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Lie Algebra Cohomology and Laplace Operators

Algebra & Discrete Mathematics

Speaker: Dmitry Fuchs, UC Davis
Location: 2112 MSB
Start time: Fri, Apr 17 2009, 1:10PM

This is a joint work with Connie Wilmarth.

Let {C_i,∂_i: C_i → C_{i-1}} be an arbitrary complex over $\bb R$, H_i= Ker(∂_i)/Im(∂_{i+1}) being its homology. Fix arbitrary Euclidean structures for C_i and let δ_{i-1}: C_{i-1} → C_i be the operator adjoint to ∂_i. The Laplace operator is Δ=Δ_i=∂_{i+1} \circ δ_i+δ_{i-1}\circ ∂_i: C_i → C_i. It is a common place that every harmonic (Δ_i c=0) chain/cochain c ∈ C_i is a cycle and a cocycle (∂_i c=0, δ_i c=0), and every homology/cohomology class from H_i is represented by a unique harmonic chain/cochain.

Let a (real) Lie algebra $\frak g$ be furnished with some Euclidean structure; this makes Euclidean the standard chain spaces C_i({\frak g}), and lets apply the previous definitions to the chain/cochain complexes {C_i({\frak g}),∂_i}, {C^i( {\frak g})=C_i({\frak g}), δ_i}$. There are a few known isolated results giving, in a nice explicit form, the full spectrum of the Laplace operators for some classical nilpotent Lie algebras. Our work extends these results to a broad class of nilp otent Lie algebras: maximal nilpotent subalgebras of arbitrary Kac-Moody Lie algebras (furnished with some special Euclidean structures).

Many things remain unclear. What is the homological/cohomological meaning of eigenvectors of the Laplace operator with non-zero eigenvalues? What is the meaning of the Euclidean structure mentioned above (it is defined uniquely)? And more.

I promise to explain the definitions of the Lie algebra homology/cohomology and of Kac-Moody. I do not promise to make the things more clear to the audience than they are clear to myself.