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The extension involves adding a singlet gluon, giving a U(n) -based theory with outer product P/sup a/(u,v) = (1/2)(d/sup a/bc + if/sup a/bc)(u/sup b/v/sup c/ - v/sup b/u/sup c/) which obeys the Jacobi identity, inner product S (u,v) = (1/2)(u/sup a/v/sup a/ + v/sup a/u/sup a/), and with the n/sup 2/ gluon fields elevated to algebraic fields over the quark color charge C* algebra. I develop an extension of the usual equations of SU(n) chromodynamics which permits the consistent introduction of classical, noncommuting quark source charges. Finally the U(2)-color algebras of a general type (N anti N) are considered briefly, and their structure is shown to be determined by that of the (N + anti N,O) color algebras with the outer products x and (,)/sub x. The type (3,0) color algebra with respect to the charge conjugation symmetric product (u * v)/sup a/ = i f/sup abc/u/sup b/v/sup c/ is studied in detail, both as an example of breakdown of the Jacobi identity and as the basis for constructing an algebraic chromodynamical theory for the interaction of three quarks. It is found that among the most general outer products one can construct from q's and Kronecker deltas, the commutators with respect to the products x and o are essentially the only ones which satisfy the Jacobi identity for color algebras of type (N,anti N) with N greater than or equal to 3 or anti N greater than or equal to 3. Two sub-ideals of it are found which solves the case of the type (3.0) color algebra. The type (N,O) color algebra with respect to the product (uov)/sup a/ = anti q/sup abc/u/sup b/v/sup c/ is considered. The structure of the type (N,O) color algebra with respect to the product (u,v)/sub x/ = uxv - vxu is partially determined. This leads to a group theoretical derivation of a complete set of identities among the U(n) structure constants q/sup abc/ = 1/2 (d/sup abc/ + more » i f/sup abc/). If the underlying gauge group is U(n), then the representation is not faithful when N greater than or equal to n.
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The type (N,O) color algebra with respect to the product (u x v)/sup a/ = q/sup abc/u/sup b/v/sup c/ is found to be nothing but a representation of the permutation group algebra S/sub N + 1/ on the N + 1 color charge Hilbert space. The color algebras appearing in algebraic chromodynamics are studied.