# Tag Info

Accepted

### Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules

I'm not quite sure this question rises to the level of MathOverflow, which is why I initially posted only a comment, but at the request of the question-asker I am converting my comment to an answer. ...
• 20.3k
Accepted

### Number of representations of a semisimple Lie algebra of any given dimension

For $\mathfrak{sl}_2\times \mathfrak{sl}_2$, the number of irreps of dimension $n$ is the number of factorizations $n=n_1n_2$ (you tensor the irreps of the two $\mathfrak{sl}_2$'s), so there's no ...
• 42.4k

• 159

• 54.8k
Accepted

### Sum of weights of an irreducible representation of $U(N)$

As discussed in the comments, your sum is a Weyl-fixed character, so trivial for $G = \operatorname{SU}(N)$ and a multiple of $\det = (1, \dotsc, 1)$ for $\operatorname U(N)$. To be concrete, as I ...
• 9,477
Accepted

### Classification of Lie group structures on $\mathbb{R}^n$

YCor claims here that the contractible Lie groups (this is equivalent to being diffeomorphic to $\mathbb{R}^n$, since a connected Lie group is diffeomorphic to the product of a Euclidean space times ...
• 111k
1 vote

• 54.8k