[Paper Review] The relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on 6-dimensional plane

Kentaro Mikami|arXiv (Cornell University)|Feb 27, 2014

Advanced Topics in Algebra6 references4 citations

TL;DR

This paper computes the relative Gel'fand-Kalinin-Fuks cohomology groups of formal Hamiltonian vector fields on the 6-dimensional plane, focusing on weights 2, 4, and 6. Using crystal base theory to overcome computational complexity in the $n=3$ case, it determines Betti numbers and explicitly constructs bases for cohomology groups, revealing nontrivial cohomology in weight 6 with Euler characteristic 3 and rank 2 for the coboundary operator.

ABSTRACT

Using Crystal basis theory, we study the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on 6-dimensional plane with weight =2,4,6.

Motivation & Objective

• To compute the relative Gel'fand-Kalinin-Fuks cohomology groups of the Lie algebra of formal Hamiltonian vector fields on $\mathbb{R}^6$, specifically for weights 2, 4, and 6.
• To overcome computational challenges in the $n=3$ case, where traditional representation theory tools like the Littlewood-Richardson rule become intractable.
• To apply crystal base theory to decompose tensor products of $\mathfrak{sp}(6,\mathbb{R})$-representations and compute cohomology spaces explicitly.
• To determine the Betti numbers and Euler characteristic of the cohomology groups, providing a complete description of the relative cohomology in low weights.

Proposed method

• The cochain complex is defined using the dual spaces of homogeneous polynomial functions on $\mathbb{R}^6$, with weight assigned as $\ell - 2$ for degree $\ell$ polynomials.
• The relative cohomology is computed via the coboundary operator $d$ on the completed Lie algebra $\mathfrak{ham}_6^0$, preserving weight.
• Crystal base theory is employed to handle the irreducible decomposition of tensor products of $\mathfrak{sp}(6,\mathbb{R})$-representations, replacing the infeasible Littlewood-Richardson rule for $n=3$.
• Concrete bases for cohomology spaces are constructed by identifying maximal vectors in $\Lambda^4\mathfrak{S}_3 \otimes \mathfrak{S}_4$ and projecting to trivial representation subspaces.
• The coboundary operator $d$ is represented as a matrix acting on a basis of $\mathfrak{C}^5_6$, with rank computed as 2.
• Explicit basis elements are provided for $\mathfrak{C}^5_6$, with over 200,000 terms in some expressions, computed via systematic reduction using crystal structure.

Experimental results

Research questions

• RQ1 What are the Betti numbers of the relative Gel'fand-Kalinin-Fuks cohomology groups for formal Hamiltonian vector fields on $\mathbb{R}^6$ at weights 2, 4, and 6?
• RQ2 How can one compute the cohomology of $\mathfrak{ham}_6^0$ with coefficients in $\mathfrak{sp}(6,\mathbb{R})$ when standard representation-theoretic tools fail for $n=3$?
• RQ3 What is the structure of the relative cochain complex $\mathfrak{C}^j_w$ for weights $w=2,4,6$, and what are the dimensions of its components?
• RQ4 Can crystal base theory be effectively used to compute explicit bases for trivial representation subspaces in high-dimensional tensor products?
• RQ5 What is the rank of the coboundary operator $d$ acting on the 5-cochain space of weight 6?

Key findings

• The Betti numbers for weight 2 are $b^0_2 = 1$, $b^1_2 = 0$, and $b^2_2 = 1$, with Euler characteristic 1.
• For weight 4, the Betti numbers are $b^0_4 = 1$, $b^4_4 = 2$, and all others zero, yielding an Euler characteristic of 2.
• In weight 6, the Betti numbers are $b^0_6 = 1$, $b^5_6 = 2$, and $b^6_6 = 5$, with all others zero, resulting in an Euler characteristic of 3.
• The dimension of the 5-cochain space $\mathfrak{C}^5_6$ is 4, and its trivial representation subspace is 4-dimensional, with a basis of 4 maximal vectors.
• The coboundary operator $d$ on $\mathfrak{C}^5_6$ has rank 2, as shown by its matrix representation acting on the basis of 4 vectors.
• Explicit basis elements for $\mathfrak{C}^5_6$ are constructed, with one example containing over 200,000 terms, and the full basis spans 1567 pages of computation.