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

Comments

In [a1] the theory related to Cartan's Theorems A and B is developed on the basis of integral representations, and not on the basis of sheaves, as in [2] or [a2], or on the basis of the Cauchy–Riemann equations, as in [3].

Generalizations to Stein manifolds are in [a2].

See also Cousin problems. For the Poincaré problem (on meromorphic functions), cf. Stein space and Meromorphic function.

References

[a1]	G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984) (Translated from Russian) MR0795028 MR0774049
[a2]	H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1977) (Translated from German) MR0513229 Zbl 0379.32001
[a3]	S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Sect. 7.1 MR0635928 Zbl 0471.32008
[a4]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. VI, Par. 6 MR0847923