Search

In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type G2, which can be obtained through the orbits of integral weights under the action of their corresponding Weyl groups and the invariant polynomial theory of the Weyl groups. We show that the characteristic polynomials can be decomposed into products of irreducible orbital factors, each of which is invariant under the action of their corresponding Weyl groups., Comment: 26pages

In this paper, we explore Fourier analysis for noncommutative $L_p$ space-valued functions on $G$, where $G$ is a totally disconnected non-abelian compact group. By additionally assuming that the value of these functions remains invariant within each conjugacy class, we establish a noncommutative maximal inequality for Fej\'er means utilizing the associated character system of $G$. This is an operator-valued version of the classical result due to G\'at. We follow essentially the classical sketch, but due to the noncommutativity, many classical arguments have to be revised. Notably, compared to the classical results. the bounds of our estimates are explicity calculated.

In this paper, the correspondence between the finite dimensional representations of a simple Lie algebra and their characteristic polynomials is established, and a monoid structure on these characteristic polynomials is constructed. Furthermore, the characteristic polynomials of sl(2, C) on some classical simple Lie algebras through adjoint representations are studied, and we present some results of Borel subalgebras and parabolic subalgebras of simple Lie algebras through characteristic polynomials., Comment: 16

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to $1$ pointwise, so that the associated Fourier multipliers on noncommutative $L_p$-spaces satisfy the pointwise convergence for all $p>1$. In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fej\'{e}r and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies., Comment: V5: minor corrections; final version to appear in Memoirs of the AMS. v4: 87pages; minor changes; some details of the proof were added in Section 5.2; the part on classical analysis has been removed and will appear separately in another forthcoming paper. v3: 83 pages; this version contains some corrections. v2: 74 pages; new results are added in Section 4, Section 5 and Section 6.3

Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected coideal *-subalgebras. Several one-to-one correspondences between such invariant functionals are established. Adding a positivity condition, this yields one-to-one correspondences of invariant quantum Markov semigroups acting on expected coideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces. The classical sphere $S^{N-1}$, the free sphere $S^{N-1}_+$, and the half-liberated sphere $S^{N-1}_*$ are considered as examples and the generators of Markov semigroups on these spheres a classified. We compute spectral dimensions for the three families of spheres based on the asymptotic behaviour of the eigenvalues of their Laplace operator., Comment: Added computation of spectral dimension in Section 7.4