- November 30th, 2006
- 2:28 pm

As a first unknown (kinda, sorta, it still falls under the category of path algebra quotients treated by Keller) A_{∞}-calculation, I shall find the A_{∞}-structure of .

To do this, I fix the group algebra

and the cohomology ring

with ,

Furthermore, we pick a canonical nice resolution P, continuing the one I calculated previously. This has the i:th component Λ^{i+1}, and the differentials looking like

for differentials starting in odd degree, and

for differentials starting in even degree. The first few you can see on the previous calculation, or if you don’t want to bother, they are

Now, armed with this, we can get cracking. By lifting, we get canonical representating chain maps for x,y,z described, loosly, by the following:

x takes an element in , keeps the first, third, et.c. elements and throws out the even ordered elements; so

For an element in , the last element gets extra treatment, so

For the lowest degrees, we also have

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An A_{∞}-structure on the cohomology of D_{8}

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- November 23rd, 2006
- 3:48 pm

In which the author, after a long session sweating blood with his advisor, manages to calculate the A_{∞}-structures on the cohomology algebras and .

We will find the A_{∞}-structures on the group cohomology ring by establishing an A_{∞}-quasi-isomorphism to the endomorphism dg-algebra of a resolution of the base field. We’ll write m_{i} for operations on the group cohomology, and μ_{i} for operations on the endomorphism dg-algebra. The endomorphism dg-algebra has μ_{1}=d and μ_{2}=composition of maps, and all higher operations vanishing, in all our cases.

## Elementary abelian 2-group

Let’s start with the easy case. Following to a certain the notation used in Dag Madsen’s PhD thesis appendix (the Canonical Source of the A_{∞}-structures of cyclic group cohomology algebras), and the recipe given in A-infinity algebras in representation theory, we may start by stating what we know as we start:

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A_{∞}-algebras and group cohomology

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- November 16th, 2006
- 4:38 pm

The Community College Dean has written about why he blogs, and asks any and all readers to tack on to his effort.

My blog is not very anonymous. It is occasionally personal, occasionally political and throughout a venting location for thoughts, and a place where I formulate myself in higher detail – so to speak a scratchpad, but public enough for me to allow others to read it.

I write it to formulate my own thoughts further, find possible errors, start discussions, or just jot down the viewpoints that illuminated some point of some argument for me. I do it in public because I thouroughly enjoy the conversations it sparks.

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Why Blog?

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- November 7th, 2006
- 4:18 pm

I recently had reason to describe my PhD work in complete laymans terms while writing letters to my grandparents. This being a good thing to do in order to digest your ideas properly, I thought I might try and write it up here as well.

It will, however, push through some 100-odd years of mathematical development rather swiftly. Try to keep up – I will keep it as light as I can while not losing what I want to say.

## Algebra

In the 19th century, a number of different mathematical efforts ended up using more or less precisely the same structures, though not really recognizing that they were the same. This recognition came with Cayley, who first brought the first abstract definition of a group.

A group is a set G of elements, with a binary operation *, such that the following relations hold:

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A_{∞} for the layman

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