Transformation Groups最新文献

We use combinatorics of qq-characters to study extensions of deformed W-algebras. We describe additional currents and part of the relations in the cases of (mathfrak {gl}(n|m)) and (mathfrak {osp}(2|2n)).

We study complex solvmanifolds (Gamma backslash G) with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of G. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form (psi ) canonically associated to ((mathfrak {g},J)), where (mathfrak {g}) is the Lie algebra of G, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of (psi ), for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold ((M^{4n},{J_1,J_2,J_3})) such that the canonical bundle of ((M,J_{alpha })) is trivial only for (alpha =1), so that M is not an ({text {SL}}(n,mathbb {H}))-manifold.

The quantum max-flow is a linear algebraic version of the classical max-flow of a graph, used in quantum many-body physics to quantify the maximal possible entanglement between two regions of a tensor network state. In this work, we calculate the quantum max-flow exactly in the case of the bridge graph. The result is achieved by drawing connections to the theory of prehomogenous tensor spaces and the representation theory of quivers. Further, we highlight relations to invariant theory and to algebraic statistics.

We show that each connected group scheme of finite type over an arbitrary ground field is isomorphic to the component of the identity inside the automorphism group scheme of some projective, geometrically integral scheme. The main ingredients are embeddings into smooth group schemes, equivariant completions, blow-ups of orbit closures, Fitting ideals for Kähler differentials, and Blanchard’s Lemma.

We construct indefinite Einstein solvmanifolds that are standard, but not of pseudo-Iwasawa type. Thus, the underlying Lie algebras take the form (mathfrak {g}rtimes _Dmathbb {R}), where (mathfrak {g}) is a nilpotent Lie algebra and D is a nonsymmetric derivation. Considering nonsymmetric derivations has the consequence that (mathfrak {g}) is not a nilsoliton, but satisfies a more general condition. Our construction is based on the notion of nondiagonal triple on a nice diagram. We present an algorithm to classify nondiagonal triples and the associated Einstein metrics. With the use of a computer, we obtain all solutions up to dimension 5, and all solutions in dimension (le 9) that satisfy an additional technical restriction. By comparing curvatures, we show that the Einstein solvmanifolds of dimension (le 5) that we obtain by our construction are not isometric to a standard extension of a nilsoliton.

Let (R) be a principal ideal domain. In this paper we investigate generic coordinate systems of the polynomial (R)-algebra (A=R^{[2]}). As an application we prove that for every locally nilpotent (R)-derivation (xi ) of (A) the automorphism (exp (xi )) is 1-stably tame in an appropriate coordinate system of (A). This shows that the well-known result due to Smith, asserting that the Nagata automorphism is 1-stably tame, actually holds in full generality.

The extended T-systems are a number of relations in the Grothendieck ring of the category of finite-dimensional modules over the quantum affine algebras of types (A_n^{(1)}) and (B_n^{(1)}), introduced by Mukhin and Young as a generalization of the T-systems. In this paper we establish the extended T-systems for more general modules, which are constructed from an arbitrary strong duality datum of type A. Our approach does not use the theory of q-characters, and so also provides a new proof to the original Mukhin–Young’s extended T-systems.

We give an affine analogue of the Robinson-Schensted-Knuth (RSK) correspondence, which generalizes the affine Robinson-Schensted correspondence by Chmutov-Pylyavskyy-Yudovina. The affine RSK map sends a generalized affine permutation of period (m, n) to a pair of tableaux (P, Q) of the same shape, where P belongs to a tensor product of level one perfect Kirillov-Reshetikhin crystals of type (A_{m-1}^{(1)}), and Q belongs to a crystal of extremal weight module of type (A_{n-1}^{(1)}) when (m,ngeqslant 2). We consider two affine crystal structures of types (A_{m-1}^{(1)}) and (A_{n-1}^{(1)}) on the set of generalized affine permutations, and show that the affine RSK map preserves the crystal equivalence. We also give a dual affine Robison-Schensted-Knuth correspondence.

Let (G_n) be an inner form of a general linear group over a non-Archimedean local field. We fix an arbitrary irreducible representation (sigma ) of (G_n). Building on the work of Lapid-Mínguez on the irreducibility of parabolic inductions, we show how to define a full subcategory of the category of smooth representations of some (G_m), on which the parabolic induction functor (tau mapsto tau times sigma ) is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.

We establish classical and categorical Howe dualities between the Lie superalgebras (mathfrak {p}(m)) and (mathfrak {p}(n)), for (m,n ge 1). We also describe a presentation via generators and relations as well as a Kostant (mathbb {Z})-form for the universal enveloping superalgebra (U(mathfrak {p}(m))).