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Assume that ${\mathbb F}$ is an algebraically closed field withcharacteristic zero. The Racah algebra $\Re$ is the unital associative${\mathbb F}$-algebra defined by generators and relations in the following way.The generators are $A$, $B$, $C$, $D$ and the relations assert that$[A,B]=[B,C]=[C,A]=2D$ and that each of $[A,D]+AC-BA$, $[B,D]+BA-CB$,$[C,D]+CB-AC$ is central in $\Re$. In this paper we discuss thefinite-dimensional irreducible $\Re$-modules in detail and classify them up toisomorphism. To do this, we apply an infinite-dimensional $\Re$-module and itsuniversal property. We additionally give the necessary and sufficientconditions for $A$, $B$, $C$ to be diagonalizable on finite-dimensionalirreducible $\Re$-modules.

Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero