Project with V. STRASSEN:

Tensor Factorization of Algebras

Let k be any field of characteristic zero. Consider the set U of isomorphism classes of finite-dimensional, schurian, associative algebras with unit element over k, and endow it with operations induced by the direct sum (direct product) and the tensor product (over k) of algebras. (If k is algebraically closed all finite-dimensional k-algebras are schurian.) Clearly, U is a commutative semi-ring. It turns out that this universal semiring is of the form N[X], where X is the set of isoclasses of additively and multiplicatively indecomposable algebras. In other words, it is the positive cone in the polynomial ring Z[X] .

The major part of the work is to prove that additively indecomposable algebras have a unique (tensor) factorization. The main ingredients are the Fundamental Theorem of Arithmetics, a unique factorization theorem for local algebras by C. HORST and a unique factorization theorem for graphs by G. SABIDUSSI with methods by W. IMRICH. Our theorem can be viewed as a common generalization of all these.

Even in finite characteristic, this is still true for a large class of algebras.

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Literature

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FEIGENBAUM, Joan (1986), ``Directed Cartesian-product graphs have unique factorizations that can be computed in polynomial time'', Discrete Appl. Math., 15, 105-110. Zbl. 637.05018.

HORST, Camilla (1987), ``A cancellation theorem for artinian local algebras'', Math. Ann., 276, 657-662. Zbl. 595.13007.

HORST, Camilla (1990), ``On product decompositions of complex spaces'', Note Mat., 10(1), 157-213. Zbl. 790.32010.

IMRICH, W. (1967), ``Kartesisches Produkt von Mengensystemen und Graphen'', Stud. Sci. Math. Hungar., 2, 285-290. Zbl. 153.32301.

MARYLAND, Wallace jun. (1978), ``Decomposition of finite connected partially ordered sets.'', J. Comb. Inf. Syst. Sci., 3, 238-244. Zbl. 404.06002.

NÜSKEN, Michael (1998), ``Unique Tensor Factorization of Algebras'', Dissertation.

NAKAYAMA, Tadasi & HASHIMOTO, Junji (1950), ``On a problem of G. Birkoff'', Proc. Amer. Math. Soc., 1, 141-142. Zbl. 036.29604.

POWER, S. C. (1990), ``Classifications of tensor products of triangular operator algebras'', Proc. Lond. Math. Soc., 61(3), 571-614. Zbl. 783.47060.

SABIDUSSI, Gert (1960), ``Graph multiplication'', Math. Z., 72, 446-457. Zbl. 093.37603.