Quasi-split symmetric pairs of 𝑈(𝔰𝔩_{𝔫}) and Steinberg varieties of classical type

We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type A n − 1 A_{n-1} . This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of n n -step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and a geometric realization of rational modules.

Precise optical control of gene expression in C. elegans using improved genetic code expansion and Cre recombinase

Synthetic strategies for optically controlling gene expression may enable the precise spatiotemporal control of genes in any combination of cells that cannot be targeted with specific promoters. We develop an improved genetic code expansion system in C. elegans and use it to create a photo-activatable Cre recombinase. We laser-activate Cre in single neurons within a bilaterally symmetric pair to selectively switch on expression of a loxP controlled optogenetic channel in the targeted neuron. We use the system to dissect, in freely moving animals, the individual contributions of the mechanosensory neurons PLML/PLMR to the C. elegans touch response circuit, revealing distinct and synergistic roles for these neurons. We thus demonstrate how genetic code expansion and optical targeting can be combined to break the symmetry of neuron pairs and dissect behavioural outputs of individual neurons that cannot be genetically targeted.

Vortex precession and exchange in a Bose-Einstein condensate

Vortices in a Bose-Einstein condensate are modelled as spontaneously symmetry breaking minimum energy solutions of the time dependent Gross-Pitaevskii equation, using the method of constrained optimization. In a non-rotating axially symmetric trap, the core of a single vortex precesses around the trap center and, at the same time, the phase of its wave function shifts at a constant rate. The precession velocity, the speed of phase shift, and the distance between the vortex core and the trap center, depend continuously on the value of the conserved angular momentum that is carried by the entire condensate. In the case of a symmetric pair of identical vortices, the precession engages an emergent gauge field in their relative coordinate, with a flux that is equal to the ratio between the precession and shift velocities.

A Design Approach to Wireless High-Power Transfer to Multiple Receivers with Asymmetric Circuit

Wireless Power Transfer (WPT) system commonly compensates by a symmetric pair of inductor and capacitor on the primary-secondary circuits to use the idea of resonance. It should be noticed that an additional component compensation on the common WPT circuit is able to affect the power transferred to the load. Although it is useful to wirelessly transfer power to multiple receivers, the complexity of the system will increase with the number of receivers as well as the system loses symmetry, and then, it would be difficult to design high power transfer system. This study explores the WPT circuit compensated with a single capacitor in the primary side to transfer high power to dual receivers. Using a single capacitor on the primary side makes the circuit asymmetry, so the idea of resonance cannot be used. To find operating points that maximize transferred power, this paper uses a mathematical optimization technique with several design variables. The NSGA-II (Non-dominated Sorting Genetic Algorithm II) is used to optimize the design variables of the mathematical system model. The results show that the proposed system is able to attain high power even though using only a single capacitor compensation without the idea of resonance.

Defining relations of quantum symmetric pair coideal subalgebras

We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative ${\mathbb N}$ -graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.

Weak Multiplier Hopf Algebras II: Source and Target Algebras

Let (A,Δ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Δ:A⟶M(A⊗A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps εs and εt are studied, and their symmetric pair of images, the source algebra and the target algebra εs(A) and εt(A), are also investigated. We show that the canonical idempotent E (which is eventually Δ(1)) belongs to the multiplier algebra M(B⊗C), where (B=εs(A), C=εt(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,Δ), it is possible to make C⊗B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the ’Hopf algebra part’ of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E∈M(B⊗C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known ’quantization’ of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).

A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].