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Family algebras
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## Family algebras

### A. A. Kirillov

**Abstract.**
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.

*Copyright 2000 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**06** (2000), pp. 7-20
- Publisher Identifier: S 1079-6762(00)00075-5
- 2000
*Mathematics Subject Classification*. Primary 15A30, 22E60
*Key words and phrases*. Enveloping algebras, invariants, representations of semisimple Lie algebras
- Received by the editors December 31, 1999
- Posted on March 1, 2000
- Communicated by Svetlana Katok
- Comments

**A. A. Kirillov**

Department of Mathematics, The University of Pennsylvania, Philadelphia, PA 19104

*E-mail address:* `kirillov@math.upenn.edu`

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