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arXiv1997

Generalized cohomologies and the physical subspace of the $SU(2)$ WZNW model

The zero modes of the monodromy extended SU(2) WZNW model give rise to a gauge theory with a finite dimensional state space. A generalized BRS operator $A$ such that $A^h=0 (h=k+2=3,4,...$ being the height of the current algebra representation) acts in a (2h-1)-dimensional indefinite metric space $H_I$ of quantum group invariant vectors. The generalized cohomologies $Ker A^n/ Im A^{h-n} (n=1,..., h-1)$ are 1-dimensional. Their direct sum spans the physical subquotient of $H_I$.

hep-th↗
arXiv1997

Bi-Graded Markovian Matrices as Non-Local Dirac Operators and a New Quantum Evolution

Measuring distances on a lattice in noncommutative geometry involves square, symmetric and real ``three-diagonal'' matrices, with the sum of their elements obeying a supremum condition, together with a constraint forcing the absolute value of the maximal eigenvalue to be equal to 1. In even dimensions, these matrices are unipotent of order two, while in odd dimensions only their squares are Markovian. We suggest that these bi-graded Markovian matrices (i.e. consisting in the square roots of Markovian matrices) can be thought of as non-local Dirac operators. The eigenvectors of these matrices are spinors. Treating these matrices as determining the stochastic time evolution of states might explain why one observes only left handed neutrinos. Some other physical interpretations are suggested. We end by presenting a mathematical conjecture applying to q-graded Markovian matrices.

hep-th↗
arXiv1997

The Higher Cohomologies of E_8 Lie Algebra

It is well known that anomaly cancellations for D_16 Lie algebra are at the root of the first string revolution. For E_8 Lie algebra, cancellation of anomalies is the principal fact leading to the existence of heterotic string. They are in fact nothing but the 6th order cohomologies of corresponding Lie algebras. Beyond 6th order, the calculations seem to require special care and it could be that their study will be worthwhile in the light of developments of the second string revolution. As we have shown in a recent article, for A_N Lie algebras, there is a method which are based on the calculations of Casimir eigenvalues. This is extended to E_8 Lie algebra in the present article. In the generality of any irreducible representation of E_8 Lie algebra, we consider 8th and 12th order cohomologies while emphasizing the diversities between the two. It is seen that one can respectively define 2 and 8 basic invariant polinomials in terms of which 8th and 12th order Casimir eigenvalues are always expressed as linear superpositions. All these can be easily investigated because each one of these invariant polinimials gives us a linear equation to calculate E_8 weight multiplicities. Our results beyond order 12 are not included here because they get more complicated though share the same characteristic properties with 12th order calculations.

hep-th↗
arXiv1997

Knots and Matrix Models

We consider a matrix model with d matrices NxN and show that in the limit of large N and d=0 the model describes the knot diagrams. The same limit in matrix string theory is also discussed. We speculate that a prototypical M(atrix) without matrix theory exists in void.

hep-th↗
arXiv1998

A representation theoretic approach to the WZW Verlinde formula

By exploring the description of chiral blocks in terms of co-invariants, a derivation of the Verlinde formula for WZW models is obtained which is entirely based on the representation theory of affine Lie algebras. In contrast to existing proofs of the Verlinde formula, this approach works universally for all untwisted affine Lie algebras. As a by-product we obtain a homological interpretation of the Verlinde multiplicities as Euler characteristics of complexes built from invariant tensors of finite-dimensional simple Lie algebras. Our results can also be used to compute certain traces of automorphisms on the spaces of chiral blocks. Our argument is not rigorous; in its present form this paper will therefore not be submitted for publication.

hep-th↗
arXiv1998

New Tetrads for General Relativity

Using the tools of q--differential calculus and quantum Lie algebras associated to quantum groups, we find a one--parameter family of q-gauge theories associated to the quantum group $ISO_q(3,1)$. Although the gauge fields, that is the spin--connection and the vierbeins are non--commuting objects depending on a deformation parameter, $q$, it is possible to construct out of them a metric theory which is insensitive to the deformation. The Christoffel symbols and the Riemann tensor are ordinary commuting objects. Hence it is argued that torsionless Einstein's General Relativity is the common invariant sector of a one--parameter family of deformed gauge theories.

hep-th↗
arXiv1997

Analytic Continuation of Operators -- operators acting complex s-times -- Applications: from Number Theory and Group Theory to Quantum Field and String Theories

We are used to thinking of an operator acting once, twice, and so on. However, an operator acting integer times can be consistently analytic continued to an operator acting complex times. Applications: (s,r) diagrams and an extension of Fractional Calculus where commutativity of fractional derivatives is preserved, generating integrals and non-standard derivations of theorems in Number Theory, non-integer power series and breaking of Leibniz and Chain rules, pseudo-groups and symmetry deforming models in particle physics and cosmology, non-local effect in analytic continued matrix representations, particle-physics-like scatterings of zeros of analytic continued Bernoulli polynomials (physics/9705021), analytic continuation of operators in QM, QFT and Strings.

hep-th↗
arXiv1997

A New Approach to Functional Analysis on Graphs, the Connes-Spectral Triple and its Distance Function

Continuing previous work we develop a certain piece of functional analysis on general graphs and use it to create what Connes calls a 'spectral triple', i.e. a Hilbert space structure, a representation of a certain (function) algebra and a socalled 'Dirac operator', encoding part of the geometric/algebraic properties of the graph. We derive in particular an explicit expression for the 'Connes-distance function' and show that it is in general bounded from above by the ordinary distance on graphs (being, typically, strictly smaller(!) than the latter). We exhibit, among other things, the underlying reason for this phenomenon.

hep-th↗
arXiv1997

Topology and physics-a historical essay

This is an article on the interaction between topology and physics which will appear in 1998 in a book called: A History of Topology, edited by Ioan James and published by Elsevier-North Holland.

hep-th↗
arXiv1995

Cohomology of Drinfel'd algebras: A General Nonsense Approach

In our paper [Markl, Shnider: Drinfel'd Algebra Deformations and the Associahedra, IMRN 1994, no. 4, 169-176] we announced a construction of a cohomology controlling deformations of quasi-coassociative (or Drinfel'd) bialgebras. The full version of the paper will appear as [Markl, Shnider: Drinfel'd Algebra Deformations, Homotopy Comodules and the Associahedra] in Trans. Amer. Math. Soc. The construction in the paper was based on very explicit arguments using deep combinatorial properties of the associahedra. The present paper gives an alternative, general nonsense approach to the construction. So, we just prove the existence of such a cohomology without explicitly constructing it. This should be compared with the two approaches to the cohomology of associative algebras: we either describe explicitly the Hochschild complex and say "Behold! this is the cohomology" or we prove the existence of a projective resolution and define the cohomology as the derived functor.

q-alg↗
arXiv1995

On Hopf algebras and the elimination theorem for free Lie algebras

The elimination theorem for free Lie algebras, a general principle which describes the structure of a free Lie algebra in terms of free Lie subalgebras, has been recently used by E. Jurisich to prove that R. Borcherds' ``Monster Lie algebra'' has certain large free Lie subalgebras, illuminating part of Borcherds' proof that the moonshine module vertex operator algebra obeys the Conway-Norton conjectures. In the present expository note, we explain how the elimination theorem has a very simple and natural generalization to, and formulation in terms of, Hopf algebras. This fact already follows from general results contained in unpublished 1972 work, unknown to us when we wrote this note, of R. Block and P. Leroux.

q-alg↗
arXiv1996

Topics in hidden symmetries. IV

This note being devoted to some aspects of the inverse problem of representation theory explicates the links between researches on the Sklyanin algebras and the author's (based on the noncommutative geometry) approach to the setting free of hidden symmetries in terms of "the quantization of constants". Namely, the Racah-Wigner algebra for the Sklyanin algebra is constructed. It may be considered as a result of the quantization of constants in the Racah-Wigner algebra for the Lie algebra $sl(2,C)$. The Racah-Wigner algebra for the Sklyanin algebra is an example of the noncommutative weighted shift operator algebras (NWSO-algebras), which generalize the mho-algebras introduced by the author earlier. If the Sklyanin algebra is interpreted as an algebra of anomalous spins then the Racah-Wigner algebra for it may be regarded as an enlargement of the Sklyanin algebra by operators of the anomalous spin-spin interaction (of tensor type).

q-alg↗
arXiv2019

Idempotents of Hecke algebras of type A

We use a skein-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram. In this context we give straightforward calculations for the eigenvalues of two natural central elements in the Hecke algebras, namely the full curl and the sum of the Murphy operators. We discuss their calculation also in terms of the framing factor associated to the appropriate irreducible representation of the quantum group SU(N,q).

q-alg↗
arXiv1997

On BC type basic hypergeometric orthogonal polynomials

The five parameter family of multivariable Askey-Wilson polynomials is studied with four parameters generically complex. The multivariable Askey-Wilson polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partially discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partially discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable q-Racah polynomials and multivariable big and little q-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the multivariable Askey-Wilson polynomials. In particular new proofs of several well known q-analogues of the Selberg integral are obtained.

q-alg↗
arXiv1996

Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations

Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota's equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's $Q$-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for $A_{k-1}$-type models appear as discrete time equations of motions for zeros of classical $τ$-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained.

hep-th↗
arXiv1997

The sh Lie structure of Poisson brackets in field theory

A general construction of an sh Lie algebra from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.

hep-th↗
arXiv1997

The hidden symmetry algebras of a class of quasi-exactly solvable multi dimensional operators

Let $P(N,V)$ denote the vector space of polynomials of maximal degree less than or equal to $N$ in $V$ independent variables. This space is preserved by the enveloping algebra generated by a set of linear, differential operators representing the Lie algebra $gl(V+1)$. We establish the counterpart of this property for the vector space $P(M,V) \oplus P(N,V)$ for any values of the integers $M,N,V$. We show that the operators preserving $P(M,V) \oplus P(N,V)$ generate an abstract superalgebra (non linear if $Δ=\mid M-N\mid\geq 2$). A family of algebras is also constructed, extending this particular algebra by $Δ-1$ arbitrary complex parameters.

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