$L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

In this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

Matric generators of coalgebras and bialgebras

Takeuchi asserted that if a bialgebra [Formula: see text] over a field [Formula: see text] is finitely generated as a [Formula: see text]-algebra, then [Formula: see text] is a matric bialgebra. We introduce the notion of a matric coalgebra over a commutative ring [Formula: see text]. We show that if [Formula: see text] is faithfully projective as a [Formula: see text]-module, then [Formula: see text] is a matric coalgebra. Using this, we also show that if a bialgebra [Formula: see text] over a semihereditary ring [Formula: see text] is projective as a [Formula: see text]-module, then any finite subset of [Formula: see text] is contained in some matric subbialgebra. This result is a generalization of Takeuchi’s assertion and can be regarded as a local finiteness theorem on bialgebras.

Unitary Representations with Dirac Cohomology: Finiteness in the Real Case

Let $G$ be a complex connected simple algebraic group with a fixed real form $\sigma $. Let $G({\mathbb{R}})=G^\sigma $ be the corresponding group of real points. This paper reports a finiteness theorem for the classification of irreducible unitary Harish-Chandra modules of $G({\mathbb{R}})$ (up to equivalence) having nonvanishing Dirac cohomology. Moreover, we study the distribution of the spin norm along Vogan pencils for certain $G({\mathbb{R}})$, with particular attention paid to the unitarily small convex hull introduced by Salamanca-Riba and Vogan.