A general construction of alternative algebras with three anticommuting elements and a unit is given. As an exhaustive result over the real and complex fields, we obtain the Clifford algebras H (quaternions), N1 (dihedral Clifford algebra which is related to real 2-spinors), and S1 (algebra of Pauli matrices which is related to complex 2-spinors). What is important is that the algebras N1 and S1 possess inverses everywhere except on a region akin to the light cone of the Minkowski space, while the quaternion algebra H has inverses everywhere except at the zero element. We discuss the reasons why the three algebras N1, H, and S1 are so difficult to distinguish in the representation space of 2×2 complex matrices.
It is shown that, in a boson representation, the operators whose eigenvalues serve to label representations of SO(N) in U(N) also serve to label representations of U(M) in Sp(2M). The problem of labeling U(2) in Sp(4) is considered in detail, and it is shown how to find labeling operators with rational eigenvalues, depending, however, on the representation. The solution of this problem is shown to provide a solution of the equivalent problem of the labeling of SO(3) in U(3).
The functions Lnk(r)(Φ1,⋯,Φn) are defined by Xr=∑k=1nLnk(r)Xn−k, where X is an indeterminate n × n matrix and Φ1, ⋯, Φn are the invariants of X (basic symmetric functions in the eigenvalues of X). In this paper the generalized Lucas polynomial Ln1(r) is expressed explicitly as a determinant of order r − n + 1 or as a ratio of two determinants of order n.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.