Faddeev Reshetikhin Takhtajan Research Articles

On the graded algebras associated with Hecke symmetries, II. The Hilbert series

Hecke symmetries give rise to a family of graded algebras which represent quantum groups and spaces of noncommutative geometry. The present paper continues the work aiming to understand general properties of these algebras without a restriction on the parameter q of Hecke relation used in earlier results. However, if q is a root of 1, we need a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of the initial vector space V. In this setting, we generalize known results on rationality of Hilbert series. The combinatorial nature of this problem stems from a relationship between the Grothendieck ring of the category of comodules for the Faddeev–Reshetikhin–Takhtajan bialgebra A(R) associated with a Hecke symmetry R and the ring of symmetric functions. We then improve two results on monoidal equivalences of corepresentation categories and on Gorensteinness of graded algebras from a previous article.

On the graded algebras associated with Hecke symmetries

A Hecke symmetry R on a finite dimensional vector space V gives rise to two graded factor algebras \mathbb S (V, R) and \Lambda (V, R) of the tensor algebra of V which are regarded as quantum analogs of the symmetric and the exterior algebras. Another graded algebra associated with R is the Faddeev–Reshetikhin–Takhtajan bialgebra A(R) which coacts on \mathbb S (V, R) and \Lambda (V, R) . There are also more general graded algebras defined with respect to pairs of Hecke symmetries and interpreted in terms of quantum hom-spaces. Their nice behaviour has been known under the assumption that the parameter q of the Hecke relation is such that 1 + q + \cdots + q^{n-1} \neq 0 for all n > 0 . The present paper makes an attempt to investigate several questions without this condition on q . Particularly we are interested in Koszulness and Gorensteinness of those graded algebras. For q a root of 1 positive results require a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of V .

FRT presentation of the Onsager algebras

A presentation \`a la Faddeev-Reshetikhin-Takhtajan (FRT) of the Onsager, augmented Onsager and sl(2)-invariant Onsager algebras is given, using the framework of the non-standard classical Yang-Baxter algebras. Associated current algebras are identified, and generating functions of mutually commuting quantities are obtained.

The category of Yetter-Drinfel'd Hom-modules and the quantum Hom-Yang-Baxter equation

In this paper, we introduce the category of Yetter-Drinfel'd Hom-modules which is a braided monoidal category and show that the category of Yetter-Drinfel'd Hom-modules is a full monoidal subcategory of the left center of left Hom-module category. Also we study the equivalent relationship between the category of Yetter-Drinfel'd Hom-modules and the category of Hom-modules over the Drinfel'd double. Finally, the Faddeev-Reshetikhin-Takhtajan (FRT) type theorem for the quantum Hom-Yang-Baxter equation is investigated.

Quantum affine Schubert cells and FRT-bialgebras: The [formula omitted] case

De Concini, Kac, and Procesi defined a family of subalgebras Uq+[w]⊆Uq(g) associated with elements w in the Weyl group of a simple Lie algebra g. These algebras are called quantum Schubert cell algebras. We show that, up to a mild cocycle twist, quotients of certain quantum Schubert cell algebras of types E6 and E6(1) map isomorphically onto distinguished subalgebras of the Faddeev–Reshetikhin–Takhtajan universal bialgebra associated with the braiding on the quantum half-spin representation of Uq(so10). We identify the quotients as those obtained by factoring out the quantum Schubert cell algebras by ideals generated by certain submodules with respect to the adjoint action of Uq(so10).

The FRT-construction via quantum affine algebras and smash products

For every element w in the Weyl group of a simple Lie algebra g , De Concini, Kac, and Procesi defined a subalgebra U q w of the quantized universal enveloping algebra U q ( g ) . The algebra U q w is a deformation of the universal enveloping algebra U ( n + ∩ w . n − ) . We construct smash products of certain finite-type De Concini–Kac–Procesi algebras to obtain ones of affine type; we have analogous constructions in types A n and D n . We show that the multiplication in the affine type De Concini–Kac–Procesi algebras arising from this smash product construction can be twisted by a cocycle to produce certain subalgebras related to the corresponding Faddeev–Reshetikhin–Takhtajan bialgebras.

Fusion categories in terms of graphs and relations

Every fusion category \mathcal{C} that is k -linear over a suitable field k is the category of finite-dimensional comodules of a weak Hopf algebra H . This weak Hopf algebra is finite-dimensional, cosemisimple and has commutative bases. It arises as the universal coend with respect to the long canonical functor \omega\colon\mathcal{C}\to \mathbf{Vect}_k . We show that H is a quotient H=H[\mathcal{G}]/I of a weak bialgebra H[\mathcal{G}] which has a combinatorial description in terms of a finite directed graph \mathcal{G} that depends on the choice of a generator M of \mathcal{C} and on the fusion coefficients of \mathcal{C} . The algebra underlying H[\mathcal{G}] is the path algebra of the quiver \mathcal{G}\times\mathcal{G} , and so the composability of paths in \mathcal{G} parameterizes the truncation of the tensor product of \mathcal{C} . The ideal I is generated by two types of relations. The first type enforces that the tensor powers of the generator M have the appropriate endomorphism algebras, thus providing a Schur–Weyl dual description of \mathcal{C} . If \mathcal{C} is braided, this includes relations of the form ‘ RTT=TTR ’ where R contains the coefficients of the braiding on \omega M\otimes\omega M , a generalization of the construction of Faddeev–Reshetikhin–Takhtajan to weak bialgebras. The second type of relations removes a suitable set of group-like elements in order to make the category of finite-dimensional comodules equivalent to \mathcal{C} over all tensor powers of the generator M . As examples, we treat the modular categories associated with U_q(\mathfrak{sl}_2) .

BMW algebra, quantized coordinate algebra and type 𝐶 Schur–Weyl duality

We prove an integral version of the Schur–Weyl duality between the specialized Birman–Murakami–Wenzl algebra B n ( − q 2 m + 1 , q ) \mathfrak {B}_n(-q^{2m+1},q) and the quantum algebra associated to the symplectic Lie algebra s p 2 m \mathfrak {sp}_{2m} . In particular, we deduce that this Schur–Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang in the symplectic case. As a byproduct, we show that, as a Z [ q , q − 1 ] \mathbb {Z}[q,q^{-1}] -algebra, the quantized coordinate algebra defined by Kashiwara (which he denoted by A q Z ( g ) A_q^{\mathbb {Z}}(g) ) is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev–Reshetikhin–Takhtajan construction.

PRESENTATION BY BOREL SUBALGEBRAS AND CHEVALLEY GENERATORS FOR QUANTUM ENVELOPING ALGEBRAS

AbstractWe provide an alternative approach to the Faddeev–Reshetikhin–Takhtajan presentation of the quantum group $\uqg$, with $L$-operators as generators and relations ruled by an $R$-matrix. We look at $\uqg$ as being generated by the quantum Borel subalgebras $U_q(\mathfrak{b}_+)$ and $U_q(\mathfrak{b}_-)$, and use the standard presentation of the latter as quantum function algebras. When $\mathfrak{g}=\mathfrak{gl}_n$, these Borel quantum function algebras are generated by the entries of a triangular $q$-matrix. Thus, eventually, $U_q(\mathfrak{gl}_n)$ is generated by the entries of an upper triangular and a lower triangular $q$-matrix, which share the same diagonal. The same elements generate over $\Bbbk[q,q^{-1}]$ the unrestricted $\Bbbk [q,q^{-1}]$-integral form of $U_q(\mathfrak{gl}_n)$ of De Concini and Procesi, which we present explicitly, together with a neat description of the associated quantum Frobenius morphisms at roots of 1. All this holds, mutatis mutandis, for $\mathfrak{g}=\mathfrak{sl}_n$ too.

Reflection equation, twist, and equivariant quantization

We prove that the reflection equation (RE) algebraL R associated with a finite dimensional representation of a quasitriangular Hopf algebraH is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show thatL R is a module algebra over the twisted tensor square $${\mathcal{H}}\mathop \otimes \limits^{\mathcal{R}} {\mathcal{H}}$$ and the double D( $${\mathcal{H}}$$ ). We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.

Spectral Decomposition and Baxterization of Exotic Bialgebras and Associated Noncommutative Geometries

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev–Reshetikhin–Takhtajan RTT prescription with non-triangular R-matrices which are denoted R03 and R14 in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1|1). We give the spectral decomposition which involves two, respectively, three, projectors. These projectors are then used to provide the baxterization procedure with one, respectively, two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess–Zumino prescription. In all these constructions there appear nonstandard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators.

Hopf algebra extension of a Zamolochikov algebra and its double

The particles with a scattering matrix R(x) are defined as operators Φi(z) satisfying the relation ∑i′,j′Ri,jj′,i′(x1/x2)Φi′(x1)Φj′(x2)=Φi(x2)Φj(x1). The algebra generated by those operators is called a Zamolochikov algebra. We construct a new Hopf algebra by adding half of the Faddeev–Reshetikhin–Takhtajan–Semenov-Tian-Shansky (FRTS) construction of a quantum affine algebra with this R(x). Then we double it to obtain a new Hopf algebra such that the full FRTS construction of a quantum affine algebra is a Hopf subalgebra inside. Drinfeld realization of quantum affine algebras is included as an example.

A Remark on the FRTS Realization and Drinfeld Realization of Quantum Affine Superalgebra U q (o\({\hat s} \)sp(1,2))

In this Letter, we present the hidden symmetry behind the Faddeev–Reshetikhin–Takhtajan–Semenov–Tian–Shansky (FRTS) realization of quantum affine superalgebras U(o\({\hat s} \)sp(1,2)) and add the q-Serre relation to the Drinfeld realization of U(o\({\hat s} \)sp(1,2)), derived from the FRTS realization.

Quantum group structure on the quantum plane

We show that the quantum plane introduced by Manin is a quantum group according to the Faddeev–Reshetikhin–Takhtajan approach. We give its matrix representation, R-matrix, and its Hopf structure. Some quantum properties of the quantum matrix whose elements generate the quantum plane are also discussed.

Duals of colored quantum universal enveloping algebras and colored universal 𝒯-matrices

We extend the notion of dually conjugate Hopf (super)algebras to the colored Hopf (super)algebras Hc that we recently introduced. We show that if the standard Hopf (super)algebras Hq that are the building blocks of Hc have Hopf duals Hq*, then the latter may be used to construct coloured Hopf duals Hc*, endowed with colored algebra and antipode maps, but with a standard coalgebraic structure. Next, we review the case where the Hq’s are quantum universal enveloping algebras of Lie (super)algebras Uq(g), so that the corresponding Hq*’s are quantum (super)groups Gq. We extend the Fronsdal and Galindo universal 𝒯-matrix formalism to the colored pairs (Uc(g),Gc) by defining colored universal 𝒯-matrices. We then show that together with the colored universal ℛ-matrices previously introduced, the latter provide an algebraic formulation of the colored RTT-relations, proposed by Basu-Mallick. This establishes a link between the colored extensions of Drinfeld–Jimbo and Faddeev–Reshetikhin–Takhtajan pictures of quantum groups and quantum algebras. Finally, we illustrate the construction of colored pairs by giving some explicit results for the two-parameter deformations of (U(gl(2)),Gl(2)), and (U(gl(1/1)),Gl(1/1)).

On the FRTS Approach to Quantized Current Algebras

We study the possibility to establish L-operator's formalism by Faddeev–Reshetikhin–Takhtajan–Semenov-Tian-Shansky (FRST) for quantized current algebras, that is, for quantum affine algebras in the ‘new realization’ by V. Drinfeld with the corresponding Hopf algebra structure and for their Yangian counterpart. We establish this formalism using the twisting procedure by Tolstoy and the second author and explain the problems which on FRST approach encounters for quantized current algebras. We also show that, for the case of Uq(ŝln), entries of the L-operators of the FRTS type give the Drinfeld current operators for the nonsimple roots, which we discovered recently. As an application, we deduce the commutation relations between these current operators for Uq(ŝl3).

Quantization of Lie group and algebra of G2 type in the Faddeev–Reshetikhin–Takhtajan approach

Based on the quantized universal enveloping (QUE) algebras, a quantization of the automorphism group of some nonassociative algebras is given in the formulation employing noncommuting matrix entries. A quantum group of G2 type included in this scheme is studied in detail in the Faddeev–Reshetikhin–Takhtajan (FRT) approach. Also in the formulation employing noncommuting matrix entries, the QUE-algebra of G2 type is reconstructed through the pairing induced by the R-matrix between the quantum group and the QUE-algebra.

REALIZATION OF FRT ALGEBRA RELATED TO A COLORED BRAID GROUP REPRESENTATION

A colored braid group representation (CBGR) is constructed by using some modified universal ℛ-matrix associated with U q( gl (2)) quantized algebra. Explicit realization of Faddeev–Reshetikhin–Takhtajan (FRT) algebra, involving color parameter dependent upper and lower triangular matrices, is built up for this CBGR and subsequently applied to generate nonadditive type solutions of quantum Yang–Baxter equation. Rational limit of such solutions interestingly yields 'colored' extension of known Lax operators associated with lattice nonlinear Schrödinger model and Toda chain.

Coloured FRT algebra and its Yang-Baxterization leading to integrable models with non-additive R-matrices

A 'colour' representation of Faddeev-Reshetikhin-Takhtajan (FRT) algebra, which, in contrast to the standard case, is related to the coloured braid group representation with generic values of q is presented. Explicit realizations of L(+or-)-matrices, occurring in this coloured variant of FRT algebra, are also obtained for the Uq(gl(2)) quantized algebra. Though these realizations are found to depend manifestly on the colour parameters, the underlying quantum group structure and associated co-product are interestingly free from such dependence. This allows us to perform the Yang-Baxterization of the coloured FRT algebra successfully, which leads to the construction of an ancestor Law operator associated with a new non-additive-type quantum R-matrix. Through different realizations of this Lax operator, a new class of quantum integrable models representing 'colour' generalizations of the well known models, such as the lattice sine-Gordon model, the Ablowitz-Ladik model lattice and the derivative nonlinear Schrodinger model etc, is generated.

Multiparametric quantization of the special linear superalgebra

It is shown how using the graded Faddeev–Reshetikhin–Takhtajan (FRT) formalism, one can multiparametrize the quantum superalgebra slq(n‖m). Since the naive Serre–Chevalley relations are not adequate for the presentation of this algebra, the quantization is carried out in the Cartan–Weyl basis. The final results can be generalized to any finite dimensional superalgebra with a symmetrizable Cartan matrix.