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Two subsets A, B of the plane are betweenness isomorphic if there is a bijection f : A → B such that, for every x, y, z ∈ A, the point f(z) lies on the line segment connecting f(x) and f(y) if and only if z lies on the line segment connecting x and y. In general, it is quite difficult to tell whether two given subsets of the plane are betweenness isomorphic. We concentrate on the case when each of the sets A, B is of the form C ∪ D where C is a circle and D is a finite set. We fully characterize the betweenness isomorphism classes in the family of all circles with three collinear points inside. In particular, we show that there are only countably many isomorphism classes, for which we provide an algebraic description. [ABSTRACT FROM AUTHOR]

We explore the role derivations of a Lie triple system play in the nilpotence and semisimplicity of T. In particular analogous to two theorems of Jacobson for Lie algebras, (by giving a sufficient condition for the nilpotence of T in terms of the existence of an invertible derivation, and in terms of the existence of certain automorphisms), are stated. The connections between the Lie algebra of derivations of and the semisimplicity of T are also studied. [ABSTRACT FROM AUTHOR]

Let g be a reductive Lie algebra and t ⊂ g a Cartan subalgebra. The t -stable decomposition g = t ⊕ m yields a bi-grading of the symmetric algebra S (g) . The subalgebra Z (g , t) generated by the bi-homogenous components of the symmetric invariants F ∈ S (g) g is known to be Poisson commutative. Furthermore the algebra Z ~ = alg ⟨ Z (g , t) , t ⟩ is also Poisson commutative. We investigate relations between Z ~ and Mishchenko–Fomenko subalgebras. In type A, we construct a quantisation of Z ~ making use of quantum Mishchenko–Fomenko algebras. [ABSTRACT FROM AUTHOR]

We compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups S L (∞) , O (∞) or S p (∞) . We show that the respective automorphism groups are much larger than S L (∞) , O (∞) or S p (∞) , and present the answer in terms of Mackey groups. The latter are groups of automorphisms of non-degenerate pairings of (in general infinite-dimensional) vector spaces. An explicit matrix form of the automorphism group of an arbitrary ind-variety of generalized flags is also given. The case of the Sato grassmannian is considered in detail, and its automorphism group is the projectivization of the connected component of unity in the group known as Japanese G L (∞) . [ABSTRACT FROM AUTHOR]

We show that an effective action of the one-dimensional torus G m on a normal affine algebraic variety X can be extended to an effective action of a semi-direct product G m ⋌ G a with the same general orbit closures if and only if there is a divisor D on X that consists of G m -fixed points. This result is applied to the study of orbits of the automorphism group Aut (X) on X. [ABSTRACT FROM AUTHOR]

An algebraic variety X is called a homogeneous variety if the automorphism group Aut (X) acts on X transitively, and a homogeneous space if there exists a transitive action of an algebraic group on X. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we give criteria for a Danielewski surface to be a homogeneous variety and a homogeneous space. Also, we construct affine suspensions of arbitrary dimension that are homogeneous varieties but not homogeneous spaces. [ABSTRACT FROM AUTHOR]

We construct for every proper algebraic space over a ground field an Albanese map to a para-abelian variety, which is unique up to unique isomorphism. This holds in the absence of rational points or ample sheaves, and also for reducible or non-reduced spaces, under the mere assumption that the structure morphism is in Stein factorization. It also works under suitable assumptions in families. In fact the treatment of the relative setting is crucial, even to understand the situation over ground fields. This also ensures that Albanese maps are equivariant with respect to actions of group schemes. Our approach depends on the notion of families of para-abelian varieties, where each geometric fiber admits the structure of an abelian variety, and representability of tau-parts in relative Picard groups, together with structure results on algebraic groups. [ABSTRACT FROM AUTHOR]

We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group O 2 (R) . The multigradings considered include the maximal grading associated to the decomposition of the representation into irreducibles as well as the bigrading associated to a cotangent-lifted representation, or equivalently, the bigrading associated to the holomorphic and antiholomorphic parts of the real invariants and covariants. This bigrading induces a bigrading on the algebra of on-shell invariants of the symplectic quotient, and the corresponding Hilbert series are computed as well. We also compute the first few Laurent coefficients of the univariate Hilbert series, give sample calculations of the multigraded Laurent coefficients, and give an example to illustrate the extension of these techniques to the semidirect product of the circle by other finite groups. We describe an algorithm to compute each of the associated Hilbert series. Communicated by Ellen Kirkman [ABSTRACT FROM AUTHOR]

Let g be a finite-dimensional perfect Lie algebra over a field k of characteristic 0. In infinite-dimensional Lie theory we encounter Lie algebras of the form g ⊗ k R , where R is a k-ring (usually a Laurent polynomial ring in finitely many variables over k), and étale twisted forms L of g ⊗ k R . Thus L is an R-Lie algebra that becomes isomorphic to the S-Lie algebra g ⊗ k S after some étale cover base ring extension S/R. The interesting infinite-dimensional Lie algebras are "built" out of L by adding a centre Z and a Lie algebra of derivations D (the affine Kac-Moody Lie algebras are the simplest examples). D , which determines Z , is a Lie subalgebra of Der k (L) of L . The understanding of this last Lie algebra is crucial. While the R-Lie algebra L can be given by étale descent, the same cannot be openly said about Der k (L) since it is not an R-Lie algebra. In the present paper we give such a descent presentation within the more general framework of relative R/k-sheaves of Lie algebras that we believe is of independent interest. [ABSTRACT FROM AUTHOR]

In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between orbits of graph states and orbits in the variety of binary symmetric principal minors, under the action of SL (2 , F 2) × n ⋊ S n . First we study the orbits of maximal abelian subgroups of the n-fold Pauli group under the action of C n loc ⋊ S n , where C n loc is the n-fold local Clifford group, and we show that this action corresponds to the natural action of SL (2 , F 2) × n ⋊ S n on the variety Z n ⊂ P (F 2 2 n) of principal minors of binary symmetric n × n matrices: the crucial step is in translating the action of SL (2 , F 2) × n into an action of the local symplectic group Sp 2 n loc (F 2) . We conclude by showing how the former action restricts onto stabilizer groups, stabilizer states and graph states. [ABSTRACT FROM AUTHOR]

We study torus-equivariant vector bundles E on a complex projective variety X which is either a Bott–Samelson–Demazure–Hansen variety or a wonderful compactification of a complex symmetric variety of minimal rank. We show that E is nef (respectively, ample) if and only if its restriction to every torus-invariant curve in X is nef (respectively, ample). We also compute the Seshadri constants ε (E , x) , where x ∈ X is any point fixed by the action of a maximal torus. [ABSTRACT FROM AUTHOR]

We consider several related examples of Fourier-Mukai transformations involving the quot scheme. A method of showing conservativity of these Fourier-Mukai transformations is described.

The universal centralizer of a semisimple algebraic group G is the family of centralizers of regular elements, parametrized by their conjugacy classes. When G is of adjoint type, we construct a smooth, log-symplectic fiberwise compactification Z ¯ of the universal centralizer Z by taking the closure of each fiber in the wonderful compactification G ¯ . We use the geometry of the wonderful compactification to give an explicit description of the symplectic leaves of Z ¯ . We also show that its compactified centralizer fibers are isomorphic to certain Hessenberg varieties—we apply this connection to compute the singular cohomology of Z ¯ , and to study the geometry of the corresponding universal Hessenberg family. [ABSTRACT FROM AUTHOR]

A function that is analytic on a domain of C n is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein–Sato polynomial of a holonomic function on a smooth algebraic variety. We analyze the structure of certain sheaves of holonomic functions, such as the algebraic functions along a hypersurface, determining their direct sum decompositions into indecomposables, that further respect decompositions of Bernstein–Sato polynomials. When the space is endowed with the action of a linear algebraic group G, we study the class of G-finite analytic functions, i.e. functions that under the action of the Lie algebra of G generate a finite dimensional rational G-module. These are automatically algebraic functions on a variety with a dense orbit. When G is reductive, we give several representation-theoretic techniques toward the determination of Bernstein–Sato polynomials of G-finite functions. We classify the G-finite functions on all but one of the irreducible reduced prehomogeneous vector spaces, and compute the Bernstein–Sato polynomials for distinguished G-finite functions. The results can be used to construct explicitly equivariant D -modules. [ABSTRACT FROM AUTHOR]

We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of cohomogeneity one (which is the same as being multiplicity free of rank one). The group acting is a compact connected Lie group (simply connected in the quasi-Hamiltonian case). This work is a concretization of a more general classification of multiplicity free manifolds in the special case of rank one. As a result we obtain numerous new concrete examples of multiplicity free quasi-Hamiltonian manifolds or, equivalently, Hamiltonian loop group actions. [ABSTRACT FROM AUTHOR]

For each n ≥ 3 we give examples of infinitesimally rigid projective manifolds of general type of dimension n with non-contractible universal cover. We provide examples with projective and examples with non-projective universal cover. [ABSTRACT FROM AUTHOR]

We compute the Hilbert series of the graded algebra of real regular functions on a linear symplectic quotient by the 2-torus as well as the first four coefficients of the Laurent expansion of this Hilbert series at t = 1 . We describe an algorithm to compute the Hilbert series as well as the Laurent coefficients in explicit examples. [ABSTRACT FROM AUTHOR]

The bandwidth of a C ∗ -action of a polarized pair (X, L) is a natural measure of its complexity. In this paper, we study C ∗ -actions on rational homogeneous spaces, determining which provide minimal bandwidth. We prove that the minimal bandwidth is linked to the smallest coefficient of the fundamental weight, in a base of simple roots, which describes the variety as a marked Dynkin diagram. As a direct application of the results we study the Chow groups of the Cayley plane E 6 (6) . [ABSTRACT FROM AUTHOR]

Let G be a semisimple algebraic group defined over an algebraically closed field. We provide some criteria for normality and rational singularities of G-saturation under certain circumstances. Our results are applied to determine when the commuting variety over simple Lie algebra of low rank is normal and Cohen-Macaulay. We also present some interesting connections between injective modules and normality (resp. rational singularities) of their G-saturations. Finally, we generalize a machinery used to study singularities of nilpotent orbit closures. [ABSTRACT FROM AUTHOR]

Let g be a semisimple Lie algebra, ϑ ∈ Aut (g) a finite order automorphism, and g 0 the subalgebra of fixed points of ϑ . Recently, we noticed that using ϑ one can construct a pencil of compatible Poisson brackets on S (g) , and thereby a ‘large’ Poisson-commutative subalgebra Z (g , ϑ) of S (g) g 0 . In this article, we study invariant-theoretic properties of (g , ϑ) that ensure good properties of Z (g , ϑ) . Associated with ϑ one has a natural Lie algebra contraction g (0) of g and the notion of a good generating system (=g.g.s.) in S (g) g . We prove that in many cases the equality ind g (0) = ind g holds and S (g) g has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), ϑ can be represented by a Kac diagram, K (ϑ) , and our results often use this presentation. The most surprising observation is that g (0) depends only on the set of nodes in K (ϑ) with nonzero labels, and that if ϑ is inner and a certain label is nonzero, then g (0) is isomorphic to a parabolic contraction of g . [ABSTRACT FROM AUTHOR]

We study the existence of a paving by affine spaces of Springer fibers. In particular, we prove that in the case of simple groups not of type E8 such pavings exist. [ABSTRACT FROM AUTHOR]

We show that every action of a smooth algebraic group on a variety admits a normal projective model. Along the way, we present new proofs of some basic results on algebraic transformation groups, including Weil’s regularization theorem. [ABSTRACT FROM AUTHOR]

In this paper, we work over C . Criterions for an abelian surface to be products of elliptic curves are known. We give a new criterion from the view point of the Galois cover. [ABSTRACT FROM AUTHOR]

A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension 3 (resp. 4) into the space S n of symmetric n × n matrices. We study the geometries of Jordan nets and webs: we classify the congruence orbits of Jordan nets (resp. webs) in S n for n ≤ 7 (resp. n ≤ 5 ), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in S n for n ≤ 5 , these obstructions show that our list of degenerations is complete. For n = 6 , the existence of one degeneration is still undetermined. To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions and then used it to compute the degenerations between Jordan nets in S 7 and Jordan webs in S n for n = 4 , 5 . [ABSTRACT FROM AUTHOR]

We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in terms of linear algebra. This result enables us to show that any direct limit of admissible embeddings of flag varieties is isomorphic to an ind-variety of generalized flags as defined in Dimitrov and Penkov (IMRN No 55:2935–2953, 2004). These latter ind-varieties have been introduced in terms of the ind-group S L (∞) (respectively, O (∞) or S p (∞) for isotropic generalized flags), and the current paper constructs them in purely algebraic–geometric terms. [ABSTRACT FROM AUTHOR]

We show that the pseudoeffective cone of k-cycles on a complete complexity-one T-variety is rational polyhedral for any k, generated by classes of T-invariant subvarieties. When X is also rational, we give a presentation of the Chow groups of X in terms of generators and relations, coming from the combinatorial data defining X as a T-variety. [ABSTRACT FROM AUTHOR]

Let X be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products Sym n X when the cohomology of X is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials μ X n S n t , u , v , codifying the permutation action of S n as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products μ Sym n X t , u , v . These formulas are then applied to the case of linear algebraic groups. [ABSTRACT FROM AUTHOR]

We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group G acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle E of homogeneous spaces for a group bundle G , all over a reduced Stein space X with compatible actions of a reductive complex group on E, G , and X. Application to the classification of generalised principal bundles with a group action. Finally, an equivariant version of Gromov's Oka principle based on a notion of a G-manifold being G-Oka. [ABSTRACT FROM AUTHOR]

We give the equations of the n-th symmetric product X n / S n of a flat affine scheme X = Spec A over a commutative ring F. As a consequence, we find a closed immersion into the coarse moduli space parameterizing n-dimensional linear representations of A. This is done by exhibiting an isomorphism between the ring of symmetric tensors over A and the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices giving representations of A. Using this we derive an isomorphism of the associated reduced schemes over an infinite field. When the characteristic is zero we show that this isomorphism is an isomorphism of schemes and we express it in term of traces. [ABSTRACT FROM AUTHOR]

We extend the Altmann-Hausen presentation of normal affine algebraic C -varieties endowed with effective torus actions to the real setting. In particular, we focus on actions of quasi-split real tori, in which case we obtain a simpler presentation. [ABSTRACT FROM AUTHOR]

Let FF be a finite group and XX be a complex quasi-projective FF-variety. For r∈Nr\in {\mathbb{N}}, we consider the mixed Hodge-Deligne polynomials of quotients Xr/F{X}^{r}\hspace{-0.15em}\text{/}\hspace{-0.08em}F, where FF acts diagonally, and compute them for certain classes of varieties XX with simple mixed Hodge structures (MHSs). A particularly interesting case is when XX is the maximal torus of an affine reductive group GG, and FF is its Weyl group. As an application, we obtain explicit formulas for the Hodge-Deligne and EE-polynomials of (the distinguished component of) GG-character varieties of free abelian groups. In the cases G=GL(n,C)G=GL\left(n,{\mathbb{C}}\hspace{-0.1em}) and SL(n,C)SL\left(n,{\mathbb{C}}\hspace{-0.1em}), we get even more concrete expressions for these polynomials, using the combinatorics of partitions.

For a commutative ring R, we define the notions of deformed Picard algebroids and deformed twisted differential operators on a smooth, separated, locally of finite type R-scheme and prove these are in a natural bijection. We then define the pullback of a sheaf of twisted differential operators that reduces to the classical definition when R = ℂ. Finally, for modules over twisted differential operators, we prove a theorem for the descent under a locally trivial torsor. [ABSTRACT FROM AUTHOR]

We establish basic properties of a sheaf of graded algebras canonically associated to every relative affine scheme f : X → S endowed with an action of the additive group scheme G a , S over a base scheme or algebraic space S, which we call the (relative) Rees algebra of the G a , S -action. In the case of affine algebraic varieties defined over a field of characteristic zero, we establish further properties of the Rees algebra of a G a -action in terms of its associated locally nilpotent derivation. We give an algebro-geometric characterization of pairs consisting of an affine algebraic variety and a G a -action on it whose associated Rees algebras are finitely generated and provide an algorithm extending van den Essen's kernel algorithm for locally nilpotent derivations to compute generators of these Rees algebras. We illustrate these properties on several examples which played important historical roles in the development of the algebraic theory of locally nilpotent derivations and give applications to the construction of new families of affine threefolds with G a -actions. [ABSTRACT FROM AUTHOR]

Let F q be a finite field and E b : y 2 = x 3 + b be an ordinary (i.e., non-supersingular) elliptic curve (of j-invariant 0) such that b ∈ F q and q ≢ 1 (mod 27) . For example, these conditions are fulfilled for the curve BLS12-381 ( b = 4 ). It is a de facto standard in the real-world pairing-based cryptography at the moment. This article provides a new constant-time hash function H : { 0 , 1 } ∗ → E b (F q) indifferentiable from a random oracle. Its main advantage is the fact that H computes only one exponentiation in F q . In comparison, the previous fastest constant-time indifferentiable hash functions to E b (F q) compute two exponentiations in F q . In particular, applying H to the widely used BLS multi-signature with m different messages, the verifier should perform only m exponentiations rather than 2m ones during the hashing phase. [ABSTRACT FROM AUTHOR]