Institute of Mathematics of NAS of Ukraine,

3 Tereshchenkivs'ka Str.,

01601 Kyiv,

UKRAINE

E-mail: popova_n@yahoo.com, mariyka@imath.kiev.ua

**On projections with Temperley-Lieb relations**

**Abstract:**

With simple connected graph we associate *-algebra. It is generated
by projections with relations of orhogonality or Temperley-Lieb type. We
study set of values of parameter for which algebra associated to tree has
*-representations. Also we construct the embedding of algebra associated
to cycle in algebra of n*n matrices with elements from ring of Laurent
polynomials on circle.