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This may well be something of a silly question, but if so, then all the more reason I get it straightened out.

I have in the past been working with representations of both groups and algebras, and in the literature I've been using, the traces of the representation matrices of both have always been referred to as characters of the representation.

Now, I'm looking properly into Lie algebras for the first time since far too long ago, and I come across a note that says that the weight of a representation of an algebra is the analogue of the character of a representation of a group.

Are weights of representations of algebras and characters of representations of algebras then the very same thing, and its just that people who work in associative algebra and people who work with Lie algebras uses differing terminology, or is there something of much importance that I am missing here?

Much look forward to getting this sorted out!

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Caveat: not a representation theorist here, and would love to be corrected if this is wrong.

It's a pretty weak (non-mathematical) analogy, and you should not think of them as being the same. What they have in common is that they are combinatorial objects which are relatively easy to compute with, and they tell you everything about the representation, so they are super-important.

So they are analogous in the human sense of the word analogous. But as far as I know, there isn't a nice uber-theory that unifies group representations and Lie algebra representations such that characters and weights become one-and-the-same.

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  • $\begingroup$ A case in point: Whereas a group representation has a unique character associated to it, a Lie algebra representation usually has many weights. (If anything, an irreducible representation is well-encoded in its highest weight.) $\endgroup$ – Torsten Schoeneberg Jan 6 at 18:51

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