arXiv subjects
Publications and source records attributed to Takuma Aihara.
We study a homological epimorphism of proper dg algebras and show that if the source is a finite dimensional piecewise hereditary algebra, then the target is derived equivalent to some piecewise hereditary algebra.
On-chip erbium-doped optical amplifiers and lasers are essential for realizing fully integrated active silicon photonic circuits, but their performance has been limited by the low gain of amorphous host materials and the difficulty of direct integration on silicon. Here, we demonstrate optical amplification and lasing from erbium-doped single-crystal gadolinium oxide (Er:Gd$_2$O$_3$) thin films epitaxially grown on silicon. Optical gain measurements on waveguides fabricated on this platform exhibit a giant material gain of $78.3\pm2.1$ dB/cm and an on-chip net gain exceeding 13 dB in a 6-mm-long waveguide at 2.3 K, while a measurable gain is maintained up to room temperature. Continuous-wave lasing with low threshold, narrow linewidth, and large side-mode suppression ratio is also demonstrated in Er:Gd$_2$O$_3$ microring resonators. These results establish Er:Gd$_2$O$_3$ as the first monolithic crystalline gain medium directly integrated on silicon, providing a scalable route toward high-performance cryogenic and quantum photonic integrated circuits.
We give several examples of tilting-discrete symmetric algebras; in particular, one explores which algebra has tilting-discrete trivial extension. We provide a counter example of the conjecture stating any τ -tilting finite symmetric algebra is tiltingdiscrete. Also, we discuss the tilting-disconnectedness of symmetric algebras and give new examples of tilting-disconnected symmetric algebras.
Aperiodic systems such as quasiperiodic systems exhibit unique properties different from periodic structures. In 2023, Smith et al. discovered a new aperiodic structure: a single-shaped tile that can only tile space aperiodically, known as an aperiodic monotile. Although the aperiodic monotile possesses intriguing mathematical properties, its experimental investigation remains unexplored. In this study, we report an experimental investigation of diffraction patterns from a monotile lattice using a nanophotonic platform. We observed clear Bragg peaks, which is evidence of long-range order and a chiral structure of the diffraction patterns. Furthermore, we found exotic behavior in circular polarization dependence, which cannot be observed in conventional quasiperiodic structures. These findings establish the monotile lattice as a novel class of aperiodic systems, expanding the study of nonperiodic structures beyond conventional quasicrystals.
We investigate symmetry of the silting quiver of a given algebra which is induced by an anti-automorphism of the algebra. In particular, one shows that if there is a primitive idempotent fixed by the anti-automorphism, then the 2-silting quiver ($=$ the support $τ$-tilting quiver) has a bisection. Consequently, in that case, we obtain that the cardinality of the 2-silting quiver is an even number (if it is finite).
We explore when the silting-discreteness is inherited. As a result, one obtains that taking idempotent truncations and homological epimorphisms of algebras transmit the silting-discreteness. We also study classification of silting-discrete simply-connected tensor algebras and silting-indiscrete selfinjective Nakayama algebras. This paper contains two appendices; one states that every derived-discrete algebra is silting-discrete, and the other is about triangulated categories whose silting objects are tilting.
First, we give a new example of silting-discrete algebras. Second, one explores when the algebra of triangular matrices over a finite dimensional algebra is $τ$-tilting finite. In particular, we classify algebras over which triangular matrix algebras are $τ$-tilting finite. Finally, we investigate when a triangular matrix algebra is silting-discrete.
Measurement and feedback control of stochastic dynamics has been actively studied for not only stabilizing the system but also for generating additional entropy flows originating in the information flow in the feedback controller. In particular, a micromechanical system offers a great platform to investigate such non-equilibrium dynamics under measurement-feedback control owing to its precise controllability of small fluctuations. Although various types of measurement-feedback protocols have been demonstrated with linear observables (e.g., displacement and velocity), extending them to the nonlinear regime, i.e., utilizing nonlinear observables in both measurement and control, retains non-trivial phenomena in its non-equilibrium dynamics. Here, we demonstrate measurement-feedback control of a micromechanical resonator by driving the second-order nonlinearity (i.e., parametric squeezing) and directly measuring quadratic observables, which are given by the Schwinger representation of pseudo angular momentum (referred as Schwinger angular momentum). In contrast to that the parametric divergence occurs when the second-order nonlinearity is blindly driven, our measurement-feedback protocol enables us to avoid such a divergence and to achieve a strong noise reduction at the level of $-5.1\pm 0.2$ dB. This strong noise reduction originates in the effective cooling included in our measurement-feedback protocol, which is unveiled by investigating entropy production rates in a coarse-grained model. Our results open up the possibility of not only improving noise-limited sensitivity performance but also investigating entropy production in information thermodynamic machines with nonlinear measurement and feedback.
Motivated by the relation between Schur algebra and the group algebra of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang and Steffen Koenig addressed some behaviour of the endomorphism algebra of a generator over a symmetric algebra, which they called gendo-symmetric algebra. Continuing this line of works, we classify in this article the representation-finite gendo-symmetric algebras that have at most one isomorphism class of indecomposable non-injective projective module. We also determine their almost ν-stable derived equivalence classes in the sense of Wei Hu and Changchang Xi. It turns out that a representative can be chosen as the quotient of a representation-finite symmetric algebra by the socle of a certain indecomposable projective module.
We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, one studies two classes of $τ$-tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $τ$-tilting-finiteness implies representatoin-finiteness, and obtain several classes of algebras in which a $τ$-tilting-finite algebra is representation-finite.
We report an observation of phononic Schwinger angular momenta, which fully represent twomode states in a micromechanical resonator. This observation is based on simultaneous optical detection of the mechanical response at the sum and difference frequency of the two mechanical modes. A post-selection process for the measured signals allows us to extract a component of phononic Schwinger angular momenta. It also enables us to conditionally prepare two-mode squeezed (correlated) states from a randomly excited (uncorrelated) state. The phononic Schwinger angular momenta could be extended to high-dimensional symmetry (e.g. SU(N) group) for studying multipartite correlations in non-equilibrium dynamics with macroscopic objects.
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.
In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.
Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.
Happel and Unger reconstructed hereditary algebras from their posets of tilting modules. Inspired by this result, we try removing the assumption to be hereditary. However, it would be unfortunately fail in general: e.g. every selfinjective algebra has the poset consisting of only one point. Therefore, we should consider a generalization of the Happel-Unger's result for posets of support $τ$-tilting modules, which contains those of tilting modules. In this paper, we spotlight finite dimensional algebras whose support $τ$-tilting posets coincide with those of tree quiver algebras and give a full characterization of such algebras.
The notion of mutation plays crucial roles in representation theory of algebras. Two kinds of mutation are well-known: tilting/silting mutation and quiver-mutation. In this paper, we focus on tilting mutation for symmetric algebras. Introducing mutation of SB quivers, we explicitly give a combinatorial description of tilting mutation of symmetric special biserial algebras. As an application, we generalize Rickard's star theorem. We also introduce flip of Brauer graphs and apply our results to Brauer graph algebras.
In this paper, we prove that certain contravariant endofunctors of singularity categories characterize Gorenstein rings.
This paper introduces the concept of the dimension of a triangulated category with respect to a fixed full subcategory. For the bounded derived category of an abelian category, upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given. Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings.