Narrow Results

We describe the primitive ideal space of the C*-algebra of a row-finite k-graph with no sources when every ideal is gauge invariant. We characterize which spectral spaces can occur, and compute the primitive ideal space of two examples. In order to do this we prove some new results on aperiodicity. Our computations indicate that when every ideal is gauge invariant, the primitive ideal space only depends on the 1-skeleton of the k-graph in question.

Let $G=exp𝔤$ be an exponential solvable Lie group with Lie algebra $𝔤$, $K=exp𝔨$ an analytic subgroup of $G$ with Lie algebra $𝔨$ and $\pi$ an irreducible unitary representation of $G$. The focus here is on the study of the restriction $\pi {|}_K$ of discrete type, especially the algebra $D_{\pi }{(G)}^K$ of $K$-invariant differential operators in the space of $\pi$. Provided that the coadjoint orbit of $G$ corresponding to $\pi$ is of maximal dimension, we diagonalize these operators by means of Penney’s distributions so that the algebra $D_{\pi }{(G)}^K$ turns out to be commutative when $\pi {|}_K$ is of discrete type, and show that the converse may fail to hold as a developed example reveals. Furthermore, we show that $D_{\pi }{(G)}^K$ is isomorphic to the algebra $ℂ{[\mathrm{\Omega }]}^K$ of the $K$-invariant polynomial functions on $\mathrm{\Omega }$. We also produce a process to provide some generators of the kernel $ker\pi$ in the enveloping algebra of ${𝔤}_{ℂ}$.

In this paper, prime as well as primitive Kumjian–Pask algebras ${\mathrm{\text{KP}}}_R(\mathrm{\Lambda })$ of a row-finite $k$-graph $\mathrm{\Lambda }$ over a unital commutative ring $R$ are completely characterized in graph-theoretic and algebraic terms. By applying quotient $k$-graphs, these results describe prime and primitive graded basic ideals of Kumjian–Pask algebras. In particular, when $\mathrm{\Lambda }$ is strongly aperiodic and $R$ is a field, all prime and primitive ideals of a Kumjian–Pask algebra ${\mathrm{\text{KP}}}_R(\mathrm{\Lambda })$ are determined.

For the centrally extended Heisenberg double of SL₂, a classification of its prime and primitive ideals is obtained. For each primitive ideal, an explicit set of generators is given.

Let be a restricted Lie algebra over an algebraically closed field F of characteristic p > 0, the center of the universal enveloping algebra of . In this note, we study primitive ideals of . The following results are included: (1) The ideal of generated by the central character ideal associated with any irreducible -module has finite co-dimension in . Furthermore, the co-dimension is no less than , where is the maximal dimension of irreducible -modules. (2) Each annihilator ideal of irreducible -modules of maximal dimension is generated by the corresponding central character ideal in . (3) Each G-stable ideal in for contains nonzero fixed points under the action of G, where G is a connected reductive algebraic group. Additionally, the arguments on ideals help us to give an alternative description of the Azumaya locus in the Zassenhaus variety without using the normality of the Zassenhaus variety.

We construct a generator system of the annihilator of a generalized Verma module of a classical reductive Lie algebra induced from a character of a parabolic subalgebra as an analogue of the minimal polynomial of a matrix. In a classical limit it gives a generator system of the defining ideal of any semisimple co-adjoint orbit of the Lie algebra. We also give some applications to integral geometry.

A finite W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. In this survey we review recent developments in the representation theory of W-algebras. We emphasize various interactions between W-algebras and universal enveloping algebras.