Family algebras

A new class of associative algebras is introduced and studied. These algebras are related to simple complex Lie algebras (or root systems). Roughly speaking, they are finite dimensional approximations to the enveloping algebra $U(\mathfrak{g})$ viewed as a module over its center.

It seems that several important questions on semisimple algebras and their representations can be formulated, studied and sometimes solved in terms of our algebras.

Here we only start this program and hope that it will be continued and developed.