TITLE: What is the relation between characters of  a group and its lie algebra?
QUESTION [5 upvotes]: What is the relation between characters of  a group and its lie algebra?
Roughly,I know that there is a one to one correspondence between representations of a lie algebra and its simple connected lie group by the exp map,and two irreducible representations of a lie group are unequivalent if and only if their characters are different. 
Now,then,I think the above statement  will still hold for representations of lie algebras, but I am not sure about it .
If someone can give some advises?
Thanks  Kevin Buzzard. For represatatons of general  lie algebras the character (trace) don't work .
But for compact lie group,characters still work,so I restrict my question on  compact lie groups and their lie algebras.
In representation theory of semisimple lie algebras ,we have formal characters/algebra characters which are sums of some formal elements over the weights.
I wonder whether we can realize these algebra characters to be real characters ?
Or can we make them to be functions on cartan subalgebras or maximal torus?

REPLY [5 votes]: Let $G$ be a compact Lie group. Then the character of a finite-dimenstional representation $R_Λ$ of $G$, is the map $\mathrm{ch}_Λ\colon G \to \mathbb{C}$ defined by
$$
\mathrm{ch}_Λ(g):=\mathrm{Tr}(R_Λ(g))\equiv \sum_{n=1}^{\dimΛ}R^{n,n}_Λ(g), \qquad\forall g\in G,
$$
and it depends  only on the isomorphism class or the representation $R_Λ$. Furthermore, because of the cyclic invariance of the trace the character is a class function. Being class functions, the characters are completely determined by their values on the maximal torus $\mathbb{T}\subset G$, for $G$ semisimple. Now, the characer of the Lie algebra $\mathfrak{g}$ of $G$ is defined as
$$
χ_Λ(h)=\mathrm{Tr}(\exp [R_Λ(h)]),
$$
for $h$ in a Cartan subalgebra $\mathfrak{g_0}$ of $\mathfrak{g}$. If $t\in\mathbb{T}$, then $t=\exp h$ with $h\in\mathfrak{g_0}$, the Cartan subalgebra associated to $\mathbb{T}$. Therefore, one has the formula
$$
χ_Λ(h)=\mathrm{ch}_Λ(\exp h).
$$
However, this does not in general work globally on $\mathbb{T}$ because a single exponential may not be sufficient to obtain all of $\mathbb{T}$ from $\mathfrak{g_0}$ [1, Sec. 21.10] . 
References 
[1] J. Fuchs & C. Schweigert, Symmetries, Lie Algebras and Representations: A graduate course for physicists, Cambridge University Press (2003).