Search

The Bannai–Ito algebra BI(n) is viewed as the centralizer of the action of $$\mathfrak {osp}(1|2)$$ in the n-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the universal R-matrix of $$\mathfrak {osp}(1|2)$$. The specific structure of the $$\mathfrak {osp}(1|2)$$ embeddings to which the centralizing elements are attached as Casimir elements is explained. With the generators defined, the structure relations of BI(n) are derived from those of BI(3) by repeated action of the coproduct and using properties of the R-matrix and of the generators of the symmetric group $${\mathfrak {S}}_n$$.

We will introduce the notion of inductive limits of compact quantum groups as $W^*$-bialgebras equipped with some additional structures. We also formulate their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantized characters associated with a given inductive system of compact quantum groups. As a byproduct, we will give an explicit representation-theoretic interpretation to some transformations that play an important role in the analysis of $q$-central probability measures on the paths in the Gelfand-Tsetlin graph., 17 pages, second version (added an appendix)

A quantum covering group is an algebra with parameters $q$ and $\pi$ subject to $\pi^2=1$ and it admits an integral form; it specializes to the usual quantum group at $\pi=1$ and to a quantum supergroup of anisotropic type at $\pi=-1$. In this paper we establish the Frobenius-Lusztig homomorphism and Lusztig-Steinberg tensor product theorem in the setting of quantum covering groups at roots of 1. The specialization of these constructions at $\pi=1$ recovers Lusztig's constructions for quantum groups at roots of 1., Comment: v2, 21 pages, mild corrections, Lett. Math. Phys. (to appear)

We consider hamiltonian models representing an arbitrary number of spin $1/2$ fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a self-adjoint operator on a tensor product of anti-symmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated $N_\tau$ estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cut-offs., Comment: 26 pages

We revisit the derivation of Knizhnik–Zamolodchikov equations in the case of non-semisimple categories of modules of a superalgebra in the case of the generic affine level and representations parameters. A proof of existence of asymptotic solutions and their properties for the superalgebra $${\mathfrak {gl}}(1|1)$$ gives a basis for the proof of existence of associator which satisfy braided tensor categories requirements. Braided tensor category structure of $$U_h({\mathfrak {gl}}(1|1))$$ quantum algebra is calculated, and the tensor product ring is shown to be isomorphic to $${\mathfrak {gl}}(1|1)$$ ring, for the same generic relations between the level and parameters of modules. We review the proof of Drinfeld–Kohno theorem for non-semisimple category of modules suggested by Geer (Adv Math 207:1–38, 2006) and show that it remains valid for the superalgebra $${\mathfrak {gl}}(1|1)$$ . Examples of logarithmic solutions of KZ equations are also presented.

We study the tensor product decomposition of the split real quantum group $${\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))$$ from the perspective of finite-dimensional representation theory of compact quantum groups. It is known that the class of positive representations of $${\mathcal {U}}_{q\widetilde{q}}(\mathfrak {sl}(2,\mathbb {R}))$$ is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch–Gordan coefficients of the tensor product decomposition of finite-dimensional representations of the compact quantum group $${\mathcal {U}}_q(\mathfrak {sl}_2)$$ by solving certain functional equations arising from analytic continuation and using normalization arising from tensor products of canonical basis.

We define and study a certain relative tensor product of subfactors over a modular tensor category. This gives a relative tensor product of two completely rational heterotic full local conformal nets with trivial superselection structures over a common chiral representation category. In particular, we have a new realization of fusion rules of modular invariants. This also gives a mathematical definition of a composition of two gapped domain walls between topological phases., 9 pages

We construct a q-boson representation of the Zamolodchikov-Faddeev algebra whose structure function is given by the stochastic R matrix of \(U_q(A^{(1)}_n)\) introduced recently. The representation involves quantum dilogarithm type infinite products in the \(n(n-1)/2\)-fold tensor product of q-bosons. It leads to a matrix product formula of the stationary probabilities in the \(U_q(A_n^{(1)})\)-zero range process on a one-dimensional periodic lattice.

We review some recent results on properties of tensor product and fusion coefficients under complex conjugation of one of the factors. Some of these results have been proven, some others are conjectures awaiting a proof, one of them involving hitherto unnoticed observations on ordinary representation theory of finite simple groups of Lie type, 8 pages, 2 figures

Using the method of implementable one-particle Bogoliubov transformations, it is possible to explicitly define a local covariant net of quantum fields on the (universal covering of the) circle S 1 with braid group statistics. These anyon fields transform under a representation of $${\widetilde{U(1)}}$$ for arbitrary real-valued spin and their commutation relations depend on the relative winding number of localization regions. By taking the tensor product with a local covariant field theory on $${{{\rm{I\!R}}}^2}$$ , one can obtain a (non-boost covariant) cone-localized field net for anyons in two dimensions.

Two methods of constructing 2D Toda τ-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group Sn by multiplication of the conjugacy class sums Cλ ∈C(Sn) in the group algebra by elements of an abelian group of central elements. Extending the characteristic map to the tensor product C(Sn )⊗ C(Sn) leads to double expansions in terms of power sum symmetric functions, in which the coefficients count the number of such paths. Applying the same map to sums over the orthogonal idempotents leads to diagonal double Schur function expansions that are iden- tified as τ-functions of hypergeometric type. The second method is the standard construc- tion of τ-functions as vacuum-state matrix elements of products of vertex operators in a fermionic Fock space with elements of the abelian group of convolution symmetries .A homomorphism between these two group actions is derived and shown to be intertwined by the characteristic map composed with fermionization. Applications include Okounkov's generating function for double Hurwitz numbers, which count branched coverings of the Riemann sphere with specified ramification profiles at two branch points, and only simple branching at all the others, and various analogous combinatorial counting functions.

We introduce a homomorphism from the quantum affine algebras $U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n}), U_q(C^{(1)}_{n})$ to the $n$-fold tensor product of the $q$-oscillator algebra ${\mathcal A}_q$. Their action commute with the solutions of the Yang-Baxter equation obtained by reducing the solutions of the tetrahedron equation associated with the modular and the Fock representations of ${\mathcal A}_q$. In the former case, the commutativity is enhanced to the modular double of these quantum affine algebras.

The direct sum ${\cal O}_{*}$ of all Cuntz algebras has a non-cocommutative comultiplication $\Delta_{\varphi}$ such that there exists no antipode of any dense subbialgebra of the C$^{*}$-bialgebra $({\cal O}_{*},\Delta_{\varphi})$. From states equations of ${\cal O}_{*}$ with respect to the tensor product, we construct an operator $W$ for $({\cal O}_{*},\Delta_{\varphi})$ such that $W^{*}$ is an isometry, $W(x\otimes I)W^{*}=\Delta_{\varphi}(x)$ for each $x\in {\cal O}_{*}$ and $W$ satisfies the pentagon equation., Comment: 15 pages

Gaudin algebras form a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\g)$ of a semisimple Lie algebra $\g$. This family is parameterized by collections of pairwise distinct complex numbers $z_1,...,z_n$. We obtain some new commutative subalgebras in $U(\g)^{\otimes n}$ as limit cases of Gaudin subalgebras. These commutative subalgebras turn to be related to the hamiltonians of bending flows and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of spectrum in the Gaudin model for some new cases., Comment: 11 pages, references added

We introduce a nonsymmetric, associative tensor product among representations of Cuntz algebras by using embeddings. We show the decomposition formulae of tensor products for permutative representations explicitly We apply decomposition formulae to determine properties of endomorphisms., Comment: 12 pages

We give a description of the (small) quantum cohomology ring of the flag variety as a certain commutative subalgebra in the tensor product of the Nichols algebras. Our main result can be considered as a quantum analog of a result by Y. Bazlov.

We obtain the existence of a cohomological obstruction to expressing N=2 line bundles as tensor products of N=1 bundles. The motivation behind this paper is an attempt at understanding the N=2 super KP equation via Baker functions, which are special sections of line bundles on supercurves.

We develop a technique for the construction of integrable models with a ℤ2 grading of both the auxiliary (chain) and quantum (time) spaces. These models have a staggered disposition of the anisotropy parameter. The corresponding Yang–Baxter equations are written down and their solution for the gl(N) case is found. We analyze in details the N = 2 case and find the corresponding quantum group behind this solution. It can be regarded as the quantum group \(\mathcal{U}_{q\mathcal{B}} ({\text{gl(2)}})\), with a matrix deformation parameter q\(\mathcal{B}\) such that (q\(\mathcal{B}\))2 = q2. The symmetry behind these models can also be interpreted as the tensor product of the (−1)-Weyl algebra by an extension of \(\mathcal{U}\) q (gl(N)) with a Cartan generator related to deformation parameter −1.

When q is a root of unity, a triangular decomposition of Uq(s\({\hat l}{\kern 1pt} \)2) is given and irreducibility conditions concerning some tensor product representations of Uq(s\({\hat l}{\kern 1pt} \)2) are presented. Their connection with physics is also pointed out.

We show that for q ≠ -1 there is no natural tensor product for q-differential algebras. In particular, the q-graded tensor product of q-differentials fails to satisfy the q-graded Leibniz rule.

A special family of partitions occurs in two apparently unrelated contexts: the evaluation of one-dimensional configuration sums of certain RSOS models, and the modular representation theory of symmetric groups or their Hecke algebras Hm. We provide an explanation of this coincidence by showing how the irreducible Hm-modules which remain irreducible under restriction to Hm_1 (Jantzen–Seitz modules) can be determined from the decomposition of a tensor product of representations sln.

We define the crystal algebra, an algebra which has a base of elements of crystal bases of a quantum group. The multiplication is defined by the tensor product rule of crystal bases. A universal n-colored crystal algebra is defined. We study the relation between those algebras and the tensor algebras of the crystal algebra of Uq(sl(2)) and give a presentation by generators and relations for the case of Uq(sl(n)).

We discuss the notion of ‘metric dual pair’ corresponding to the ‘algebraic condition’ in Alain Connes' definition of noncommutative Poincare duality. We show that the DeRham complex of a metric dual pair is the homomorphic image of a skew tensor product, this leading to a natural definition of ‘biconnections’.

Invertible universal R-matrices of quantum Lie algebras do not exist at roots of unity. There exist however quotients for which intertwiners of tensor products of representations always exist, i.e. R-matrices exist in the representations. One of these quotients, which is finite dimensional, has a universal R-matrix. In this paper, we answer the following question: on which condition are the different quotients of U_q(sl(2)) (Hopf)-equivalent? In the case when they are equivalent, the universal R-matrix of one can be transformed into a universal R-matrix of the other. We prove that this happens only when q^4=1, and we explicitly give the expressions for the automorphisms and for the transformed universal R-matrices in this case., Comment: 11 pages, minor TeXnical revision to allow automatic TeXing

Let $U_q(\hat{\cal G})$ be a quantized affine Lie algebra. It is proven that the universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As applications, braid generators are shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight $U_q(\hat{\cal G})$-module and a spectral decomposition formula for the braid generators is obtained which is the generalization of Reshetikhin's and Gould's forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found., 11 pages (minor error corrected)

In this paper we show that in the case of noncommutative two-tori one gets in a natural way simple structures which have analogous formal properties as Hopf algebra structures but with a deformed multiplication on the tensor product.

We present a general scheme for describing su(N)_k fusion rules in terms of elementary couplings, using Berenstein-Zelevinsky triangles. A fusion coupling is characterized by its corresponding tensor product coupling (i.e. its Berenstein-Zelevinsky triangle) and the threshold level at which it first appears. We show that a closed expression for this threshold level is encoded in the Berenstein-Zelevinsky triangle and an explicit method to calculate it is presented. In this way a complete solution of su(4)_k fusion rules is obtained., 14 pages

It is shown that the braid generator is diagonalizable on arbitrary tensor product modules V ⊗ V, V an irreducible module for a quantum group. A generalization of the Reshetikhin form for the braid generator is thereby obtained in the general case. As an application, a general closed formula is determined for link polynomials.

We prove that the rings of q-differential operators on quantum planes of the GL q (n) and SO q (n) types are isomorphic to the rings of classical differential operators. Also, we construct decompositions of the rings of q-differential operators into tensor products of the rings of q-differential operators with less variables.

Up to now, the universal R-matrix for quantized Kac-Moody algebras1 is believed to be uniquely determined (for some ansatz) by properties of a quasi-cocommutativity and a quasi-triangularity. We prove here that the universal R-matrix (for the same ansatz) is uniquely determined by the property of the quasi-cocommutativity only. Thus, the quasi-triangular property (and the Yang-Baxter equation!) for the universal R-matrix is a consequence of the linear equation of the quasi-cocommutativity. The proof is based on properties of singular vectors in the tensor product of the Verma modules and the structure of extremal projector for quantized algebras. Explicit expressions of the universal R-matrix for quantized algebras Uq(Ainf1sup(1)) and Uq(Ainf2sup(2)) are given.

We write down a complete set of n-point Uq(sl(2)) invariants (using a polynomial basis for the irreducible finite dimensional Uq-modules) that are regular for all nonzero values of the deformation parameter q.

We introduce a new family of examples of ‘quantum groups’. Its irreducible representations are indexed by level l dominant integral weights of affine Lie algebra sī(n), and their branching rules with respect to the ‘truncated tensor product’ $$\bar\otimes $$ is given by the fusion rules of SU(n) l -Wess-Zumino-Witten model in conformal field theory.

Given a rational number ζ, when does the tensor product of two modular invariant Virasoro modules have central charge ζ? I show that the resulting equations define elliptic curves over the rational numbers, and determine the torsion subgroups of these curves for the case in which both tensor factors have the same central charge. This gives an alternative parametrization of the curves with torsion subgroup ℤ/8ℤ x ℤ/2ℤ. Consideration of a larger class of curves (including some which do not correspond to tensor products of modules) gives a parametrization of all curves with a point of order 8. I conclude with some examples of curves which have 16 torsion points and positive rank.

There is a natural connection and parallel transport on the Hilbert tensor product ℋ⊗ℋ (or, equivalently, the space of Hilbert-Schmidt operators), the elements of which represent density matrices in ℋ up to unitary operators. We postulate a time evolution equation, which leads to this connection after extracting a proper ‘dynamical’ unitary phase. As an example, we compute the holonomy of a loop of temperature states for the spin in a rotating magnetic field.

A Hopf *-algebra corresponding to a solution of the classical Yang-Baxter equation is introduced, providing a new quantum deformation of SL (2 ℝ).

For some family of representations of SU(n) the explicit formulae for the multiplicity of a given representation in arbitrary tensor product are derived. The adjoint representation and the representations acting in the symmetrical and antisymmetrical tensors are contained in the considered family as the simplest cases.

We prove a theorem for the tensor product of representations of the holomorphic discrete series analogous to the classical theorem of Clebsch-Gordan and give an asymptotic formula for corresponding ‘coefficients of Clebsch-Gordan.’ For small groups we compute the coefficients explicitly.

Schur's lemma and the properties of circular matrices are used to establish a necessary and sufficient condition for the finite-dimensional irreducible matrix representations of an arbitrary groupG to admit real coupling (or Clebsch-Gordan) coefficients. The Pontryagin-Van Kampen and Tannaka-Krein duality theorems are found to be of considerable value in implementing the condition, which requires that complex conjugation effects an automorphism on the group of all matrices having the same reduction of their tensor products as the matrix representations ofG. This result is noted to be relevant to a generalization of the Frobenius-Schur invariant.

Let U n (a, Λ) be a massless, helicity n/2, representation of the Poincare group in 3+1 dimensions. U n (a, Λ) is realized in an adapted nuclear space D n. We explicitly determine the various classes of cohomology for the extension of U n (a, Λ) by U n1 (a, Λ) ⊗ U n2 (a, Λ).

The problem of the decomposition of the tensor product of finite and infinite representations of a complex semigroup of a Lie group is examined by using the theory of characters of completely irreducible representations. A theorem is proved which indicates that completely irreducible representations enter into the expansion of the tensor product of a finite and elementary representation.

Bell's correlation inequality is considered in the algebraic framework as discussed by Baez. It is shown that all normal states of the tensor product of two W*-algebras satisfy Bell's inequality, if and only if every normal state lies in the closure of the convex hull of the normal product states, if and only if one of the algebras is commutative.

We verify and generalize a conjecture of Fulling [1] that the Kronecker product of a finite number of unitary representations, not all of which possess an invariant vector, of the Lorentz group SO 0(1, n), any n≥2, does not contain the trivial representation discretely.