Pentagon Equation Research Articles

SL(2)-Solution of the Pentagon Equation and Invariants of Three-Dimensional Manifolds

We construct an algebraic complex corresponding to a triangulation of a three-manifold starting with a classical solution of the pentagon equation, constructed earlier by the author and Martyushev and related to the flat geometry, which is invariant under the group SL(2). If this complex is acyclic (which is confirmed by examples), we can use it to construct an invariant of the manifold.

Higher-dimensional Mac Lane's pentagon and Zamolodchikov equations

An important ingredient of Mac Lane's coherence theorem for monoidal categories is Mac Lane's pentagon, a diagram whose commutativity is needed so that “all diagrams commute”. This paper gives a higher-dimensional generalization of Mac Lane's pentagon: a 6-dimensional diagram whose commutativity is needed in order for all diagrams in somewhat weak teisi to commute. Looping twice gives a 4-dimensional diagram in somewhat weak braided teisi, of which five 3-dimensional edges can be interpreted as proofs of five different Zamolodchikov equations in braided monoidal 2-categories. Hence higher-dimensional Mac Lane's pentagon expresses the relations between these proofs concisely.

Distinguishing Three-Dimensional Lens Spaces L(7,1) and L(7,2) by Means of Classical Pentagon Equation

We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7, 1) and L(7, 2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e. algebraic relation corresponding to a “2 tetrahedra → 3 tetrahedra” local re-building of a manifold triangulation. This solution, found earlier by one of the authors, is expressed in terms of metric characteristics of Euclidean tetrahedra.

PENTAGON EQUATION AND MATRIX BIALGEBRAS

We classify coproducts on matrix algebra in terms of solutions to some modification of pentagon equation. The construction of Baaj and Skandalis describing finite dimensional unitary solutions of the pentagon equation is extended to the non-unitary case. We establish the relation between Hopf-Galois algebras and solutions to the modified pentagon equation.

Invariants of PL Manifolds from Metrized Simplicial Complexes. Three-Dimensional Case

An invariant of three-dimensional orientable manifolds is built on the base of a solution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.

A CLASSICAL SOLUTION OF THE PENTAGON EQUATION RELATED TO THE GROUP SL(2)

A new solution of the pentagon equation related to the flat geometry is obtained; it is invariant under the action of the group SL(2).

QUANTUM EXPONENTIAL FUNCTION

A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that the deformation parameter q2=e-iℏ. The first variable r runs over non-zero real numbers; the range of the second one depends on the sign of r: ϱ=0 for r>0 and ϱ=±1 for r<0. After the holomorphic continuation the function satisfies the functional equation [Formula: see text] The name "exponential function" is justified by the formula: [Formula: see text] where R, S are selfadjoint operators satisfying certain commutation relations and [R+S] is a selfadjoint extension of the sum R+S determined by operators ρ and σ appearing in the formula. This formula will be used in a forthcoming paper to construct a unitary operator W satisfying the pentagonal equation of Baaj and Skandalis.

The Pentagon equation and mapping-class groups of punctured surfaces

In the quantum Teichmuller theory, the mapping-class groups of punctured surfaces are represented projectively based on Penner coordinates. Algebraically, the representation is based on the pentagon equation together with pair of additional relations. Two more examples of solutions of these equations are connected with matrix (or operator) generalizations of the Rogers dilogarithm. The corresponding central charges are rational. It is possible that this system of equations admits many different solutions.

Finite-dimensional analogues of the strings↔t duality and the pentagon equation

We put forward one of the forms of functional pentagon equation (FPE), known from the theory of integrable models, as an algebraic explanation to the phenomenon known in physics as s<->t duality. We present two simple geometrical examples of FPE solutions, one of them yielding in a particular case the well-known Veneziano expression for 4-particle amplitude. Finally, we interpret our solutions of FPE in terms of relations in Lie groups.

The hopf modules category and the hopf equation

Let (A,△) be a Hopf-von Neumann algebra and R be the unitary fundamental operator on A defined by Takesaki in [28]:R(a⊗ b) = △(b)(a ⊗ 1). Then R 12 R 23= R 23 R 13 R 12 (see lemma 4.9 of [28]). This operator R plays a vital role in the theory of duality for von Neumann algebras (see [28] or [2]). If V is a vector space over an arbitrary field k, we shall study what we have called the Hopf equation: R 12 R 23= R 23 R 13 R 12 in End k (V ⊗ V ⊗ V). Taking W = rRr the Hopf equation is equivalent with the pentagonal equation:W l2 W l3 W 23 = W 23 W 12 from the theory of operator algebras (see [2]), where W are viewed as map in L (K ⊗ K), for a Hilbert space K. For a bialgebra H, we shall prove that the classic category of Hopf modules plays a decisive role in describing all solutions of the Hopf equation. More precisely, if H is a bialgebra over k and is an H-Hopf module, then the natural map R = R (M, ,ρ) a solution for the Hopf equation. Conversely, the main result of this paper is a FRT type theorem:if M is a finite dimensional vector space and R ∈ End k -(M⊙M) is a solution for the Hopf equation, then there exists a bialgebra B(R) such that By applying this result, we construct now examples of noncommutative and noncocomimitative bialgebras which are different from the ones arising from quantum group theory. In particular, over a field of characteristic two, an example of five dimensional noncommutative and noiicocommutative bialgebra is given.

Recursive calculation of the R matrices of q-deformed algebras

The authors show that the pentagonal equation, relating R-matrices and vector coupling coefficients, can be used to calculate recursively general R-matrices from those containing the primitive irrep. All R-matrices for su(2)q are calculated by this method. As an illustration of the more general case the primitive vector coupling coefficients for su(3)q are obtained and used to derive formulae for the R-matrices with a certain restriction on the representations.

TETRAHEDRA AND POLYNOMIAL EQUATIONS IN TOPOLOGICAL FIELD THEORY

Some tetrahedra in SUk(2) Chern-Simons-Witten theory are computed. The results can be used to compute an arbitrary tetrahedron inductively by fusing with the fundamental representation. The results obtained are in agreement with those of quantum groups. By associating a (finite) topological field theory (FTFT) to every rational conformal field theory (RCFT), we show that the pentagon and hexagon equations in RCFT follow directly from some skein relations in FTFT. By generalizing the operation of surgery on links in FTFT, we also derive an explicit expression for the modular transformation matrix S(k) of the one-point conformal blocks on a torus in RCFT and the equations satisfied by S(k), in agreement with those required in RCFT. The implication of our results on the general program of classifying RCFT is also discussed.

Polynomial equations and modular transformations in topological field theory

By associating a (finite) topological field theory (FTFT) to every rational conformal field theory (RCFT), we show that the pentagon and hexagon equations in RCFT follow directly from some skein relations in FTFT. By generalizing the operation of surgery on links in FTFT we derive an explicit expression for the modular transformation matrix S( k) of the one point conformal blocks on a torus in RCFT. In the framework of FTFT we also derive the equations satisfied by S>( k), in agreement with those obtained in RCFT.

Regular Pentagon and Quadratic Equation

Regular Pentagon and Quadratic Equation