scholarly journals Highest-weight vectors in tensor products of Verma modules for U_q(sl_2)

2018 ◽  
Vol 36 (4) ◽  
pp. 107-119 ◽  
Author(s):  
Elizabeth Creath ◽  
Dijana Jakelic
Keyword(s):  
Tensor Product ◽  
Tensor Products ◽  
Alternative Proof ◽  
Verma Modules ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Structure Constants ◽  
Quantized Universal Enveloping Algebra

We obtain an explicit basis for the subspace spanned by highest-weight vectors in a tensor product of two highest-weight modules for the quantized universal enveloping algebra of sl_2. The structure constants provide a generalization of the Clebsh-Gordan coefficients. As a byproduct, we give an alternative proof for the decomposition of these tensor products as direct sums of indecomposable modules and supply generators for all highest weight summands.

scholarly journals Canonical bases and higher representation theory

2014 ◽  
Vol 151 (1) ◽  
pp. 121-166 ◽  
Author(s):  
Ben Webster
Keyword(s):  
General Theory ◽  
Representation Theory ◽  
Finite Type ◽  
Canonical Basis ◽  
Canonical Bases ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Natural Categories ◽  
Grothendieck Groups ◽  
Quantized Universal Enveloping Algebra

AbstractThis paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. As an application, we show that Lusztig’s canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is of finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest- and highest-weight integrable representations. This generalizes past work of the author’s in the highest-weight case.

Tensor Products of Holomorphic Discrete Series Representations

1979 ◽  
Vol 31 (4) ◽  
pp. 836-844 ◽  
Author(s):  
Joe Repka
Keyword(s):  
Discrete Series ◽  
Tensor Products ◽  
Holomorphic Functions ◽  
Verma Modules ◽  
Holomorphic Discrete Series ◽  
Spaces Of Holomorphic Functions ◽  
Highest Weight ◽  
Generalized Verma Modules ◽  
Series Representations ◽  
Discrete Series Representations

We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions.For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors.The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.

scholarly journals Some stability properties of c0-saturated spaces

1995 ◽  
Vol 118 (2) ◽  
pp. 287-301 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Tensor Products ◽  
General Version ◽  
Direct Sums ◽  
Stability Properties ◽  
Infinite Dimensional ◽  
The Stability

AbstractA Banach space is c0-saturated if all of its closed infinite-dimensional subspaces contain an isomorph of c0. In this article, we study the stability of this property under the formation of direct sums and tensor products. Some of the results are: (1) a slightly more general version of the fact that c0-sums of c0-saturated spaces are c0-saturated; (2) C(K, E) is c0-saturated if both C(K) and E are; (3) the tensor product is c0-saturated, where JH is the James Hagler space

scholarly journals On the structures of hive algebras and tensor product algebras for general linear groups of low rank

2019 ◽  
Vol 29 (07) ◽  
pp. 1193-1218
Author(s):  
Donggyun Kim ◽  
Sangjib Kim ◽  
Euisung Park
Keyword(s):  
Tensor Product ◽  
Irreducible Polynomial ◽  
Tensor Products ◽  
Low Rank ◽  
General Linear ◽  
Generators And Relations ◽  
General Linear Groups ◽  
Highest Weight ◽  
Polynomial Representations ◽  
Product Algebra

The tensor product algebra [Formula: see text] for the complex general linear group [Formula: see text], introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of [Formula: see text]. Using the hive model for the Littlewood–Richardson (LR) coefficients, we provide a finite presentation of the algebra [Formula: see text] for [Formula: see text] in terms of generators and relations, thereby giving a description of highest weight vectors of irreducible representations in the tensor products. We also compute the generating function of certain sums of LR coefficients.

Quantum Deformations of Simple Lie Algebras

1997 ◽  
Vol 40 (2) ◽  
pp. 143-148 ◽  
Author(s):  
Murray Bremner
Keyword(s):  
Lie Algebras ◽  
Simple Complex ◽  
Product Decomposition ◽  
3 Dimensional ◽  
Enveloping Algebra ◽  
Highest Weight ◽  
Quantum Deformations ◽  
Quantized Enveloping Algebra ◽  
Structure Constants ◽  
Special Case

AbstractIt is shown that every simple complex Lie algebra 𝔤 admits a 1-parameter family 𝔤q of deformations outside the category of Lie algebras. These deformations are derived from a tensor product decomposition for Uq(𝔤)-modules; here Uq(𝔤) is the quantized enveloping algebra of 𝔤. From this it follows that the multiplication on 𝔤q is Uq(𝔤)-invariant. In the special case 𝔤 = (2), the structure constants for the deformation 𝔤 (2)q are obtained from the quantum Clebsch-Gordan formula applied to V(2)q ⊗ V(2)q; here V(2)q is the simple 3-dimensional Uq(𝔤(2))-module of highest weight q2.

scholarly journals GENERALIZED GAUSS DECOMPOSITION OF TRIGONOMETRIC R-MATRICES

1995 ◽  
Vol 10 (19) ◽  
pp. 1375-1392 ◽  
Author(s):  
S.M. KHOROSHKIN ◽  
A.A. STOLIN ◽  
V.N. TOLSTOY
Keyword(s):  
Tensor Product ◽  
Bilinear Form ◽  
Central Charge ◽  
Irreducible Representations ◽  
Verma Modules ◽  
R Matrix ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Gauss Decomposition ◽  
Affine Algebras

The general formula for the universal R-matrix for quantized nontwisted affine algebras, obtained by the first and third authors, is applied to zero central charge, highest weight modules of the quantized affine algebras. It is shown how the universal R-matrix produces the Gauss decomposition of trigonometric R-matrix in tensor product of these modules. In particular, [Formula: see text] current realization of the universal R-matrix is presented. It gives a new universal presentation for the trigonometric R-matrix with a parameter in tensor product of Uq(sl2)-Verma modules. Detailed analysis of a scalar factor arising in finite-dimensional representations of the universal R-matrix for any Uq(ĝ) is given. We interpret this scalar factor as a multiplicative bilinear form on highest weight polynomials of irreducible representations and express this form in terms of infinite q-shifted factorials.

scholarly journals Fusion frames and g-frames in tensor product and direct sum of Hilbert spaces

2012 ◽  
Vol 6 (2) ◽  
pp. 287-303
Author(s):  
Amir Khosravi ◽  
Azandaryani Mirzaee
Keyword(s):  
Tensor Product ◽  
Finite Number ◽  
Hilbert Spaces ◽  
Product Space ◽  
Tensor Products ◽  
Riesz Bases ◽  
Direct Sums ◽  
Fusion Frames ◽  
Direct Sum ◽  
Tensor Product Space

In this paper we study fusion frames and g-frames for the tensor products and direct sums of Hilbert spaces. We show that the tensor product of a finite number of g-frames (resp. fusion frames, g-Riesz bases) is a g-frame (resp. fusion frame, g-Riesz basis) for the tensor product space and vice versa. Moreover we obtain some important results in tensor products and direct sums of g-frames, fusion frames, resolutions of the identity and duals.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

scholarly journals BMS modular diaries: torus one-point function

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Arjun Bagchi ◽  
Poulami Nandi ◽  
Amartya Saha ◽  
Zodinmawia
Keyword(s):  
Cosmological Solution ◽  
Three Dimensions ◽  
Field Theories ◽  
Point Function ◽  
Asymptotic Structure ◽  
Highest Weight ◽  
Structure Constants ◽  
Flat Spacetimes ◽  
Geodesic Approximation ◽  
Highest Weight Representation

Abstract Two dimensional field theories invariant under the Bondi-Metzner-Sachs (BMS) group are conjectured to be dual to asymptotically flat spacetimes in three dimensions. In this paper, we continue our investigations of the modular properties of these field theories. In particular, we focus on the BMS torus one-point function. We use two different methods to arrive at expressions for asymptotic structure constants for general states in the theory utilising modular properties of the torus one-point function. We then concentrate on the BMS highest weight representation, and derive a host of new results, the most important of which is the BMS torus block. In a particular limit of large weights, we derive the leading and sub-leading pieces of the BMS torus block, which we then use to rederive an expression for the asymptotic structure constants for BMS primaries. Finally, we perform a bulk computation of a probe scalar in the background of a flatspace cosmological solution based on the geodesic approximation to reproduce our field theoretic results.

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