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We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras UT_2(W_\lambda) or End(W_\mu) for some integral dominant weight \lambda,\mu with \mu \neq 0. In the special case L=\mathfrak {sl}_2 we prove that this is a sufficient condition too. [ABSTRACT FROM AUTHOR]

We consider how Morita equivalences are compatible with the notion of a corner subring. Namely, we outline a canonical way to replace a corner subring of a given ring with one which is Morita equivalent, and look at how such an equivalence ascends. We use the language of Morita contexts, and then specify these more general results. We give applications to trivial extensions of finite-dimensional algebras, tensor rings of pro-species, semilinear clannish algebras arising from orbifolds, and functor categories. [ABSTRACT FROM AUTHOR]

Let G be a semisimple complex algebraic group with a simple Lie algebra , and let ℳ G 0 denote the moduli stack of topologically trivial stable G -bundles on a smooth projective curve C. Fix a theta characteristic κ on C which is even in case dim is odd. We show that there is a nonempty Zariski open substack κ of ℳ G 0 such that H i (C , ad (E G) ⊗ κ) = 0 , i = 1 , 2 , for all E G ∈ κ . It is shown that any such E G has a canonical connection. It is also shown that the tangent bundle T U κ has a natural splitting, where U κ is the restriction of κ to the semi-stable locus. We also produce an isomorphism between two naturally occurring Ω M G r s 1 -torsors on the moduli space of regularly stable M G r s . [ABSTRACT FROM AUTHOR]

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e., certain affine Kazhdan-Lusztig polynomials. [ABSTRACT FROM AUTHOR]

This document provides a comprehensive review of the classification and structure of finite-dimensional simple right-alternative superalgebras. It explores the application of superalgebras in ring theory and the construction of a meaningful structural theory. The document covers various types of superalgebras, including associative superalgebras, commutative alternative superalgebras, and Abelian superalgebras, providing definitions, examples, and classifications. It also discusses the properties and decomposition of different types of superalgebras based on their even parts, as well as the structure of singular superalgebras. The document includes mathematical proofs, examples, and open questions for further research. [Extracted from the article]

The given document is a research paper titled "Quasiregular Radicals of Nonassociative Algebras" written by A. Yu. Golubkov. The paper discusses various mathematical concepts related to Lie algebras, Jordan linear pairs, and linear triple systems. It explores the construction of quasiregular radicals and their applications in the Amitsur-Procesi theorem. The document also discusses the properties and relationships of different radicals in these algebraic structures. The author acknowledges funding support and provides their contact information for further inquiries. [Extracted from the article]

This article explores the automorphisms of a Chevalley group of type G2 over a commutative ring R. The authors demonstrate that every automorphism of this group, when R is generated by the invertible elements and the ideal 2R, can be expressed as a combination of ring and inner automorphisms. The paper offers a historical overview of the study of automorphisms of classical groups and Chevalley groups, as well as the methods employed in previous research. The authors introduce definitions and main theorems related to Chevalley groups and their automorphisms. The text focuses on automorphisms of the group Gad(G2, R) and their properties, defining ring automorphisms and inner automorphisms, and establishing standard automorphisms as compositions of these two types. The primary objective is to prove that any automorphism of the group Gad(G2, R) is standard. The text also covers definitions and theorems concerning the localization of rings and modules, isomorphisms of Chevalley groups over fields, and the characteristic subgroup Ead(G2, R) in Gad(G2, R). The main theorem's proof is outlined in several steps, including the embedding of the ring R into a product of its localizations and the mapping of elements under conjugation by an element of Gad(G2, S). Additionally, a lemma is proven that demonstrates the mapping of matrices under conjugation by an element of Gad(G2, S) [Extracted from the article]

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an affine cellular algebra A is semisimple if and only if the scheme associated to A is reduced and 0-dimensional, and the bilinear forms with respect to all layers of A are invertible. Moreover, if the ground ring is a perfect field, then A is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple. [ABSTRACT FROM AUTHOR]

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, GenestierLafforgue and Fargues-Scholze have attached a semisimple parameter Lss to each irreducible representation π. Our first result shows that the Genestier-Lafforgue parameter of a tempered π can be uniquely refined to a tempered L-parameter L(π), thus giving the unique local Langlands correspondence which is compatible with the GenestierLafforgue construction. Our second result establishes ramification properties of Lss (π) for unramified G and supercuspidal π constructed by induction from an open compact (modulo center) subgroup. If Lss is pure in an appropriate sense, we show that Lss (π) is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show Lss is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is P1 and a simple application of Deligne’s Weil II. [ABSTRACT FROM AUTHOR]

Let 퐺 be a real semisimple Lie group with finite centre and without compact factors, Q < G a parabolic subgroup and 푋 a homogeneous space of 퐺 admitting an equivariant projection on the flag variety G / Q with fibres given by copies of lattice quotients of a semisimple factor of 푄. Given a probability measure 휇, Zariski-dense in a copy of H = SL 2 ⁡ (R) in 퐺, we give a description of 휇-stationary probability measures on 푋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case G = Q , the type of stationary measures that 휇 admits depends strongly on the position of 퐻 relative to 푄. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss. [ABSTRACT FROM AUTHOR]

Consider a symmetric space G/H with simple Lie group G. We demonstrate that when G/H is not irreducible, it is necessarily even dimensional and noncompact. Furthermore, the subgroup H is also both noncompact and non-semisimple. Additionally, we establish that the only G-invariant connection on G/H is the canonical connection. On the other hand, we show that if G/H has an odd dimension, it must be irreducible, and the subgroup H must be semisimple. Finally, we present an explicit example, and we show that there exists no other torsion-free G-invariant connection on a symmetric space G/H with semisimple Lie group G which has the same curvature as the canonical one. [ABSTRACT FROM AUTHOR]

Let Γ be a weakly irreducible lattice in a higher rank semisimple algebraic Lie group G without property (T) such as in . In this paper, for such Γ, we prove a cocycle superrigidity, Dynamical cocycle superrigidity, for Γ-actions under L2 integrability and irreducibility assumptions. It gives a similar result to Zimmer's cocycle superrigidity, so we can apply it instead of Zimmer's cocycle superrigidity in many situations. For instance, in this paper, we obtain global rigidity of Anosov Γ-actions on nilmanifolds under the irreducibility assumption on a fully supported invariant measure. [ABSTRACT FROM AUTHOR]

A result of Nayak asserts that lim m → ∞ | A m | 1 / m exists for each n × n complex matrix A , where | A | = (A ⁎ A) 1 / 2 , and the limit is given in terms of the spectral decomposition. We extend the result of Nayak, namely, we prove that the limit of lim m → ∞ | B A m C | 1 / m exists for any n × n complex matrices A , B , and C , where B and C are nonsingular; the limit is obtained and is independent of B. We then provide generalization in the context of real semisimple Lie groups. [ABSTRACT FROM AUTHOR]

This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contains that of all others? (ii) Do expanding left-invariant Ricci solitons exhibit such maximal symmetry? Question (i) is addressed both for semisimple and for solvable Lie groups. Building on previous work of the authors on Einstein metrics, a complete answer is given to (ii): expanding homogeneous Ricci solitons have maximal isometry algebras although not always maximal isometry groups. As a consequence of the tools developed to address these questions, partial results of Böhm, Lafuente, and Lauret are extended to show that left-invariant Ricci solitons on solvable Lie groups are unique up to scaling and isometry. [ABSTRACT FROM AUTHOR]

k-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant k-symplectic structures on Lie groups having a bi-invariant pseudo-Riemannian metric. We show that compact semi-simple Lie groups and a large class of Lie groups having a bi-invariant pseudo-Riemannian metric does not carry any left invariant k-symplectic structure. This class contains the oscillator Lie groups which are the only solvable non abelian Lie groups having a bi-invariant Lorentzian metric. However, we built a natural left invariant n-symplectic structure on SL (n , R) . Moreover, up to dimension 6, only three connected and simply connected Lie groups have a bi-invariant indecomposable pseudo-Riemannian metric and a left invariant k-symplectic structure, namely, the universal covering of SL (2 , R) with a 2-symplectic structure, the universal covering of the Lorentz group SO (3 , 1) with a 2-symplectic structure, and a 2-step nilpotent 6-dimensional connected and simply connected Lie group with both a 1-symplectic structure and a 2-symplectic structure. [ABSTRACT FROM AUTHOR]

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216. [ABSTRACT FROM AUTHOR]

Quantum circuits of a general quantum gate acting on multiple d$d$‐level quantum systems play a prominent role in multi‐valued quantum computation. A recursive Cartan decomposition of semi‐simple unitary Lie group U(3n)$U(3^n)$ (arbitrary n$n$‐qutrit gate) is first proposed with a rigorous proof, which completely decomposes an n$n$‐qutrit gate into local and non‐local operations. On this basis, an explicit quantum circuit is designed for implementing arbitrary two‐qutrit gates, and the cost of the construction is 21 generalized controlled X$X$ (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Furthermore, the program is extended to the n$n$‐qutrit system, and the quantum circuit of generic n$n$‐qutrit gates contained 4196·32n−4·3n−1−(n22+n4−2932)$\frac{41}{96}\cdot 3^{2n}-4\cdot 3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})$ GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far and its strength becomes more remarkable as n$n$ increases, for example, when n=5$n=5$, the program saves 7146 GCXs compared to the previous best result. In addition, concrete recursive decomposition expressions is given for each non‐local operation instead of only quantum circuit diagrams. [ABSTRACT FROM AUTHOR]

Complementary pairs of symmetric 2-designs are equivalent to coherent configurations of type (2, 2; 2). D. G. Higman studied these coherent configurations and adjacency algebras of coherent configurations over a field of characteristic zero. These are always semisimple. We investigate these algebras over fields of any characteristic prime and the structures. [ABSTRACT FROM AUTHOR]

The 3-cyclic quantum Weyl algebra A=A(α,β,γ;q²), where α,β,γ,q²∈K (a ground field), is a quadratic Noetherian domain of Gelfand–Kirillov dimension 3 that is generated by three subalgebras each of them is either a quantum plane or the quantum Weyl algebra. For the algebras A, their prime, completely prime, primitive and maximal spectra are described together with containments of prime ideals (the Zariski–Jacobson topology on the spectrum) and simple A-modules are classified when q2 is not a root of unity. For each prime ideal, an explicit set of ideal generators is given. The centre Z(A) of A is K[Ω] where Ω is a cubic element. A semisimplicity criterion for the category of finite dimensional A-modules is given. Criteria are presented for all ideals of the algebra A to commute and for each ideal of A to be a unique product of primes (up to order). [ABSTRACT FROM AUTHOR]

Let V be a simple vertex algebra of countable dimension, G be a finite automorphism group of V and σ be a central element of G. Assume that S is a finite set of inequivalent irreducible σ -twisted V -modules such that S is invariant under the action of G. Then there is a finite dimensional semisimple associative algebra A α (G , S) for a suitable 2-cocycle α naturally determined by the G -action on S such that (A α (G , S) , V G) form a dual pair on the sum M of σ -twisted V -modules in S in the sense that (1) the actions of A α (G , S) and V G on M commute, (2) each irreducible A α (G , S) -module appears in M , (3) the multiplicity space of each irreducible A α (G , S) -module is an irreducible V G -module, (4) the multiplicity spaces of different irreducible A α (G , S) -modules are inequivalent V G -modules. As applications, every irreducible V -module is a direct sum of finitely many irreducible V G -modules and irreducible V G -modules appearing in different G -orbits are inequivalent. This result generalizes many previous results. We also establish a bijection between subgroups of G and subalgebras of V containing V G. [ABSTRACT FROM AUTHOR]

Let g be a complex simple Lie algebra with a Borel subalgebra b. Consider the semidirect product I b = b ⋉ b ⁎ , where the dual b ⁎ of b is equipped with the coadjoint action of b and is considered as an abelian ideal of I b. We describe the automorphism group Aut (I b) of the Lie algebra I b. In particular we prove that it contains the automorphism group of the extended Dynkin diagram of g. In type A n , the dihedral subgroup was recently proved to be contained in Aut (I b) by Dror Bar-Natan and Roland van der Veen in [1] (where I b is denoted by I u n). Their construction is ad hoc and they asked for an explanation which is provided by this note. Let n denote the nilpotent radical of b. We obtain similar results for I b ‾ = b ⋉ n ⁎ that is both an Inönü-Wigner contraction of g and the quotient of I b by its center. [ABSTRACT FROM AUTHOR]

We investigate three aspects of weak* convergence of the n-step distributions of random walks on finite volume homogeneous spaces $ G/\Gamma $ G / Γ of semisimple real Lie groups. First, we look into the obvious obstruction to the upgrade from Cesàro to non-averaged convergence: periodicity. We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish a strong uniformity property for the Cesàro convergence towards Haar measure for uniquely ergodic random walks. [ABSTRACT FROM AUTHOR]

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification G C of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case. [ABSTRACT FROM AUTHOR]

We consider chiral-type systems admitting a Lax representation with values in a real or complex semisimple Lie algebra such that an additional regularity condition is satisfied (one of the matrices is a regular element of the Lie algebra). We prove that for a chiral-type system with vanishing torsion and a nonvanishing curvature, the existence of at least one pointwise cosymmetry is a necessary condition for the regular Lax representation. [ABSTRACT FROM AUTHOR]

We give the exact contributions of Harish-Chandra transform, (1-lf)(>-.), of Schwartz functions f to the harmonic analysis of spherical convolutions and the corresponding V-Schwartz algebras on a connected semisimple Lie group G (with finite center). One of our major results gives the proof of how the TrombiVaradarajan Theorem enters into the spherical convolution transform of V-Schwartz functions and the generalization of this Theorem under the full spherical convolution map. [ABSTRACT FROM AUTHOR]

The central concept in the harmonic analysis of a compact gronp is the completeness of Peter-Weyl orthonormal basis as constructed from the matrix coefficients of a maximal set of irreducible unitary representations of the group, leading ultimately to the direct sum decomposition of its L2-space. A PeterWeyl theory for a semicomplete orthonormal set is also possible and is here developed in this paper for compact groups. Existence of semicomplete orthonormal sets on a compact group is proved by an explicit construction of the standard Riemann-Lebesgue semicomplete orthonormal set on the Torus, T. This approach gives an insight into the role played by the LL space of a compact group, which is discovered to be just an example (indeed the largest example for every semicomplete orthonormal set) of what is called a primeParseval subspace, which we proved to be dense in the usual L2- space, serves as the natural domain of the Fourier transform and breaks up into a direct-sum decomposition. This paper essentially gives the harmonic analysis of the primeParseval subspace of a compact group corresponding to any semicomplete orthonormal set, with an introduction to what is expected for all connected semisimple Lie groups through the notion of a K-semicomplete orthonormal set. [ABSTRACT FROM AUTHOR]

Let X = X 1 × X 2 be a product of two rank one symmetric spaces of non-compact type and Γ a torsion-free discrete subgroup in G 1 × G 2 . We show that the spectrum of Γ \ (X 1 × X 2) is related to the asymptotic growth of Γ in the two directions defined by the two factors. We obtain that L 2 (Γ \ (G 1 × G 2)) is tempered for a large class of Γ . [ABSTRACT FROM AUTHOR]

Some generalizations of spin Sutherland models descend from 'master integrable systems' living on Heisenberg doubles of compact semisimple Lie groups. The master systems represent Poisson–Lie counterparts of the systems of free motion modeled on the respective cotangent bundles and their reduction relies on taking quotient with respect to a suitable conjugation action of the compact Lie group. We present an enhanced exposition of the reductions and prove rigorously for the first time that the reduced systems possess the property of degenerate integrability on the dense open subset of the Poisson quotient space corresponding to the principal orbit type for the pertinent group action. After restriction to a smaller dense open subset, degenerate integrability on the generic symplectic leaves is demonstrated as well. The paper also contains a novel description of the reduced Poisson structure and a careful elaboration of the scaling limit whereby our reduced systems turn into the spin Sutherland models. [ABSTRACT FROM AUTHOR]

A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p , q). In this paper, we study pseudo-Riemannian Lie groups (G , 〈 ⋅ , ⋅ 〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g , then dim ⁡ h ≤ max ⁡ { 0 , min ⁡ { p , q } − 2 }. Secondly, we classify trans-Lorentzian Lie groups (i.e., min ⁡ { p , q } = 2) with non-Killing left-invariant conformal vector fields, and prove that [ g , g ] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat. [ABSTRACT FROM AUTHOR]

We classify discrete quantum subgroups in the quantum double of the |$q$| -deformation of a compact semisimple Lie group, regarded as the complexification. We also record their classifications in some variants of quantum groups. Along the way, we show that quantum doubles of non-Kac-type compact quantum groups do not admit the quantum analog of lattices considered by Brannan–Chirvasitu–Viselter. [ABSTRACT FROM AUTHOR]

We introduce supergroup analogs of 3-manifold invariants Z ^ , also known as homological blocks, which were previously considered for ordinary compact semisimple Lie groups. We focus on superunitary groups and work out the case of SU(2|1) in details. Physically these invariants are realized as the index of BPS states of a system of intersecting fivebranes wrapping a 3-manifold in M-theory. As in the original case, the homological blocks are q-series with integer coefficients. We provide an explicit algorithm to calculate these q-series for a class of plumbed 3-manifolds and study quantum modularity and resurgence properties for some particular 3-manifolds. Finally, we conjecture a formula relating the Z ^ invariants and the quantum invariants constructed from a non-semisimple category of representation of the unrolled version of a quantum supergroup. [ABSTRACT FROM AUTHOR]

Motivated by Haviar and Ploščica's 2021 characterisation of Boolean products of simple De Morgan algebras, we investigate Boolean products of simple algebras in filtral varieties. We provide two main theorems. The first yields Werner's Boolean-product representation of algebras in a discriminator variety as an immediate application. The second, which applies to algebras in which the top congruence is compact, yields a generalisation of the Haviar–Ploščica result to semisimple varieties of Ockham algebras. The property of having factor principal congruences is fundamental to both theorems. While major parts of our general theorems can be derived from results in the literature, we offer new, self-contained and essentially elementary proofs. [ABSTRACT FROM AUTHOR]

In this article, we investigate Hopf actions on vertex algebras. Our first main result is that every finite-dimensional Hopf algebra that inner faithfully acts on a given π 2 -injective vertex algebra must be a group algebra. Secondly, under suitable assumptions, we establish a Schur-Weyl type duality for semisimple Hopf actions on Hopf modules of vertex algebras. [ABSTRACT FROM AUTHOR]

It is proven that finite idempotent left non-degenerate set-theoretic solutions |$(X,r)$| of the Yang–Baxter equation on a set |$X$| are determined by a left simple semigroup structure on |$X$| (in particular, a finite union of isomorphic copies of a group) and some maps |$q$| and |$\varphi _{x}$| on |$X$|⁠ , for |$x\in X$|⁠. This structure turns out to be a group precisely when the associated Yang–Baxter monoid |$M(X,r)$| is cancellative and all the maps |$\varphi _{x}$| are equal to an automorphism of this group. Equivalently, the Yang–Baxter algebra |$K[M(X,r)]$| is right Noetherian, or in characteristic zero it has to be semiprime. The Yang–Baxter algebra is always a left Noetherian representable algebra of Gelfand–Kirillov dimension one. To prove these results, it is shown that the Yang–Baxter semigroup |$S(X,r)$| has a decomposition in finitely many cancellative semigroups |$S_{u}$| indexed by the diagonal, each |$S_{u}$| has a group of quotients |$G_{u}$| that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that |$X$| equals the diagonal is fully described by a single permutation on |$X$|⁠. [ABSTRACT FROM AUTHOR]

In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group G in which the control functions belong to the Lie algebra of a compact Lie subgroup K of G and we investigate conditions under which the time optimal solutions of this compactified system are “approximately” time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control ( [1], [2]). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair (G, K) under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of K. In the second part of the paper we applied our results to the quantum systems known as Icing n-chains (introduced in [2]). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by ( [3], [4]) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains. [ABSTRACT FROM AUTHOR]

Let G be a non-compact classical semisimple Lie group and let G / V be the adjoint orbit with respect to a fixed element in G. These manifolds can be equipped with an almost-Kähler structure and we provide explicit formulae for the existence of special almost-complex structures on G / V purely in terms of the combinatorics of the associated Vogan diagram. The formulae are given separately for Lie groups whose Lie algebras are of type A ℓ , B ℓ , C ℓ , D ℓ , where ℓ denotes the rank of the Lie algebra. [ABSTRACT FROM AUTHOR]

This document is an editors' introduction to a special issue of the Journal of Spectral Theory dedicated to honoring Edward Brian Davies on his 80th birthday. It provides a brief biography of Davies, highlighting his academic achievements and contributions to the field of mathematics, particularly in spectral theory and the study of non-self-adjoint operators. The document also mentions Davies' role in attracting talented mathematicians from the former Soviet Union to the UK and his influence on the development of the theory of linear partial differential equations in the UK. The introduction concludes by listing the topics covered in the special issue and expressing the editors' hope that readers will find joy in reading the volume. [Extracted from the article]

Every stationary action of a strongly irreducible lattice or commensurator of such a lattice in a general semisimple group, with at least one higher-rank connected factor, either has finite stabilizers almost surely or finite index stabilizers almost surely. Consequently, every minimal action of such a lattice on an infinite compact metric space is topologically free. [ABSTRACT FROM AUTHOR]

We classify all different composite Higgs models (CHMs) characterised by the coset space G / H of compact semi-simple Lie groups G and H involving up to 13 Nambu-Goldstone bosons (NGBs), together with mild phenomenological constraints. As a byproduct of this work, we prove several simple yet, to the best of our knowledge, mostly unknown results: (1) under certain conditions, a given set of massless scalars can be UV completed into a CHM in which they arise as NGBs; (2) the set of all CHMs with a fixed number of NGBs is finite, and in particular there are 642 of them with up to 13 massless scalars (factoring out models that differ by extra U(1)'s); (3) any scalar representation of the Standard Model group can be realised within a CHM; (4) Certain symmetries of the scalar sector allowed from the IR perspective are never realised within CHMs. On top of this, we make a simple statistical analysis of the landscape of CHMs, determining the frequency of models with scalar singlets, doublets, triplets and other multiplets of the custodial group as well as their multiplicity. We also count the number of models with a symmetric coset. [ABSTRACT FROM AUTHOR]

In a previous paper, the author and his collaborator studied the problem of lifting Hamiltonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups. Only even quantizations were considered there. In this paper, these results are generalized to the case of general quantizations with arbitrary periods. The key step is to introduce an enhanced version of the (truncated) period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth symplectic variety X, with values in the space of Picard Lie algebroid over X. As an application, we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition. [ABSTRACT FROM AUTHOR]

In this paper, we establish a structural inequality of the ∞-subLaplacian ▵ 0 , ∞ in a class of the semi-simple Lie group endowed with the horizontal vector fields X 1 , ... , X 2 n . When 1 < p ≤ 4 with n = 1 and 1 < p < 3 + 1 n − 1 with n ≥ 2 , we apply the structural inequality to obtain the local horizontal W 2 , 2 -regularity of weak solutions to p-Laplacian equation in the semi-simple Lie group. Compared to Euclidean spaces R 2 n with n ≥ 2 , the range of this p obtained is already optimal. [ABSTRACT FROM AUTHOR]

In this paper, we study the cohomology of semisimple local systems in the spirit of classical Hodge theory. On the one hand, we construct a generalized Weil operator from the complex conjugate of the cohomology of a semisimple local system to the cohomology of its dual local system, which is functorial with respect to smooth restrictions. This operator allows us to study the Poincaré pairing, usually not positive definite, in terms of a positive definite Hermitian pairing. On the other hand, we prove a global invariant cycle theorem for semisimple local systems. As an application, we give a new proof of Sabbah's Decomposition Theorem for the direct images of semisimple local systems under proper algebraic maps, by adapting the method of de Cataldo-Migliorini, without using the category of polarizable twistor D -modules. This answers a question of Sabbah. [ABSTRACT FROM AUTHOR]

Quotient grading classes are essential participants in the computation of the intrinsic fundamental group π 1 (A) of an algebra A. In order to study quotient gradings of a finite-dimensional semisimple complex algebra A it is sufficient to understand the quotient gradings of twisted group algebra gradings. We establish the graded structure of such quotients using Mackey's obstruction class. Then, for matrix algebras A = M n (C) we tie up the concepts of braces, group-theoretic Lagrangians and elementary crossed products. We also manage to compute the intrinsic fundamental group of the diagonal algebras A = C 4 and A = C 5. [ABSTRACT FROM AUTHOR]

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSL n (q) (n ≥ 4) , PSL 3 (q) (q > 2) , or PSp 2 n (q) (n ≥ 3) , is isomorphic to a group algebra. [ABSTRACT FROM AUTHOR]

We prove Wiener Tauberian theorem type results for various spaces of radial functions, which are Banach algebras on a real-rank-one semisimple Lie group G. These are natural generalizations of the Wiener Tauberian theorem for the commutative Banach algebra of the integrable radial functions on G. [ABSTRACT FROM AUTHOR]

The Witten r-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin–Frobenius manifold using the Givental–Teleman reconstruction theorem. We show that the R-matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Using this, we prove that the descendant intersection theory of the shifted Witten classes satisfies topological recursion on two 1-parameter families of spectral curves. By taking the limit as the parameter goes to zero, we prove that the descendant intersection theory of the Witten r-spin class can be computed by topological recursion on the r-Airy spectral curve. We finally show that this proof suffices to deduce Witten's r-spin conjecture, already proved by Faber, Shadrin and Zvonkine, which claims that the generating series of r-spin intersection numbers is the tau function of the r-KdV hierarchy that satisfies the string equation. [ABSTRACT FROM AUTHOR]

The solution of Shareshian–Wachs conjecture by Brosnan–Chow linked together the cohomology of regular semisimple Hessenberg varieties and graded chromatic symmetric functions on unit interval graphs. On the other hand, it is known that unicellular LLT polynomials have similar properties to graded chromatic symmetric functions. In this paper, we link together the unicellular LLT polynomials and twin of regular semisimple Hessenberg varieties introduced by Ayzenberg–Buchstaber. We prove the palindromicity of LLT polynomials from topological viewpoint. We also show that modules of a symmetric group generated by faces of a permutohedron are related to a shifted unicellular LLT polynomial and observe the |$e$| -positivity of shifted unicellular LLT polynomials, which is established by Alexandersson–Sulzgruber in general, for path graphs and complete graphs through the cohomology of the twins. [ABSTRACT FROM AUTHOR]

Let \mathcal {O}_p denote the characteristic p>0 version of the ordinary category \mathcal {O} for a semisimple complex Lie algebra. In this paper we give some (formal) character formulas in \mathcal {O}_p. First we concentrate on the irreducible characters. Here we give explicit formulas for how to obtain all irreducible characters from the characters of the finitely many restricted simple modules as well as the characters of a small number of infinite dimensional simple modules in \mathcal {O}_p with specified highest weights. We next prove a strong linkage principle for Verma modules which allow us to split \mathcal {O}_p into a finite direct sum of linkage classes. There are corresponding translation functors and we use these to further cut down the set of irreducible characters needed for determining all others. Then we show that the twisting functors on \mathcal {O} carry over to twisting functors on \mathcal {O}_p, and as an application we prove a character sum formula for Jantzen-type filtrations of Verma modules with antidominant highest weights. Finally, we record formulas relating the characters of the two kinds of tilting modules in \mathcal {O}_p. [ABSTRACT FROM AUTHOR]