# Cartan theorem

Cartan's theorem on the highest weight vector. Let $\mathfrak g$ be a complex semi-simple Lie algebra, let $e _{i} ,\ f _{i} ,\ h _{i}$ , $i = 1 \dots r$ , be canonical generators of it, that is, linearly-independent generators for which the following relations hold: $$[ e _{i} ,\ f _{j} ] = \delta _{ij} h _{i} , [ h _{i} ,\ e _{j} ] = a _{ij} e _{j} , [ h _{i} ,\ f _{j} ] = - a _{ij} f _{j} ,$$ $$[ h _{i} ,\ h _{j} ] = 0 ,$$ where $a _{ii} = 2$ , $a _{ij}$ are non-positive integers when $i \neq j$ , $i ,\ j = 1 \dots r$ , $a _{ij} = 0$ implies $a _{ji} = 0$ , and let $\mathfrak t$ be the Cartan subalgebra of $\mathfrak g$ which is the linear span of $h _{1} \dots h _{r}$ . Also let $\rho$ be a linear representation of $\mathfrak g$ in a complex finite-dimensional space $V$ . Then there exists a non-zero vector $v \in V$ for which $$\rho ( e _{i} ) v = 0 , \rho ( h _{i} ) v = k _{i} v , i = 1 \dots r ,$$ where the $k _{i}$ are certain numbers. This theorem was established by E. Cartan [1]. The vector $v$ is called the highest weight vector of the representation $\rho$ and the linear function $\Lambda$ on $\mathfrak t$ defined by the condition $\Lambda ( h _{i} ) = k _{i}$ , $i = 1 \dots r$ , is called the highest weight of the representation $\rho$ corresponding to $v$ . The ordered set $( k _{1} \dots k _{r} )$ is called the set of numerical marks of the highest weight $\Lambda$ . Cartan's theorem gives a complete classification of irreducible finite-dimensional linear representations of a complex semi-simple finite-dimensional Lie algebra. It asserts that each finite-dimensional complex irreducible representation of $\mathfrak g$ has a unique highest weight vector (up to proportionality), and that the numerical marks of the corresponding highest weight are non-negative integers. Two finite-dimensional irreducible representations are equivalent if and only if the corresponding highest weights are the same. Any set of non-negative integers is the set of numerical marks of the highest weight of some finite-dimensional complex irreducible representation.

#### References

 [1] E. Cartan, "Les tenseurs irréductibles et les groupes linéaires simples et semi-simples" Bull. Sci. Math. , 49 (1925) pp. 130–152 Zbl 51.0322.01 [2] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201 [3] , Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Ecole Norm. Sup. (1955) Zbl 0068.02102 [4] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013 [5] J. Dixmier, "Enveloping algebras" , North-Holland (1977) (Translated from French) MR0498737 MR0498740 MR0498742 Zbl 0346.17010 Zbl 0339.17007 [6] A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , Seminar on algebraic groups and related finite groups , Lect. notes in math. , 131 , Springer (1970) Zbl 0192.36201

#### References

 [a1] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) MR0323842 Zbl 0254.17004

Cartan's theorem in the theory of functions of several complex variables. These are the so-called theorems A and B on coherent analytic sheaves on Stein manifolds, first proved by H. Cartan [1]. Let ${\mathcal O}$ be the sheaf of germs of holomorphic functions on a complex manifold $X$ . A sheaf ${\mathcal S}$ of ${\mathcal O}$ - modules on $X$ is called a coherent analytic sheaf if there exists in a neighbourhood of each point $x \in X$ an exact sequence of sheaves $${\mathcal O} ^{p} \rightarrow {\mathcal O} ^{q} \rightarrow {\mathcal S} \rightarrow 0$$ for some natural numbers $p ,\ q$ . Examples are all locally finitely-generated subsheaves of ${\mathcal O} ^{p}$ .

Theorem A. Let ${\mathcal S}$ be a coherent analytic sheaf on a Stein manifold $X$ . Then there exists for each point $x \in X$ a finite number of global sections $s _{1} \dots s _{N}$ of ${\mathcal S}$ such that any element $s$ of the fibre ${\mathcal S} _{x}$ is representable in the form $$s = h _{1} ( s _{1} ) _{x} + \dots + h _{N} ( s _{N} ) _{x} ,$$ with all $h _{j} \in {\mathcal O} _{x}$ . (In other words, locally ${\mathcal S}$ is finitely generated over ${\mathcal O}$ by its global sections.)

Theorem B. Let ${\mathcal S}$ be a coherent analytic sheaf on a Stein manifold $X$ . Then all cohomology groups of $X$ of order $p \geq 1$ with coefficients in ${\mathcal S}$ are trivial: $$H ^{p} ( X ,\ {\mathcal S} ) = 0 \textrm{ for } p \geq 1 .$$ These Cartan theorems have many applications. From Theorem A, various theorems can be obtained on the existence of global analytic objects on Stein manifolds. The main corollary of Theorem B is the solvability of the $\overline \partial$ - problem: On a Stein manifold, the equation $\overline \partial$ with the compatibility condition $\overline \partial f = g$ is always solvable.

The scheme of application of Theorem B is as follows: If $\overline \partial g = 0$ is an exact sequence of sheaves on $$0 \rightarrow {\mathcal S} \rightarrow F \rightarrow G \rightarrow 0,$$ then the sequence $X$ $$\dots \rightarrow H ^{p} ( X ,\ {\mathcal S} ) \rightarrow H ^{p} ( X ,\ F \ ) \stackrel{ {\phi _{p}}} \rightarrow H ^{p} ( X ,\ G ) \rightarrow$$ is also exact. If $$\rightarrow H ^{p+1} ( X ,\ {\mathcal S} ) \rightarrow \dots$$ is a Stein manifold, then $X$ and hence, $$H ^{p} ( X ,\ {\mathcal S} ) = 0 , p \geq 1 ,$$ is mapping onto and the $\phi _{0}$ , $\phi _{p}$ , are isomorphisms.

Theorem B is best possible: If on a complex manifold $p \geq 1$ the group $X$ for every coherent analytic sheaf $H ^{1} ( X ,\ {\mathcal S} ) = 0$ , then ${\mathcal S}$ is a Stein manifold. Theorems A and B together with their numerous corollaries constitute the so-called Oka–Cartan theory of Stein manifolds. A corollary of these theorems is the solvability on Stein manifolds of all the classical problems of multi-dimensional complex analysis, such as the Cousin problem, the Levi problem, the Poincaré problem and others. Theorems A and B generalize verbatim to Stein spaces (cf. Stein space).

#### References

 [1] H. Cartan, "Variétés analytiques complexes et cohomologie" R. Remmert (ed.) J.-P. Serre (ed.) , Collected works , Springer (1979) pp. 669–683 MR0064154 Zbl 0053.05301 [2] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601 [3] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) MR0344507 Zbl 0271.32001

E.M. Chirka