We classify all the subalgebras of the Galilei algebra g(3) into conjugacy classes under the associated group G(3) and under the Galilei–similitude group Gd(3), which includes a simultaneous time and space dilation. We also classify the subalgebras of gd(3) under Gd(3). All the classifications are given up to an isomorphism, and the results are summarized in normalized lists.
Uniformly describing the conjugacy classes of the unipotent upper triangular groups UT n (F q ) (for all or many values of n and q) is a nearly impossible task. This paper takes on the related problem of describing the normal subgroups of UT n (F q ). For q a prime, a bijection will be established between these subgroups and pairs of combinatorial objects with labels from F × q . Each pair comprises a loopless binary matroid and a tight splice, an apparently new kind of combinatorial object which interpolates between nonnesting set partitions and shortened polyominoes. For arbitrary q, the same approach describes a natural subset of normal subgroups: those which correspond to the ideals of the Lie algebra ut n (F q ) under an approximation of the exponential map.
This paper constructs a novel Hopf algebra cfpUT ‚ q on the class functions of the unipotent upper triangular groups UT n pF q q over a finite field. This construction is representation theoretic in nature and uses the machinery of Hopf monoids in the category of vector species. In contrast with a similar known construction, this Hopf algebra has the property that induction to the finite general linear group induces a homomorphism to Zelevinsky's Hopf algebra of GL n pF q q class functions. Furthermore, cfpUT ‚ q contains a Hopf subalgebra which is isomorphic to a known combiantorial Hopf algebra, previously used to prove a conjecture about chromatic quasisymmetric functions. Some additional Hopf algebraic properties are also established.
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