Lie algebra cohomology

In mathematics , the Lie algebra cohomology is a technical aid that is used in particular in differential geometry, mathematical physics and the theory of Lie groups . It is defined as the cohomology of the Koszul complex . For compact Lie groups, the algebraically defined Lie algebra cohomology of the Lie algebra is isomorphic to the De Rham cohomology of the Lie group.

definition

Be a Lie algebra . On the outer algebra of the dual vector space we define an operator for all of them ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \Lambda {\mathfrak {g}}^{*}=\bigoplus _{k}\Lambda ^{k}{\mathfrak {g}}^{*}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathfrak {g}}^{*}}$ ${\displaystyle k\in \mathbb {N} }$

${\displaystyle d^{k}:\Lambda ^{k}{\mathfrak {g}}^{*}\rightarrow \Lambda ^{k+1}{\mathfrak {g}}^{*}}$ as follows.

Be

${\displaystyle f\in \operatorname {Hom} (\Lambda ^{k}{\mathfrak {g}},\mathbb {R} )\cong \Lambda ^{k}{\mathfrak {g}}^{*}}$,

then we define

${\displaystyle d^{k}f\in \operatorname {Hom} (\Lambda ^{k+1}{\mathfrak {g}},\mathbb {R} )\cong \Lambda ^{k+1}{\mathfrak {g}}^{*}}$

by

${\displaystyle d^{k}f(g_{1}\wedge \ldots \wedge g_{k+1})=\sum _{1\leq i<j\leq k+1}\left(-1\right)^{i+j-1}f(\left[g_{i},g_{j}\right]\wedge g_{1}\wedge \ldots {\hat {g_{i}}}\ldots {\hat {g_{j}}}\ldots \wedge g_{k+1})+\sum _{1\leq i\leq k}\left(-1\right)^{i}f(g_{1}\wedge \ldots {\hat {g_{i}}}\ldots \wedge g_{k+1})}$.

The complex is called the Koszul complex . For all true ${\displaystyle (\Lambda {\mathfrak {g}}^{*},d^{\bullet })}$ ${\displaystyle k\in \mathbb {N} }$

${\displaystyle d^{k}d^{k-1}=0}$.

The Lie algebra cohomology of is defined as the cohomology of the Koszul complex, i.e. as ${\displaystyle {\mathfrak {g}}}$

${\displaystyle H^{k}({\mathfrak {g}}):=\ker(d^{k})/\operatorname {im} (d^{k-1})}$.

Lie groups and Lie algebra cohomology

For a Lie group with Lie algebra , the Koszul complex is canonically isomorphic to the complex of -invariant differential forms on : ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle G}$ ${\displaystyle G}$

${\displaystyle (\Lambda {\mathfrak {g}}^{*},d^{\bullet })=(\Omega ^{G}(G),d)}$,

the Lie algebra comology of is isomorphic to the cohomology of the complex . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle (\Omega ^{G}(G),d)}$

Élie Cartan has proven that for compact Lie groups, inclusion

${\displaystyle \Omega ^{G}(G)\subset \Omega ^{\bullet }(G)}$

induces an isomorphism of the De Rham cohomology groups. So for compact Lie groups we have ${\displaystyle G}$

${\displaystyle H_{dR}^{\bullet }(G)=H^{\bullet }({\mathfrak {g}})}$.

Lie algebra cohomology with respect to a representation

C. Chevalley and S. Eilenberg have carried out the following cohomology construction for a Lie algebra representation . ${\displaystyle \pi :{\mathfrak {g}}\rightarrow \mathrm {gl} (V)}$

For let the space of -linear, alternating maps , for k = 0 . Furthermore, be through ${\displaystyle k>0}$ ${\displaystyle C_{\pi }^{k}}$ ${\displaystyle k}$ ${\displaystyle {\mathfrak {g}}^{k}\rightarrow V}$ ${\displaystyle C_{\pi }^{0}=V}$ ${\displaystyle \delta ^{k}:C_{\pi }^{k}\rightarrow C_{\pi }^{k+1}}$

${\displaystyle (\delta ^{k}f)(g_{1}\wedge \ldots \wedge g_{k+1})=\sum _{1\leq i<j\leq k+1}\left(-1\right)^{i+j-1}f(\left[g_{i},g_{j}\right]\wedge g_{1}\wedge \ldots {\hat {g_{i}}}\ldots {\hat {g_{j}}}\ldots \wedge g_{k+1})+\sum _{1\leq i\leq k+1}\left(-1\right)^{i}\pi (g_{i})f(g_{1}\wedge \ldots {\hat {g_{i}}}\ldots \wedge g_{k+1})}$.

In the cited work by C. Chevalley and S. Eilenberg, division is still made by what is not the case in the textbook by Hilgert and Neeb cited below. It shows that there is a coquette complex , which is also called the Chevalley-Eilenberg complex . The elements out ${\displaystyle k+1}$ ${\displaystyle \delta ^{k}\circ \delta ^{k+1}=0}$

${\displaystyle Z_{\pi }^{k}=\mathrm {ker} (\delta ^{k})}$

as usual, they are called k-coccycles

${\displaystyle B_{\pi }^{k}=\mathrm {im} (\delta ^{k-1})}$

are called k-Korander. With that there are the cohomology groups

${\displaystyle H_{\pi }^{k}:=Z_{\pi }^{k}/B_{\pi }^{k}}$

defined, where in the case of the Korand operator is defined as 0. One speaks more accurately of the Chevalley cohomology of values in respect. . ${\displaystyle k=0}$ ${\displaystyle \delta ^{-1}}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle V}$ ${\displaystyle \pi }$

Elements from the Chevalley-Eilenberg complex occur naturally. So is for example by the formula

${\displaystyle (\rho (g)f)(h):=\pi (h)(f(g)),\quad g,h\in {\mathfrak {g}},f\in Z_{\pi }^{1}}$

defines a representation that can be used for further investigations of Lie algebra. Further conclusions can be drawn if is, that is, if the 1st Chevalley cohomology disappears. Therefore, the following two so-called Whitehead lemmas are of particular interest: ${\displaystyle \rho :{\mathfrak {g}}\rightarrow \mathrm {gl} (Z_{\pi }^{1})}$ ${\displaystyle Z_{\pi }^{1}=B_{\pi }^{1}}$

1. Whitehead's Lemma : Is a semi-simple , finite-dimensional, real or complex Lie algebra and is a finite-dimensional representation, then is . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \pi :{\mathfrak {g}}\rightarrow \mathrm {gl} (V)}$ ${\displaystyle H_{\pi }^{1}=\{0\}}$

2. Whitehead's Lemma : Is a semi-simple, finite-dimensional, real or complex Lie algebra and is a finite-dimensional representation, then is . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \pi :{\mathfrak {g}}\rightarrow \mathrm {gl} (V)}$ ${\displaystyle H_{\pi }^{2}=\{0\}}$

The following theorem is a consequence of the 1st lemma of Whitehead and the above construction of representations on : ${\displaystyle Z_{\pi }^{1}}$

• Is a semi-simple, finite, real or complex Lie algebra and is a short exact sequence of finite - modules , it separates them, that is, there is a module-morphism with . ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle 0\rightarrow U\rightarrow V\,{\xrightarrow {\varphi }}\,W\rightarrow 0}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle \psi :W\rightarrow V}$ ${\displaystyle \varphi \circ \psi =\mathrm {id} _{W}}$

This theorem can be seen as an essential step in the proof of Weyl's theorem .

Whitehead's 2nd Lemma is an important building block for Levi's Theorem .

literature

• C. Chevalley, S. Eilenberg: Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63: 85-124 (1948).
• JL Koszul: Homologie et cohomologie des algèbres de Lie. Bull. Soc. Math. France, 78 (1950) pp. 65-127
• Gerhard Hochschild , Jean-Pierre Serre : Cohomology of Lie algebras. Ann. of Math. (2) 57, (1953). 591-603. JSTOR 1969740
• JC Jantzen, Representations of Algebraic groups, Pure and Applied Mathematics, vol. 131, Boston, etc., 1987 (Academic).
• JC Jantzen: Restricted Lie algebra cohomology. Lecture Notes in Math. 1271 (1986), 91-108.
• Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , Chapter II.5: Lie algebra cohomology
• AW Knapp, Lie groups, Lie algebras and cohomology, Mathematical Notes, Princeton University Press, 1988, 509 pp.

Individual evidence

1. ^ C. Chevalley, S. Eilenberg: Cohomology theory of Lie groups and Lie algebras , Trans. Amer. Math. Soc. 63 (1948), Chapter IV: Cohomology Groups associated with a representation
2. ↑ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , definition II.5.3
3. ↑ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , II.5.12, II.5.14
4. ^ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , II.4.16, II.5.5, II.5.12
5. ↑ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , II.4.8, II.5.7, II.5.14