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Algebras of curvature forms on homogeneous manifolds. (English) Zbl 0971.53034
Astashkevich, Alexander (ed.) et al., Differential topology, infinite-dimensional Lie algebras, and applications. D. B. Fuchs’ 60th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 194(44), 227-235 (1999).
In this very interesting paper the authors prove three important theorems cited as Theorem 2, Theorem 3, and Theorem 4.
If $$G$$ is a connected complex semisimple Lie group and $$B$$ a Borel subgroup, then $$X=G/B$$ is a compact homogeneous complex manifold. If $$K$$ is a maximal compact subgroup of $$G$$, then $$K$$ acts transitively on $$X$$ and $$X$$ can be identified with the quotient space $$K/T$$ where $$T=K\cap B$$ is the maximal torus of $$K$$.
Let $$C(X)$$ denote the subalgebra in the algebra of differential forms on $$X$$ that is generated by the curvature forms $$\theta (L_\lambda)$$ of line bundles $$L_\lambda$$, $$\lambda \in \widehat T$$. $$C(X)=C^0 (X)\oplus C^1(X)\oplus C^2(X) \oplus\dots$$ where $$C^k (X)$$ is the subspace of $$2k$$-forms in $$C(X)$$.
In Theorem 2, the authors prove that the dimension of $$C(X)$$ is equal to the number $$\text{ind} (\Delta_+)$$ of independent subsets in the set of positive roots $$\Delta_+$$ and the dimension of $$C^k(X)$$ is equal to the number of independent subsets $$S\subset\Delta_+$$ such that $$k=N-|S|-\text{act}(S)$$, where $$N=|\Delta_+|$$ and $$\text{act}(S)$$ denotes the number of externally active vectors with respect to $$S$$. In Theorems 3 and 4 the authors consider a more general algebra $$C_v$$, associated with an element $$P$$ of the Grassmannian $$G(n,N)$$ of $$n$$-dimensional planes in $$\mathbb{C}^N$$, where $$N\geq n$$, and show that the dimension of the algebra $$C_v$$ is equal to number $$\text{ind}(V)$$ of independent subsets in $$V$$ and exhibit the algebra $$C_v$$ in terms of its gererators and relations.
Finally, some very interesting open problems for further study and the possible interrelations of this work with that of H. Tamvakis and of Konstant are presented in section 5 of the paper.
For the entire collection see [Zbl 0921.00044].

##### MSC:
 53C30 Differential geometry of homogeneous manifolds 32M10 Homogeneous complex manifolds