User:Maximilian Janisch/latexlist/Algebraic Groups/Cartan theorem

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Cartan's theorem on the highest weight vector. Let $8$ be a complex semi-simple Lie algebra, let $e _ { i } , f _ { i } , h _ { i }$, $\dot { i } = 1 , \ldots , r$, be canonical generators of it, that is, linearly-independent generators for which the following relations hold:

\begin{equation} f _ { j } ] = \delta _ { i j } h _ { i } , \quad [ h _ { i } , e _ { j } ] = \alpha _ { i j } e _ { j } , \quad [ h _ { i } , f _ { j } ] = - \alpha _ { j } f _ { j } \end{equation}

\begin{equation} [ h _ { i } , h _ { j } ] = 0 \end{equation}

where $\alpha _ { i j } = 2$, $a _ { i }$ are non-positive integers when $i \neq j$, $j = 1 , \dots , r$, $\alpha _ { \xi j } = 0$ implies $\alpha _ { j i } = 0$, and let $1$ be the Cartan subalgebra of $8$ which is the linear span of $h _ { 1 } , \ldots , h _ { r }$. Also let $0$ be a linear representation of $8$ in a complex finite-dimensional space $V$. Then there exists a non-zero vector $v \in V$ for which

\begin{equation} \rho ( e _ { i } ) v = 0 , \quad \rho ( h _ { i } ) v = k _ { i } v , \quad i = 1 , \dots , r \end{equation}

where the $k$ are certain numbers. This theorem was established by E. Cartan [1]. The vector $v$ is called the highest weight vector of the representation $0$ and the linear function $1$ on $1$ defined by the condition $\Lambda ( h _ { i } ) = k _ { i }$, $\dot { i } = 1 , \ldots , r$, is called the highest weight of the representation $0$ corresponding to $v$. The ordered set $( k _ { 1 } , \ldots , k _ { r } )$ is called the set of numerical marks of the highest weight $1$. 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 $8$ 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 $0$ be the sheaf of germs of holomorphic functions on a complex manifold $x$. A sheaf $5$ of $0$-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

\begin{equation} O ^ { p } \rightarrow O ^ { q } \rightarrow S \rightarrow 0 \end{equation}

for some natural numbers $p , q$. Examples are all locally finitely-generated subsheaves of $O ^ { p }$.

Theorem A. Let $5$ 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 $5$ such that any element $5$ of the fibre $S _ { x }$ is representable in the form

\begin{equation} s = h _ { 1 } ( s _ { 1 } ) _ { x } + \ldots + h _ { N } ( s _ { N } ) _ { x } \end{equation}

with all $h _ { j } \in O _ { x }$. (In other words, locally $5$ is finitely generated over $0$ by its global sections.)

Theorem B. Let $5$ be a coherent analytic sheaf on a Stein manifold $x$. Then all cohomology groups of $x$ of order $p \geq 1$ with coefficients in $5$ are trivial:

\begin{equation} H ^ { p } ( X , S ) = 0 \quad \text { for } p \geq 1 \end{equation}

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 $0$-problem: On a Stein manifold, the equation $\overline { \partial } f = g$ with the compatibility condition $\overline { \partial } g = 0$ is always solvable.

The scheme of application of Theorem B is as follows: If

\begin{equation} 0 \rightarrow S \rightarrow F \rightarrow G \rightarrow 0 \end{equation}

is an exact sequence of sheaves on $x$, then the sequence

\begin{equation} \rightarrow H ^ { p } ( X , S ) \rightarrow H ^ { p } ( X , F ) \stackrel { \phi p } { \rightarrow } H ^ { p } ( X , G ) \rightarrow \end{equation}

\begin{equation} H ^ { p + 1 } ( X , S ) \rightarrow \end{equation}

is also exact. If $x$ is a Stein manifold, then

\begin{equation} H ^ { p } ( X , S ) = 0 , \quad p \geq 1 \end{equation}

and hence, $90$ is mapping onto and the $\phi _ { p }$, $p \geq 1$, are isomorphisms.

Theorem B is best possible: If on a complex manifold $x$ the group $H ^ { 1 } ( X , S ) = 0$ for every coherent analytic sheaf $5$, then $x$ 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