Quantum Function Algebra Research Articles

Poisson Trace Orders

Abstract The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras.

Quantum function algebras from finite-dimensional Nichols algebras

We describe how to find quantum determinants and antipode formulas from braided vector spaces using the FRT-construction and finite-dimensional Nichols algebras. It improves the construction of quantum function algebras using quantum grassmanian algebras. Given a finite-dimensional Nichols algebra \mathfrak B , our method provides a Hopf algebra H such that \mathfrak B is a braided Hopf algebra in the category of H -comodules. It also serves as source to produce Hopf algebras generated by cosemisimple subcoalgebras, which are of interest for the generalized lifting method. We give several examples, among them quantum function algebras from Fomin–Kirillov algebras associated with the symmetric group on three letters.

Relations in Quantized Function Algebras

We develop a method to give presentations of quantized function algebras of complex reductive groups. In particular, we give presentations of quantized function algebras of automorphism groups of finite dimensional simple complex Lie algebras.

Tetrahedron Equation, Weyl Group, and Quantum Dilogarithm

We derive a family of solutions to the tetrahedron equation using the RTT presentation of a two parametric quantized algebra of regular functions on an upper triangular subgroup of GL(n). The key ingredients of the construction are the longest element of the Weyl group, the quantum dilogarithm function, and central elements of the quantized division algebra of rational functions on the subgroup in question.

An Introductory Informatics Theoretical Framework, based on an Integrable Lattice Model using Quantized Function Algebras, for Studying DNA-Ionized Gas Interaction System

Usually we observe that Bio-physical systems or Bio-chemical systems are many a time based on nanoscale phenomenon in different host environments, which involve many particles can often not be solved explicitly. Instead a physicist, biologist or a chemist has to rely either on approximate or numerical methods. For a certain type of systems, called integrable in nature, there exist particular mathematical structures and symmetries which facilitate the exact and explicit description. Most integrable systems, we come across are low-dimensional, for instance, a one-dimensional chain of coupled atoms in DNA molecular system with a particular direction or exist as a vector in the environment. This theoretical research paper aims at bringing one of the pioneering ‘Reaction-Diffusion’ aspects of the DNA-plasma material system based on an integrable lattice model approach utilizing quantized functional algebras, to disseminate the new developments, initiate novel computational and design paradigms.

Generalized sine-Gordon models and quantum braided groups

Abstract We determine the quantized function algebras associated with various examples of generalized sine-Gordon models. These are quadratic algebras of the general Freidel-Maillet type, the classical limits of which reproduce the lattice Poisson algebra recently obtained for these models defined by a gauged Wess-Zumino-Witten action plus an integrable potential. More specifically, we argue based on these examples that the natural framework for constructing quantum lattice integrable versions of generalized sine-Gordon models is that of affine quantum braided groups.

Tetrahedron and 3D reflection equations from quantized algebra of functions

Soibelman’s theory of quantized function algebra Aq(SLn) provides a representation theoretical scheme to construct a solution of the Zamolodchikov tetrahedron equation. We extend this idea originally due to Kapranov and Voevodsky to Aq(Sp2n) and obtain the intertwiner K corresponding to the quartic Coxeter relation. Together with the previously known three-dimensional (3D) R matrix, the K yields the first ever solution to the 3D analogue of the reflection equation proposed by Isaev and Kulish. It is shown that matrix elements of R and K are polynomials in q and that there are combinatorial and birational counterparts for R and K. The combinatorial ones arise either at q = 0 or by tropicalization of the birational ones. A conjectural description for type B and F4 cases is also given.

Braided differential operators on quantum algebras

We propose a general scheme of constructing braided differential algebras via algebras of “quantum exponentiated vector fields” and those of “quantum functions”. We treat a reflection equation algebra as a quantum analog of the algebra of vector fields. The role of a quantum function algebra is played by a general quantum matrix algebra. As an example we mention the so-called RTT algebra of quantized functions on the linear matrix group G L ( m ) . In this case our construction essentially coincides with the quantum differential algebra introduced by S. Woronowicz. If the role of a quantum function algebra is played by another copy of the reflection equation algebra we get two different braided differential algebras. One of them is defined via a quantum analog of (co)adjoint vector fields, the other algebra is defined via a quantum analog of right-invariant vector fields. We show that the former algebra can be identified with a subalgebra of the latter one. Also, we show that “quantum adjoint vector fields” can be restricted to the so-called “braided orbits” which are counterparts of generic G L ( m ) -orbits in g l ∗ ( m ) . Such braided orbits endowed with these restricted vector fields constitute a new class of braided differential algebras.

Regular Representations of Quantum Groups at Roots of Unity

We study the bimodule structure of the quantum function algebra for type A at roots of unity and prove that it admits a filtration with factors isomorphic to the tensor products of the dual of the Weyl modules . As an application, we compute the HH 0 Hochschild cohomology of the function algebra at roots of unity.

F q [M 2], F q [GL 2], and F q [SL 2] as Quantized Hyperalgebras

Within the quantum function algebra F q [SL 2], we study the subset ℱ q [SL 2]—introduced in Gavarini (1998a)—of all elements of F q [SL 2] which are Z opf [q, q −1]-valued when paired with 𝒰 q (𝔰 𝔩 2), the unrestricted ℤ [q, q −1] -integral form of U q (𝔰 𝔩 2) introduced by De Concini, Kac, and Procesi. In particular, we yield a presentation of it by generators and relations, and a nice ℤ [q, q −1]–spanning set (of PBW type). Moreover, we give a direct proof that ℱ q [SL 2] is a Hopf subalgebra of F q [SL 2], and that . We describe explicitly its specializations at roots of 1, say ϵ, and the associated quantum Frobenius (epi)morphism (also introduced in Gavarini, 1998a) from ℱ ϵ[SL 2] to . The same analysis is done for ℱ q [GL 2], with similar results, and also (as a key, intermediate step) for ℱ q [M 2].

PBW THEOREMS AND FROBENIUS STRUCTURES FOR QUANTUM MATRICES

AbstractLet $G \in \{{\it Mat}_n(\C), {GL}_n(\C), {SL}_n(\C)\}$, let $\Oqg$ be the quantum function algebra – over $\Z [q,q^{-1}]$ – associated to G, and let $\Oeg$ be the specialisation of the latter at a root of unity ϵ, whose order ℓ is odd. There is a quantum Frobenius morphism that embeds $\Og,$ the function algebra of G, in $\Oeg$ as a central Hopf subalgebra, so that $\Oeg$ is a module over $\Og$. When $G = {SL}_n(\C)$, it is known by [3], [4] that (the complexification of) such a module is free, with rank ℓdim(G). In this note we prove a PBW-like theorem for $\Oqg$, and we show that – when G is Matn or GLn – it yields explicit bases of $\Oeg $ over $ \Og$ over $\Og,$. As a direct application, we prove that $\Oegl$ and $\Oem$ are free Frobenius extensions over $\Ogl$ and $\Om$, thus extending some results of [5].

Formula omitted], [formula omitted] and [formula omitted] as quantized hyperalgebras

Within the quantum function algebra F q [ GL n ] , we study the subset F q [ GL n ] —introduced in [F. Gavarini, Quantization of Poisson groups, Pacific J. Math. 186 (1998) 217–266]—of all elements of F q [ GL n ] which are Z [ q , q −1 ] -valued when paired with U q ( gl n ) , the unrestricted Z [ q , q −1 ] -integral form of U q ( gl n ) introduced by De Concini, Kac and Procesi. In particular we obtain a presentation of it by generators and relations, and a PBW-like theorem. Moreover, we give a direct proof that F q [ GL n ] is a Hopf subalgebra of F q [ GL n ] , and that F q [ GL n ] | q = 1 ≅ U Z ( gl n ∗ ) . We describe explicitly its specializations at roots of 1, say ε, and the associated quantum Frobenius (epi)morphism from F ε [ GL n ] to F 1 [ GL n ] ≅ U Z ( gl n ∗ ) , also introduced in [F. Gavarini, Quantization of Poisson groups, Pacific J. Math. 186 (1998) 217–266]. The same analysis is done for F q [ SL n ] and (as key step) for F q [ M n ] .

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.

On the Quantum Function Algebra at Roots of One#

In this note we show that certain properties of the quantum function algebra at roots of unity hold over any algebraically closed field of characteristic zero.

Poisson Geometrical Symmetries Associated to Non-Commutative Formal Diffeomorphisms

Let Open image in new window be the group of all formal power series starting with x with coefficients in a field Open image in new window of zero characteristic (with the composition product), and let F [ Open image in new window ] be its function algebra. In [BF] a non-commutative, non-cocommutative graded Hopf algebra Open image in new window was introduced via a direct process of ‘‘disabelianisation’’ of F [ Open image in new window ], taking the like presentation of the latter as an algebra but dropping the commutativity constraint. In this paper we apply a general method to provide four one-parameter deformations of Open image in new window, which are quantum groups whose semiclassical limits are Poisson geometrical symmetries such as Poisson groups or Lie bialgebras, namely two quantum function algebras and two quantum universal enveloping algebras. In particular the two Poisson groups are extensions of Open image in new window, isomorphic as proalgebraic Poisson varieties but not as proalgebraic groups.

Real spectra of quantum groups

The only noncommutative ring for which the real spectrum has been described so far is the quantum affine ring R q [x,y] [J. Algebra 212 (1999) 190]. The aim of this paper is to describe the real spectra of quantum affine rings k q [x 1 ,…,x n ] where k is a formally real affine R -algebra and q ∈M n ( R + ) . As a by-product we describe the real spectra of quantized enveloping algebra U q ( s l 2 ( R )) and quantum special linear group O q (SL 2 ( R )) . Formal reality and semireality is characterized for the following classes of quantum groups: quantum affine rings, quantized enveloping algebras, quantized function algebras, quantized Weyl algebras.

Real spectra of quantum groups

The only noncommutative ring for which the real spectrum has been described so far is the quantum affine ring R q[x,y] [J. Algebra 212 (1999) 190]. The aim of this paper is to describe the real spectra of quantum affine rings k q [x 1,…,x n] where k is a formally real affine R -algebra and q ∈M n( R +) . As a by-product we describe the real spectra of quantized enveloping algebra U q( sl 2( R)) and quantum special linear group O q(SL 2( R)) . Formal reality and semireality is characterized for the following classes of quantum groups: quantum affine rings, quantized enveloping algebras, quantized function algebras, quantized Weyl algebras.

Representations of affine quantum function algebras

Let $C$ be a symmetrizable generalized Cartan Matrix, and $q$ an indeterminate. ${\fg}(C)$ is the Kac-Moody Lie algebra and $U=U_q({\fg}(C))$ the associated quantum enveloping algebra over $ k={\Bbb Q}(q)$. The quantum function algebra ${\Bbb C}_{q}[G]$ is defined as a suitable $U$-bisubalgebra of the dual space $\hom_{k}(U,k)$ which can be described using matrix elements of integrable $U$-modules. For $\fg$ affine, the highest weight modules of $C_q[G]$ are constructed and, assuming a minimality condition, their (unitarizable) irreducible quotients are shown to be in a 1-1 correspondence with the reduced elements of the Weyl group of ${\frak g}(C)$. Further, these simple module are described in terms of the $C_q[SL_2]$-modules obtained by restriction, and they satisfy a Tensor Product theorem, similar to the finite type case.

Weakly multiplicative coactions of quantized function algebras

A condition is identified which guarantees that the coinvariants of a coaction of a Hopf algebra on an algebra form a subalgebra, even though the coaction may fail to be an algebra homomorphism. A Hilbert Theorem (finite generation of the subalgebra of coinvariants) is obtained for such coactions of a cosemisimple Hopf algebra. This is applied for two coactions α,β : A→ A⊗ O , where A is the coordinate algebra of the quantum matrix space associated with the quantized coordinate algebra O of a classical group, and α, β are quantum analogues of the conjugation action on matrices. Provided that O is cosemisimple and coquasitriangular, the α-coinvariants and the β-coinvariants form two finitely generated, commutative, graded subalgebras of A , having the same Hilbert series. Consequently, the cocommutative elements and the S 2-cocommutative elements in O form finitely generated subalgebras. A Hopf algebra monomorphism from the quantum general linear group to Laurent polynomials over the quantum special linear group is found and used to explain the strong relationship between the corepresentation (and coinvariant) theories of these quantum groups.

An ℏ-adic valuation property of universal R-matrices

We prove that if U ℏ ( g ) is a quasitriangular QUE algebra with universal R -matrix R , and O ℏ (G ∗ ) is the quantized function algebra sitting inside U ℏ ( g ) , then ℏlog( R ) belongs to the tensor square O ℏ (G ∗ ) ⊗ ̄ O ℏ (G ∗ ) . This gives another proof of the results of Gavarini and Halbout, saying that R normalizes O ℏ (G ∗ ) ⊗ ̄ O ℏ (G ∗ ) and therefore induces a braiding of the formal group G ∗ (in the sense of Weinstein and Xu, or Reshetikhin).