Van Est's Theorem

In the mathematical theory of Lie groups , the van Est isomorphism, or van Est's theorem, enables the computation of the continuous cohomology of semi-simple Lie groups . He was proven by Willem Titus van Est .

statement

The continuous cohomology of a non-compact semi-simple Lie group can be calculated as ${\displaystyle H_{c}^{*}(G)}$ ${\displaystyle G}$

${\displaystyle H_{c}^{*}(G)=H_{dR}^{*}(G^{u}/K)}$.

Here a maximally compact subgroup of and denotes the compact dual of the symmetrical space , as well as the De-Rham cohomology of . ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle G^{u}/K}$ ${\displaystyle G/K}$ ${\displaystyle H_{dR}^{*}(G^{u}/K)}$ ${\displaystyle G^{u}/K}$

Examples

• For is the hyperbolic space , its dual symmetrical space is the sphere and with van Est's theorem one obtains ${\displaystyle G=SO(n,1)}$ ${\displaystyle G/K=H^{n}}$ ${\displaystyle G^{u}/K=S^{n}}$

${\displaystyle H_{c}^{i}(SO(n,1)=H_{dR}^{i}(S^{n})={\begin{cases}\mathbb {R} ,&{\text{für }}i=0,n\\0&{\text{sonst}}\end{cases}}}$

• For is with compact dual , with van Est's theorem one gets ${\displaystyle G=SL(n,\mathbb {C} )}$ ${\displaystyle G/K=SL(n,\mathbb {C} )/SU(n)}$ ${\displaystyle G^{u}/K=(SU(n)\times SU(n))/SU(n)\simeq SU(n)}$

${\displaystyle H_{c}^{*}(SL(n,\mathbb {C} ))=H_{dR}^{*}(SU(n))=\Lambda _{\mathbb {Z} }(b_{3},b_{5},\ldots ,b_{2n-1}),}$

where denotes the i-th Borel class . ${\displaystyle b_{i}\in H_{c}^{i}(SL(n,\mathbb {C} ))}$

literature

• WT van Est: Group cohomology and Lie algebra cohomology in Lie groups I, II, Proc. Con. Ned. Akad. 56: 484-504 (1953)
• WT van Est: On the algebraic cohomology concepts in Lie groups I, II, Proc. Con. Ned. Akad. 58: 225-233, 286-294 (1955)
• WT van Est: Une application d'une méthode de Cartan-Leray, Proc. Con. Ned. Akad. 58: 542-544 (1955)