Results 1  10
of
108
On the classification of finitedimensional pointed Hopf algebras
, 2006
"... We classify finitedimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are onedimensional, and whose group of grouplike elements G(A) is abelian such that all prime divisors of the order of G(A) are> 7. Since these Hopf algebras turn out to be def ..."
Abstract

Cited by 115 (15 self)
 Add to MetaCart
(Show Context)
We classify finitedimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are onedimensional, and whose group of grouplike elements G(A) is abelian such that all prime divisors of the order of G(A) are> 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.
Instability in cosmological topologically massive gravity at the chiral point
, 2008
"... Abstract: We demonstrate that cosmological topologically massive gravity at the chiral point exhibits a negative energy bulk mode that grows linearly in time. Unless there are physical reasons to discard this mode, this theory is unstable. To address this issue we prove that the mode is not pure gau ..."
Abstract

Cited by 102 (19 self)
 Add to MetaCart
(Show Context)
Abstract: We demonstrate that cosmological topologically massive gravity at the chiral point exhibits a negative energy bulk mode that grows linearly in time. Unless there are physical reasons to discard this mode, this theory is unstable. To address this issue we prove that the mode is not pure gauge and that its negative energy is timeindependent and finite. The isometry generators L0 and ¯ L0 have nonunitary matrix representations like in a logarithmic CFT. While the new mode obeys boundary conditions that are slightly weaker than the ones by Brown and Henneaux, its falloff behavior is compatible with spacetime being asymptotically AdS3. We employ holographic renormalization to show that the variational principle is welldefined. The corresponding Brown–York stress tensor coincides with that of global AdS3. Finally we address possibilities to eliminate the instability and prospects for chiral gravity.
Logarithmic minimal models
"... Working in the dense loop representation, we use the planar TemperleyLieb algebra to build integrable lattice models called logarithmic minimal models LM(p,p ′). Specifically, we construct YangBaxter integrable TemperleyLieb models on the strip acting on link states and consider their associated ..."
Abstract

Cited by 88 (25 self)
 Add to MetaCart
(Show Context)
Working in the dense loop representation, we use the planar TemperleyLieb algebra to build integrable lattice models called logarithmic minimal models LM(p,p ′). Specifically, we construct YangBaxter integrable TemperleyLieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the TemperleyLieb algebra are inherently nonlocal and not (timereversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1 − 6(p−p ′ ) 2 pp ′ where p,p ′ = 1,2,... are coprime. The first few members of the principal series LM(m,m + 1) are critical dense polymers (m = 1, c=−2), critical percolation (m = 2, c=0) and logarithmic Ising model (m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights, r,s = 1,2,.... The associated conformal partition functions are given in terms of Virasoro characters of highestweight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion. ∆r,s = ((m+1)r−ms)2 −1
Modular invariance of vertex operator algebra satisfying C2cofiniteness
 Duke Math. J
"... We investigate trace functions of modules for vertex operator algebras satisfying C2cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Zh]: A(V) is semisimple and C2cofiniteness. We show that C2cofiniteness is enough to prove a modular invariance property. For exampl ..."
Abstract

Cited by 76 (5 self)
 Add to MetaCart
(Show Context)
We investigate trace functions of modules for vertex operator algebras satisfying C2cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Zh]: A(V) is semisimple and C2cofiniteness. We show that C2cofiniteness is enough to prove a modular invariance property. For example, if a VOA V = ⊕ ∞ m=0 Vm is C2cofinite, then the space spanned by generalized characters (pseudotrace functions of the vacuum element) of Vmodules is a finite dimensional SL2(Z)invariant space and the central charge and conformal weights are all rational numbers. Namely we show that C2cofiniteness implies “rational conformal field theory” in a sense as expected in [GN]. Viewing a trace map as one of symmetric linear maps and using a result of symmetric algebras, we introduce “pseudotraces ” and pseudotrace functions and then show that the space spanned by such pseudotrace functions has a modular invariance property. We also show that C2cofiniteness is equivalent to the condition that every weak module is an Ngraded weak module which is a direct sum of generalized eigenspaces of L(0). 1
Finite tensor categories
 Moscow Math. Journal
"... These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We wil ..."
Abstract

Cited by 75 (12 self)
 Add to MetaCart
(Show Context)
These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We will also assume the basics of the theory of abelian categories (for a more detailed treatment see the book [F]). If C is a category, the notation X ∈ C will mean that X is an object of C, and the set of morphisms between X, Y ∈ C will be denoted by Hom(X, Y). Throughout the notes, for simplicity we will assume that the ground field k is algebraically closed unless otherwise specified, even though in many cases this assumption will not be needed. 1. Monoidal categories 1.1. The definition of a monoidal category. A good way of thinking
A Logarithmic Generalization of Tensor Product Theory for Modules for a Vertex Operator Algebra
, 2005
"... ..."
Wextended fusion algebra of critical percolation
 J. Phys. A: Math. Theor
"... We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasirational in ..."
Abstract

Cited by 37 (13 self)
 Add to MetaCart
(Show Context)
We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasirational in the sense that the fusion of a finite number of these representations decomposes into a finite direct sum of these representations. The fusion rules are commutative, associative and exhibit an sℓ(2) structure. They involve representations which we call Kac representations of which some are reducible yet indecomposable representations of rank 1. In particular, the identity of the fusion algebra is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the recent results of EberleFlohr and ReadSaleur. Notably, in agreement with EberleFlohr, we find the appearance of indecomposable representations of rank 3. Our fusion rules are supported by extensive numerical studies of an integrable lattice model of critical percolation. Details of our lattice findings and numerical results will be presented elsewhere.
The logarithmic triplet theory with boundary
, 2006
"... The boundary theory for the c = −2 triplet model is investigated in detail. In particular, we show that there are four different boundary conditions that preserve the triplet algebra, and check the consistency of the corresponding boundary operators by constructing their OPE coefficients explicitly. ..."
Abstract

Cited by 35 (4 self)
 Add to MetaCart
The boundary theory for the c = −2 triplet model is investigated in detail. In particular, we show that there are four different boundary conditions that preserve the triplet algebra, and check the consistency of the corresponding boundary operators by constructing their OPE coefficients explicitly. We also compute the correlation functions of two bulk fields in the presence of a boundary, and verify that they are consistent with factorisation.