Because algebra

In mathematics , the Weil algebra (originally introduced by Henri Cartan and named after André Weil ) is an aid in calculating characteristic classes .

definition

Be a Lie algebra . The Weil algebra is the graduated algebra ${\displaystyle {\mathfrak {g}}}$

${\displaystyle W^{n}({\mathfrak {g}})=\bigoplus _{2p+q=n}S^{p}({\mathfrak {g}}^{*})\otimes \Lambda ^{q}({\mathfrak {g}}^{*})}$,

where is the dual vector space , the polynomial algebra and the Graßmann algebra . ${\displaystyle {\mathfrak {g}}^{*}}$ ${\displaystyle S^{*}({\mathfrak {g}}^{*})}$ ${\displaystyle \Lambda ^{*}({\mathfrak {g}}^{*})}$

Explicit definition of the differential

Be a base of . Let be the structural constants , so . Weil algebra has generators of degree 1 and degree 2. Then the differential is defined by ${\displaystyle e_{1},\ldots ,e_{m}}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle c_{ij}^{k}}$ ${\displaystyle \left[e_{i},e_{j}\right]=c_{ij}^{k}e_{k}}$ ${\displaystyle W^{*}({\mathfrak {g}})}$ ${\displaystyle \theta _{1},\ldots ,\theta _{m}}$ ${\displaystyle \mu _{1},\ldots ,\mu _{m}}$

${\displaystyle d(\theta _{i})=\mu _{i}-{\frac {1}{2}}\sum _{j,k}c_{jk}^{i}\theta _{j}\theta _{k}}$

${\displaystyle d(\mu _{i})=-\sum _{jk}c_{jk}^{i}\theta _{j}\mu _{k}}$.

The cohomology of is trivial (except in degree 0). ${\displaystyle (W({\mathfrak {g}}),d)}$

Construction of characteristic classes

A connection on a - principal bundle induces homomorphism ${\displaystyle \omega \in \Omega ^{1}(P,{\mathfrak {g}})}$ ${\displaystyle G}$ ${\displaystyle \pi \colon P\to M}$

${\displaystyle f_{\omega }\colon \Omega ^{1}({\mathfrak {g}})\to \Omega ^{1}(P,\mathbb {R} )}$,

which leads to a homomorphism of differential graduated algebras

${\displaystyle f_{\omega }\colon W({\mathfrak {g}})\to \Omega ^{*}(P,\mathbb {R} )}$

lets continue. The -invariant elements of are mapped onto (the archetype of) and define characteristic classes in . (This construction is called the Chern-Weil homomorphism .) ${\displaystyle G}$ ${\displaystyle W^{2k}({\mathfrak {g}})}$ ${\displaystyle \Omega ^{2k}(M,\mathbb {R} )}$ ${\displaystyle H_{dR}^{2k}(M;\mathbb {R} )}$

Relative Weil algebra

Let be a Lie group and a maximally compact subgroup with Lie algebra . The relative Weil algebra is defined as ${\displaystyle G}$ ${\displaystyle K\subset G}$ ${\displaystyle {\mathfrak {k}}\subset {\mathfrak {g}}}$

${\displaystyle W(G,K)=(S({\mathfrak {g}}^{*})\otimes \Lambda ({\mathfrak {g}}/{\mathfrak {k}}))^{K}}$.

Be the classifying space and the universal bundle of the Lie group . One has canonical isomorphisms of the cohomology groups ${\displaystyle BG}$ ${\displaystyle EG\to BG}$ ${\displaystyle G}$

${\displaystyle H^{*}(W(G,K))=H^{*}(EG/K)=H^{*}(BG)=I(K)}$

with the ring of invariant polynomials . ${\displaystyle I(K)}$

The relative Weil algebra is used in the computation of secondary characteristic classes of locally symmetric spaces .

literature

• W. Greub, S. Halperin, R. Vanstone, "Connections, curvature, and cohomology. Volume III: Cohomology of principal bundles and homogeneous spaces." Pure and Applied Mathematics, Vol. 47-III. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976 (Chapter VI).
• H. Cartan, "Cohomologie réelle d'un espace fiber principal differentiable", Sem. H. Cartan 1949/50, Exp. 19-20 (1950).
• H. Cartan, "Notions d'algébre différentielle; application aux groupes de Lie et aux variétés ou opère un groupe de Lie", Colloque de topologie (espaces fibers), Bruxelles, (1950), pp. 15-27.
• JL Dupont, FW Kamber, "On a generalization of Cheeger-Chern-Simons classes" Illinois J. Math. 34 (1990), no. 2, 221-255.
• FW Kamber, Ph. Tondeur, "Foliated bundles and characteristic classes", Lecture Notes in Mathematics, 493, Springer (1975).
• FW Kamber, Ph. Tondeur, "Semi-simplicial Because algebras and characteristic classes" Tôhoku Math. J. 30 (1978) pp. 373-422 ( pdf ).
• JL Dupont, FW Kamber, "Cheeger-Chern-Simons classes of transversally symmetric foliations: dependence relations and eta-invariants." Math. Ann. 295 (1993), no. 3, 449-468, doi: 10.1007 / BF01444896 .

Web links

• Because Algebra of a Lie Algebra (Encyclopedia of Mathematics)
• Because algebra (nLab)

Individual evidence

1. ^ Theorem 5.18 in: JI Burgos Gil: "The regulators of Beilinson and Borel." CRM Monograph Series, 15th American Mathematical Society, Providence, RI, 2002. ISBN 0-8218-2630-1