Cohomology of Heisenberg Lie Superalgebras

Suppose the ground field to be algebraically closed and of characteristic different from $2$ and $3$. All Heisenberg Lie superalgebras consist of two super versions of the Heisenberg Lie algebras, $\frak{h}_{2m,n}$ and $\frak{ba}_n$ with $m$ a nonnegative integer and $n$ a positive integer. The space of a "classical" Heisenberg Lie superalgebra $\frak{h}_{2m,n}$ is the direct sum of a superspace with a non-degenerate anti-supersymmetric even bilinear form and a one-dimensional space of values of this form constituting the even center. The other super analog of the Heisenberg Lie algebra, $\frak{ba}_n$, is constructed by means of a non-degenerate anti-supersymmetric odd bilinear form with values in the one-dimensional odd center. In this paper, we study the cohomology of $\frak{h}_{2m,n}$ and $\frak{ba}_n$ with coefficients in the trivial module by using the Hochschild-Serre spectral sequences relative to a suitable ideal. In characteristic zero case, for any Heisenberg Lie superalgebra, we determine completely the Betti numbers and associative superalgebra structure for their cohomology. In characteristic $p>3$ case, we determine the associative superalgebra structures for the divided power cohomology of $\frak{ba}_n$ and we also make an attempt to determine the cohomology of $\frak{h}_{2m,n}$ by computing it in a low-dimensional case.Comment: 19 page

A Web Aggregation Approach for Distributed Randomized PageRank Algorithms

The PageRank algorithm employed at Google assigns a measure of importance to each web page for rankings in search results. In our recent papers, we have proposed a distributed randomized approach for this algorithm, where web pages are treated as agents computing their own PageRank by communicating with linked pages. This paper builds upon this approach to reduce the computation and communication loads for the algorithms. In particular, we develop a method to systematically aggregate the web pages into groups by exploiting the sparsity inherent in the web. For each group, an aggregated PageRank value is computed, which can then be distributed among the group members. We provide a distributed update scheme for the aggregated PageRank along with an analysis on its convergence properties. The method is especially motivated by results on singular perturbation techniques for large-scale Markov chains and multi-agent consensus.Comment: To appear in the IEEE Transactions on Automatic Control, 201