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I've always been secretly fascinated with the rich structure and applications of finite-dimensional associative unital algebras over complete fields. In particular, I am very much interested in the structure and representations of commutative ones and their central extensions. My background is number theory, but I am somewhat ashamed to admit that I know very little about these objects as a whole and that little knowledge is rather scattered in bits and pieces one picks up from algebraic geometry, Lie theory, and google...

I will try to make my inquiry as specific as possible:

  1. What would be a relatively concise introduction focused on the structure and representation theory of finite-dimensional commutative associative unital algebras over $\mathbb{R}$ and $\mathbb{C}$ (or even other complete fields like $\mathbb{Q}_p$ too)? In that line of thoughts I would like to learn more about algebraic, geometric and analytic aspects of these algebras in a structured manner.

  1. After that, where should I look in order to learn about the theory of their central extensions? (as I'm also interested in finite-dimensional real or complex non-commutative associative unital algebras with rich interesting non-trivial centers).

  1. Are there any surveys outlining current progress and open problems in the area of finite-dimensional commutative associative unital algebras over complete fields and their central extensions?

  1. Whose research / research groups should I follow closely, that focus on the structure and geometric and analytic applications of finite-dimensional real or complex commutative associative unital algebras and their central extensions?

Please note that the focus of the question is really finite-dimensional commutative associative real or complex unital algebras.

Thank you!

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