Crystal bases for the quantum queer superalgebra

Dimitar Grantcharov; Ji Hye Jung; Seok-Jin Kang; Masaki Kashiwara; Myungho Kim

Displaying similar documents to “Crystal bases for the quantum queer superalgebra”

Exponentiations over the quantum algebra $U_{q} (s l_{2} (ℂ))$

Sonia L’Innocente, Françoise Point, Carlo Toffalori (2013)

Confluentes Mathematici

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We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra $U_{q} (s l_{2} (ℂ))$ . We discuss two cases, according to whether the parameter $q$ is a root of unity. We show that the universal enveloping algebra of $s l_{2} (ℂ)$ embeds in a non-principal ultraproduct of $U_{q} (s l_{2} (ℂ))$ , where $q$ varies over the primitive roots of unity.

Right coideal subalgebras of $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$

V. K. Kharchenko (2011)

Journal of the European Mathematical Society

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We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group $U_{q}^{+} ({𝔰𝔬}_{2 n + 1})$ provided that $q$ is not a root of 1. If $q$ has a finite multiplicative order $t > 4$ ; this classification remains valid for homogeneous right coideal subalgebras of the Frobenius–Lusztig kernel $u_{q}^{+} ({𝔰𝔬}_{2 n + 1})$ . In particular, the total number of right coideal subalgebras that contain the coradical equals $(2 n)!!$ ; the order of the Weyl group defined by the root system of type $B_{n}$ .

Effective Hamiltonians and Quantum States

Lawrence C. Evans (2000-2001)

Séminaire Équations aux dérivées partielles

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We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function $u$ solving the eikonal equation aėȧnd a probability measure $σ$ solving a related transport equation. We present some elementary formal identities relating certain quantum states $ψ$ and $u, σ$ . We show also how...

Addendum to "Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces" (Colloq. Math. 127 (2012), 105-109)

Aydin Sh. Shukurov (2014)

Colloquium Mathematicae

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It is well known that if φ(t) ≡ t, then the system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is a basis in some Lebesgue space $L_{p}$ . The aim of this short note is to show that the answer to this question is negative.

Quantum expanders and geometry of operator spaces

Gilles Pisier (2014)

Journal of the European Mathematical Society

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We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of $M_{N}$ -spaces needed to represent (up to a constant $C > 1$ ) the $M_{N}$ -version of the $n$ -dimensional operator Hilbert space $O H_{n}$ as a direct sum of copies of $M_{N}$ . We show that, when $C$ is close to 1, this multiplicity grows...

Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces

Aydin Sh. Shukurov (2012)

Colloquium Mathematicae

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A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_{p}$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_{p}$ (1 ≤ p < +∞).

Three-space problems and bounded approximation properties

Wolfgang Lusky (2003)

Studia Mathematica

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Let ${R ₙ}_{n = 1}^{\infty}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_{p}$ -space, then both X and A have bases. We apply these results to show that the spaces $C_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset C ()$ and $L_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset L ₁ ()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

On a cubic Hecke algebra associated with the quantum group $U_{q} (2)$

Janusz Wysoczański (2010)

Banach Center Publications

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We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_{q} (2)$ , which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_{j} : = I_{j} \otimes α \otimes I_{n - 2 - j}$ on ${(ℂ ³)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra $ℋ_{q, n} (2)$ associated with the quantum group $U_{q} (2)$ . The purpose of this note is to present the construction.

On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Xuwen Chen, Justin Holmer (2016)

Journal of the European Mathematical Society

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We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$ -particle Schrödinger equation. We assume the pair interaction is $N^{3 β - 1} V (B^{β})$ . For the interaction parameter $β \in (0, 2 / 3)$ , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N \to \infty$ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $β \in (0, 3 / 5)$ in [28]. This allows, in the case $β \in (0, 3 / 5)$ , for the application of the Klainerman–Machedon...

Hall algebras of two equivalent extriangulated categories

Shiquan Ruan, Li Wang, Haicheng Zhang (2024)

Czechoslovak Mathematical Journal

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For any positive integer $n$ , let $A_{n}$ be a linearly oriented quiver of type $A$ with $n$ vertices. It is well-known that the quotient of an exact category by projective-injectives is an extriangulated category. We show that there exists an extriangulated equivalence between the extriangulated categories $ℳ_{n + 1}$ and $ℱ_{n}$ , where $ℳ_{n + 1}$ and $ℱ_{n}$ are the two extriangulated categories corresponding to the representation category of $A_{n + 1}$ and the morphism category of projective representations of $A_{n}$ , respectively. As a...

The basic construction from the conditional expectation on the quantum double of a finite group

Qiaoling Xin, Lining Jiang, Zhenhua Ma (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $H$ a subgroup. Denote by $D (G; H)$ (or $D (G)$ ) the crossed product of $C (G)$ and $ℂ H$ (or $ℂ G$ ) with respect to the adjoint action of the latter on the former. Consider the algebra $〈 D (G), e 〉$ generated by $D (G)$ and $e$ , where we regard $E$ as an idempotent operator $e$ on $D (G)$ for a certain conditional expectation $E$ of $D (G)$ onto $D (G; H)$ . Let us call $〈 D (G), e 〉$ the basic construction from the conditional expectation $E : D (G) \to D (G; H)$ . The paper constructs a crossed product algebra $C (G / H \times G) ⋊ ℂ G$ , and proves that there is an algebra isomorphism between...

Normal number constructions for Cantor series with slowly growing bases

Dylan Airey, Bill Mance, Joseph Vandehey (2016)

Czechoslovak Mathematical Journal

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Let $Q = {(q_{n})}_{n = 1}^{\infty}$ be a sequence of bases with $q_{i} \geq 2$ . In the case when the $q_{i}$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$ -Cantor series expansion is both $Q$ -normal and $Q$ -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$ , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

Explicit construction of a unitary double product integral

R. L. Hudson, Paul Jones (2011)

Banach Center Publications

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In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral $\prod^{\to \to} (1 + d r)$ with generator $d r = λ (d A^{†} \otimes d A - d A \otimes d A^{†})$ . The method of construction is to approximate the product integral by a discrete double product $\prod_{(j, k) \in ℕ_{m} \times ℕ ₙ}^{\to \to} Γ (R_{m, n}^{(j, k)}) = Γ (\prod_{(j, k) \in ℕ_{m} \times ℕ ₙ}^{\to \to} (R_{m, n}^{(j, k)}))$ of second quantised rotations $R_{m, n}^{(j, k)}$ in different planes using the embedding of $ℂ^{m} \oplus ℂ ⁿ$ into L²(ℝ) ⊕ L²(ℝ) in which the standard orthonormal bases of $ℂ^{m}$ and ℂⁿ are mapped to the orthonormal sets consisting of normalised indicator...

On category $𝒪$ for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

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We study equivalences for category $𝒪_{p}$ of the rational Cherednik algebras $𝐇_{p}$ of type $G_{ℓ} (n) = {(μ_{ℓ})}^{n} ⋊ 𝔖_{n}$ : a highest weight equivalence between $𝒪_{p}$ and $𝒪_{σ (p)}$ for $σ \in 𝔖_{ℓ}$ and an action of $𝔖_{ℓ}$ on an explicit non-empty Zariski open set of parameters $p$ ; a derived equivalence between $𝒪_{p}$ and $𝒪_{p^{'}}$ whenever $p$ and $p^{'}$ have integral difference; a highest weight equivalence between $𝒪_{p}$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$ . As a consequence, we confirm special cases...

A basis of Zₘ

Min Tang, Yong-Gao Chen (2006)

Colloquium Mathematicae

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Let $σ_{A} (n) = | (a, a^{'}) \in A ² : a + a^{'} = n |$ , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, $σ_{A} (n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which $σ_{A} (n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and $σ_{A} (n ̅) \leq 768$ for all n̅ ∈ Zₘ.

Matrix subspaces of L₁

Gideon Schechtman (2013)

Studia Mathematica

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If $E = e_{i}$ and $F = f_{i}$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i, j}$ with norm $| | a_{i, j} {| |}_{E (F)} = | | \sum_{k} | | \sum_{l} a_{k, l} f_{l} | | e_{k} | |$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

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We show that the dimer model on a bipartite graph $Γ$ on a torus gives rise to a quantum integrable system of special type, which we call a. The phase space of the classical system contains, as an open dense subset, the moduli space $Ł_{Γ}$ of line bundles with connections on the graph $Γ$ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs $Γ_{1}$ and $Γ_{2}$ areif the Newton polygons of the corresponding partition functions coincide up to translation....

General Haar systems and greedy approximation

Anna Kamont (2001)

Studia Mathematica

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We show that each general Haar system is permutatively equivalent in $L^{p} ([0, 1])$ , 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^{p} ([0, 1])$ , 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^{p} ({[0, 1]}^{d})$ , 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases...

Some results on $(n, d)$ -injective modules, $(n, d)$ -flat modules and $n$ -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $n, d$ be two non-negative integers. A left $R$ -module $M$ is called $(n, d)$ -injective, if ${Ext}^{d + 1} (N, M) = 0$ for every $n$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called $(n, d)$ -flat, if ${Tor}_{d + 1} (V, N) = 0$ for every $n$ -presented left $R$ -module $N$ . A left $R$ -module $M$ is called weakly $n$ - $F P$ -injective, if ${Ext}^{n} (N, M) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called weakly $n$ -flat, if ${Tor}_{n} (V, N) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . In this paper, we give some characterizations and properties of $(n, d)$ -injective modules and $(n, d)$ -flat modules in...

On almost complex structures from classical linear connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $ℳ f_{m}$ be the category of $m$ -dimensional manifolds and local diffeomorphisms and let $T$ be the tangent functor on $ℳ f_{m}$ . Let $𝒱$ be the category of real vector spaces and linear maps and let $𝒱_{m}$ be the category of $m$ -dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F : 𝒱_{m} \to 𝒱$ admitting $ℳ f_{m}$ -natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$ -dimensional manifolds $M$ into almost complex structures $\tilde{J} (\nabla)$ on $F (T) M = ⋃_{x \in M} F (T_{x} M)$ .