Unique tensor factorization of algebras
By an algebra the author means a finite-dimensional, associative, unitary \(\mathbb{C}\)-algebra, where \(\mathbb{C}\) is the field of complex numbers. An algebra \(A\) is called \(\oplus\)-indecomposable if in each direct decomposition \(A=A_1\oplus A_2\) exactly one of the summands is trivial, i.e. \(A_i=0\). Similarly an algebra is called \(\otimes\)-indecomposable if in each tensor product decomposition \(A=A_1\otimes A_2\) exactly one of the factors is trivial, i.e. \(A_i=\mathbb{C}\). The main result established by the author is: A \(\oplus\)-indecomposable, finite-dimensional \(\mathbb{C}\)-algebra \(A\) has an essentially unique tensor factorization \(A=A_1\otimes\cdots\otimes A_r\) into non-trivial, \(\otimes\)-indecomposable factors \(A_i\). Thus the semiring of isomorphism classes of finite-dimensional \(\mathbb{C}\)-algebras is a polynomial semiring \(N[X]\). Also \(\mathbb{C}\) can be replaced by an arbitrary field of characteristic zero if one restricts oneself to Schurian algebras.
finite-dimensional \(\mathbb{C}\)-algebras
scientific article; zbMATH DE number 1384373