2. Are the two $G^{∨}$ and $G^{'}$ exactly the same?

3. What are the constraints on $G$ to give such groups: $G^{∨}$ and $G^{'}$? (eg, compact or not, simple, semi-simple, connected, simply-connected, Lie group or not, topological group, etc.)

A Reference on Goddard-Nyuts-Olive dual group: P. Goddard(Cambridge U.), J. Nuyts(UMH, Mons), David I. Olive(CERN and Bohr Inst.) Dec 1, 1976, Published in: Nucl.Phys.B 125 (1977) 1-28 DOI: 10.1016/0550-3213(77)90221-8 https://www.sciencedirect.com/science/article/pii/0550321377902218?via%3Dihub

Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.

Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.

1. Are the two $G^{∨}$ and $G^{'}$ defined in different settings? Or are their definitions related? What are the intuitions behind their definitions?

2. Are the two $G^{∨}$ and $G^{'}$ exactly the same?

3. What are the constraints on $G$ to give such groups: $G^{∨}$ and $G^{'}$?

A Reference on Goddard-Nyuts-Olive dual group: P. Goddard(Cambridge U.), J. Nuyts(UMH, Mons), David I. Olive(CERN and Bohr Inst.) Dec 1, 1976, Published in: Nucl.Phys.B 125 (1977) 1-28 DOI: 10.1016/0550-3213(77)90221-8 https://www.sciencedirect.com/science/article/pii/0550321377902218?via%3Dihub

Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.

Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.

1. Are the two $G^{∨}$ and $G^{'}$ defined in different settings? Or are their definitions related? What are the intuitions behind their definitions?

2. Are the two $G^{∨}$ and $G^{'}$ exactly the same?

3. What are the constraints on $G$ to give such groups: $G^{∨}$ and $G^{'}$? (eg, simple, semi-simple, connected, simply-connected, Lie group or not, topological group, etc.)

A Reference on Goddard-Nyuts-Olive dual group: P. Goddard(Cambridge U.), J. Nuyts(UMH, Mons), David I. Olive(CERN and Bohr Inst.) Dec 1, 1976, Published in: Nucl.Phys.B 125 (1977) 1-28 DOI: 10.1016/0550-3213(77)90221-8 https://www.sciencedirect.com/science/article/pii/0550321377902218?via%3Dihub