Structure of finite dimensional exact estimation algebra on state dimension 3 and linear rank 2

Author(s):  
Xiaopei Jiao ◽  
Stephen S.-T. Yau
10.37236/933 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Gregg Musiker ◽  
James Propp

Fomin and Zelevinsky show that a certain two-parameter family of rational recurrence relations, here called the $(b,c)$ family, possesses the Laurentness property: for all $b,c$, each term of the $(b,c)$ sequence can be expressed as a Laurent polynomial in the two initial terms. In the case where the positive integers $b,c$ satisfy $bc < 4$, the recurrence is related to the root systems of finite-dimensional rank $2$ Lie algebras; when $bc>4$, the recurrence is related to Kac-Moody rank $2$ Lie algebras of general type. Here we investigate the borderline cases $bc=4$, corresponding to Kac-Moody Lie algebras of affine type. In these cases, we show that the Laurent polynomials arising from the recurence can be viewed as generating functions that enumerate the perfect matchings of certain graphs. By providing combinatorial interpretations of the individual coefficients of these Laurent polynomials, we establish their positivity.


1993 ◽  
Vol 05 (02) ◽  
pp. 345-361 ◽  
Author(s):  
J. R. LINKS ◽  
M. D. GOULD ◽  
R. B. ZHANG

Unlike the quantum group case, it is shown that the braid generator σ is not always diagonalizable on V ⊗ V, V an irreducible module for a quantum supergroup. Nevertheless a generalization of the Reshetikhin form of the braid generator, obtained previously for quantum groups, is determined corresponding to every finite dimensional standard cyclic module V of a quantum supergroup. This result is applied to obtain a general closed formula for link polynomials arising from standard cyclic modules of a quantum supergroup belonging to a certain class. As explicit examples we determine link polynomials corresponding to the rank 2 symmetric tensor representation of Uq [gl(m|m)] and the defining representation of Uq [osp(2n|2n)].


2014 ◽  
Vol 14 (15&16) ◽  
pp. 1308-1337
Author(s):  
Daniel Cariello

This paper is devoted to the study of the separability problem in the field of Quantum information theory. We focus on the bipartite finite dimensional case and on two types of matrices: SPC and PPT matrices (see definitions 32 and 33). We prove that many results hold for both types. If these matrices have specific Hermitian Schmidt decompositions then they are separable in a very strong sense (see theorem 38 and corollary 39). We prove that both types have what we call \textbf{split decompositions} (see theorems 41 and 42). We also define the notion of weakly irreducible matrix (see definition 43), based on the concept of irreducible state defined recently in \cite{chen1}, \cite{chen} and \cite{chen2}.}{These split decomposition theorems imply that every SPC $($PPT$)$ matrix can be decomposed into a sum of $s+1$ SPC $($PPT$)$ matrices of which the first $s$ are weakly irreducible, by theorem 48, and the last one has a further split decomposition of lower tensor rank, by corollary 49. Thus the SPC $($PPT$)$ matrix is decomposed in a finite number of steps into a sum of weakly irreducible matrices. Different components of this sum have support on orthogonal local Hilbert spaces, therefore the matrix is separable if and only if each component is separable. This reduces the separability problem for SPC $($PPT$)$ matrices to the case of weakly irreducible SPC $($PPT$)$ matrices. We also provide a complete description of weakly irreducible matrices of both types (see theorem 46).}{Using the fact that every positive semidefinite Hermitian matrix with tensor rank 2 is separable (see theorem 58), we found sharp inequalites providing separability for both types (see theorems 61 and 62).


2008 ◽  
Vol 60 (4) ◽  
pp. 923-957 ◽  
Author(s):  
F. Okoh ◽  
F. Zorzitto

AbstractThe Kronecker modules , where m is a positive integer, h is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X over an algebraically closed field K, aremodels for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial f in K(X)[Y]. When the endomorphism algebra of is commutative and non-trivial, the regulator f must be quadratic in Y. If f has one repeated root in K(X), the endomorphismalgebra is the trivial extension for some vector space S. If f has distinct roots in K(X), then the endomorphisms forma structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End that are domains have zero radical. In addition, each semi-local End must be either a trivial extension or the product K × K.


2018 ◽  
Vol 29 (05) ◽  
pp. 1850034 ◽  
Author(s):  
Andrew Schopieray

There is a long-standing belief that the modular tensor categories [Formula: see text], for [Formula: see text] and finite-dimensional simple complex Lie algebras [Formula: see text], contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels [Formula: see text]. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture when [Formula: see text] has rank 2, contributing proofs and explicit bounds when [Formula: see text] is of type [Formula: see text] or [Formula: see text], adding to the previously known positive results for types [Formula: see text] and [Formula: see text].


2017 ◽  
Vol 24 (1) ◽  
pp. 15-34 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Iván Angiono ◽  
Fiorela Rossi Bertone

Author(s):  
Yuri Bahturin ◽  
Abdallah Shihadeh

In this paper, we explore the possibility of endowing simple infinite-dimensional [Formula: see text]-modules by the structure of graded modules. The gradings on the finite-dimensional simple modules over simple Lie algebras have been studied in 7, 8.


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