# Weight space

From Encyclopedia of Mathematics

A finite-dimensional space carrying a representation of a Lie algebra over a field and satisfying the following condition: there exists a function such that for any , ,

for some positive integer . The function is called the weight. The tensor product of two representations of with weight spaces which have weights and , respectively, is the representation of in the space , which is also a weight space and has the weight . On passing from the representation to the contragredient representation , the space is replaced by the adjoint space , and is replaced by .

#### Comments

See also Weight of a representation of a Lie algebra.

**How to Cite This Entry:**

Weight space. E.N. Kuz'min (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Weight_space&oldid=14327

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098