Affine group
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Ghosts in metric-affine higher order curvature gravity Open
We disprove the widespread belief that higher order curvature theories of gravity in the metric-affine formalism are generally ghost-free. This is clarified by considering a sub-class of theories constructed only with the Ricci tensor and …
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Affine inflation Open
Affine gravity, a gravity theory based on affine connection with no notion of metric, supports scalar field dynamics only if scalar fields have nonvanishing potential. The nonvanishing vacuum energy ensures that the cosmological constant i…
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Homogeneous affine surfaces: affine Killing vector fields and gradient Ricci solitons Open
The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature f…
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WILDER MCKAY CORRESPONDENCES Open
A conjectural generalization of the McKay correspondence in terms of stringy invariants to arbitrary characteristics, including the wild case, was recently formulated by the author in the case where the given finite group acts linearly on …
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On Representations of Affine Temperley--Lieb Algebras Open
We study the finite-dimensional simple modules, over an algebraically closed field, of the affine Temperley--Lieb algebra corresponding to the affine Weyl group of type $A$. These turn out to be closely related to the simple modules for a …
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Families of commuting automorphisms, and a characterization of the affine space Open
In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of auto…
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Homogeneous geodesics and g.o. manifolds Open
Let $M$ be a homogeneous pseudo-Riemannian manifold, affine manifold, or Finsler space. A homogeneous geodesic is an orbit of a $1$-parameter group of isometries, respectively, of affine diffeomorphisms. A homogeneous manifold is called a …
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Bianchi type cosmological models in f(T) tele-parallel gravity Open
Symmetry assumptions on the geometrical framework have provided successful mechanisms to develop physically meaningful solutions to many problems. In tele-parallel gravity, invariance of the frame and spin-connection under a group of motio…
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Maximal Newton Points and the Quantum Bruhat Graph Open
We discuss a surprising relationship between the partially ordered set of Newton points associated to an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the…
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On compact affine quaternionic curves and surfaces. Open
This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only on…
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Quantum Representation of Affine Weyl Groups and Associated Quantum Curves Open
We study a quantum (non-commutative) representation of the affine Weyl group mainly of type $E_8^{(1)}$, where the representation is given by birational actions on two variables $x$, $y$ with $q$-commutation relations. Using the tau variab…
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A natural linear equation in affine geometry: The affine quasi-Einstein Equation Open
We study the affine quasi-Einstein equation, a second-order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite dimensional and its dimension is a strongly projec…
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A Drinfeld type presentation of affine $\imath$quantum groups I: split ADE type Open
We establish a Drinfeld type new presentation for the $\imath$quantum groups arising from quantum symmetric pairs of split affine ADE type, which includes the $q$-Onsager algebra as the rank 1 case. This presentation takes a form which can…
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Mirror symmetry for extended affine Weyl groups Open
We give a uniform, Lie-theoretic mirror symmetry construction for the Frobenius manifolds defined by Dubrovin–Zhang in [21] on the orbit spaces of extended affine Weyl groups, including exceptional Dynkin types. The B-model mirror is given…
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Distances on a masure (affine ordered hovel) Open
A masure (a.k.a affine ordered hovel) I is a generalization of the Bruhat-Tits building that is associated to a split Kac-Moody group G over a non-archimedean local field. This is a union of affine spaces called apartments. When G is a red…
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Characterization of affine surfaces with a torus action by their automorphism groups Open
In this paper we prove that if two normal affine surfaces $S$ and $S'$ have isomorphic automorphism groups, then every connected algebraic group acting regularly and faithfully on $S$ acts also regularly and faithfully on $S'$. Moreover, i…
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Miura transformations for discrete Painlevé equations coming from the affine E8 Weyl group Open
We derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine Weyl group E8(1). By deautonomisation we obtain two hitherto unknown systems, one of which turns…
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Affine classification of $n-$curves Open
The classification of curves up to affine transformations in a finite dimensional space was studied by some different methods. In this paper, we obtain the exact formulas of affine invariants via the equivalence problem in view of Cartan’s…
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On a purely inseparable analogue of the Abhyankar conjecture for affine curves Open
Let $U$ be an affine smooth curve defined over an algebraically closed field of positive characteristic. The Abhyankar conjecture (proved by Raynaud and Harbater in 1994) describes the set of finite quotients of Grothendieck’s étale fundam…
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Characterizing smooth affine spherical varieties via the automorphism group Open
Let be a connected reductive algebraic group. We prove that for a quasi-affine -spherical variety the weight monoid is determined by the weights of its non-trivial -actions that are homogeneous with respect to a Borel subgroup of . As an a…
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A Tannakian Framework for<i>G</i>-displays and Rapoport–Zink Spaces Open
We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport–Zink functors associated to triples $(G,\{\mu \},[b]…
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Phase Retrieval From the Magnitudes of Affine Linear Measurements Open
In this paper, we consider the phase retrieval problem in which one aims to recover a signal from the magnitudes of affine measurements. Let $\{{\mathbf a}_j\}_{j=1}^m \subset {\mathbb H}^d$ and ${\mathbf b}=(b_1, \ldots, b_m)^\top\in{\mat…
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Duality cascades and affine Weyl groups Open
A bstract Brane configurations in a circle allow subsequent applications of the Hanany-Witten transitions, which are known as duality cascades. By studying the process of duality cascades corresponding to quantum curves with symmetries of …
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Characterization of affine surfaces with a torus action by their\n automorphism groups Open
In this paper we prove that if two normal affine surfaces $S$ and $S'$ have\nisomorphic automorphism groups, then every connected algebraic group acting\nregularly and faithfully on $S$ acts also regularly and faithfully on $S'$.\nMoreover…
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Complete algebraic vector fields on affine surfaces Open
Let [Formula: see text] be the subgroup of the group [Formula: see text] of holomorphic automorphisms of a normal affine algebraic surface [Formula: see text] generated by elements of flows associated with complete algebraic vector fields.…
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Flat affine symplectic Lie groups Open
We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symp…
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Perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian Open
For a reductive group over an algebraically closed field of characteristic $p > 0$ we construct the abelian category of perverse $\mathbb{F}_p$-sheaves on the affine Grassmannian that are equivariant with respect to the action of the posit…
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Affine surfaces and their Veech groups Open
We introduce a class of objects which we call 'affine surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in …
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Nilpotent orbits of height 2 and involutions in the affine Weyl group Open
Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B⊂G be a Borel subgroup. Then B acts with finitely many orbits on the variety N2⊂g of the nilpotent elements whos…
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Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group Open
We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category O is equivalent to the category of twisted …