arXiv subjects
Publications and source records attributed to I. Heckenberger.
Yetter--Drinfel'd modules of diagonal type admit an equivalence relation which conjecturally preserves dimension and Gel'fand--Kirillov dimension of the corresponding Nichols algebras. This relation is determined explicitly for all rank 2 Yetter--Drinfeld modules where the Gel'fand--Kirillov dimension is known to be finite. Key Words: Brandt groupoid, Hopf algebra, pseudo-reflections, Weyl group
The theory of Nichols algebras of diagonal type is known to be closely related to that of semisimple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semisimple Lie algebra. They give rise to the definition of a Brandt groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding. Key Words: Brandt groupoid, Hopf algebra, pseudo-reflections, Weyl group
Nichols algebras naturally appear in the classification of finite dimensional pointed Hopf algebras. Assuming only that the base field has characteristic zero several new finite dimensional rank 2 Nichols algebras of diagonal type are listed. Each of them is described in terms of generators and relations. A Poincaré--Birkhoff--Witt basis of all of these Nichols algebras is given and their dimension is computed. Together with the continuation of this paper this gives an answer to a question of Andruskiewitsch. Key Words: Hopf algebra, Lyndon words, full binary tree
The method of subquotients is developed and used to determine all finite dimensional rank 2 Nichols algebras of diagonal type over an arbitrary field of characteristic zero. Key Words: Hopf algebra, Nichols algebra
It is shown that quantized irreducible flag manifolds possess a canonical $q$-analogue of the de Rham complex. Generalizing the well known situation for the standard Podleś' quantum sphere this analogue is obtained as the universal differential calculus of a distinguished first order differential calculus. The corresponding differential $\dif$ can be written as a sum of differentials $\del$ and $\delbar$. The universal differential calculus corresponding to the first order differential calculi $\dif$, $\del$, and $\delbar$ are given in terms of generators and relations. Relations to well known quantized exterior algebras are established. The dimensions of the homogeneous components are shown to be the same as in the classical case. The existence of a volume form is proven.
The notion of locally finite part of the dual coalgebra of certain quantized coordinate rings is introduced. In the case of irreducible flag manifolds this locally finite part is shown to coincide with a natural quotient coalgebra V of U_q(g). On the way the coradical filtration of V is determined. A graded version of the duality between V and the quantized coordinate ring is established. This leads to a natural construction of several examples of quantized vector spaces. As an application covariant first order differential calculi on quantized irreducible flag manifolds are classified. Keywords: quantum groups, quantized flag manifolds
The dual coalgebra of Podleś' quantum sphere O_q(S^2_c) is determined explicitly. This result is used to classify all finite dimensional covariant first order differential calculi over O_q(S^2_c) for all but exceptional values of the parameter c.
For differential calculi over certain right coideal subalgebras of quantum groups the notion of quantum tangent space is introduced. In generalization of a result by Woronowicz a one to one correspondence between quantum tangent spaces and covariant first order differential calculi is established. This result is used to classify differential calculi over quantum Grassmann manifolds. It turns out that up to special cases in low dimensions there exists exactly one such calculus of classical dimension 2r(N-r). Keywords: Quantum groups, quantum spaces, quantum Grassmann manifolds, differential calculus
We study the q-Clifford algebras Cl_q(N,c), called FRT-Clifford algebras, introduced by Faddeev, Reshetikhin and Takhtajan. It is shown that Cl_q(N,c) acts on the q-exterior algebra Λ(O_q^N). Moreover, explicit formulas for the embedding of U_q(so_N) into Cl_q(N,c) and its relation to the vector and spin representations of U_q(so_N) are given and proved. Key Words: q-Clifford algebra, Drinfeld-Jimbo algebra, spin representation
Let G=G(t,z) be one of the N^2-dimensional bicovariant first order differential calculi for the quantum groups GL_q(N), SL_q(N), O_q(N), or Sp_q(N), where q is a transcendental complex number and z is a regular parameter. It is shown that the de Rham cohomology of Woronowicz' external algebra G^ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (twosided) coinvariant subcomplexes. In the cases GL_q(N) and SL_q(N) the cohomology ring is isomorphic to the coinvariant external algebra G^_{inv} and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. Keywords: quantum groups, bicovariant differential calculi, de Rham cohomology, Laplace-Beltrami operator, Hodge theory
Let A be a cosemisimple Hopf *-algebra with antipode S and let $Γ$ be a left-covariant first order differential *-calculus over A such that $Γ$ is self-dual and invariant under the Hopf algebra automorphism S^2. A quantum Clifford algebra $\Cl(Γ,σ,g)$ is introduced which acts on Woronowicz' external algebra $Γ^\wedge$. A minimal left ideal of $\Cl(Γ,σ,g)$ which is an A-bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on $Γ$ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SL_q(2) and its bicovariant $4D_\pm$-calculi these concepts are studied in detail. A generalization of Bochner's theorem is given. All invariant differential operators over a given spinor module are determined. The eigenvalues of the Dirac operator are computed. Keywords: quantum groups, covariant differential calculus, spin geometry
For transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group $\SLq 2$ are given. All such differential calculi $Γ$ are determined and investigated for which the left-invariant differential one-forms $ω(u^1_2)$, $ω(u^2_1)$ and $ω(u^1_1-u^2_2)$ generate $Γ$ as a bimodule and the universal higher order differential calculus has the same dimension as in the classical case. Important properties (cohomology spaces, *-structures, braidings, generalized Lie brackets) of these calculi are examined as well. Keywords: quantum groups, noncommutative differential calculus, quantum tangent space
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite dimensional. Using Jucys-Murphy elements of the Hecke algebra the eigenvalues of the Laplace-Beltrami operator for the Hopf algebra O(SLq(N)) are computed.
For the 9-dimensional bicovariant differential calculi on the Hopf algebra $O(O_q(3))$ several kinds of exterior algebras are examined. The corresponding dimensions, bicovariant subbimodules and eigenvalues of the antisymmetrizer are given. Exactly one of the exterior algebras studied by the authors has a unique left invariant form with maximal degree.
For transcendental values of $q$ all bicovariant first order differential calculi on the coordinate Hopf algebras of the quantum groups $SL_q(n+1)$ and $Sp_q(2n)$ are classified. It is shown that the irreducible bicovariant first order calculi are determined by an irreducible corepresentation of the quantum group and a complex number $ζ$ such that $ζ^{n+1}=1$ for $SL_q(n+1)$ and $ζ^2=1$ for $Sp_q(2n)$. Any bicovariant calculus is inner and its quantum Lie algebra is generated by a central element. The main technical ingredient is a result of the Hopf algebra $R(G_q)^0$ for arbitrary simple Lie algebras.
Let $Γ$ be an $N^2$-dimensional bicovariant first order differential calculus on a Hopf algebra $SL_q(N)$. There are three possibilities to construct a differential Z-graded Hopf algebra $Γ^\wedge$ which contains $Γ$ as its first order part. Let $q$ be a transcendental complex number. For $N>2$ these three Z-graded Hopf algebras coincide. For Woronowicz' external algebra we calculate the dimensions of the spaces of left-invariant and bi-invariant $k$-forms. In this case each bi-invariant form is closed. In case of $4D_\pm$ calculi on $SL_q(2)$ the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed.
For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, Levi-Civita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on $SL_{q}(N)$, $O_{q}(N)$ and $Sp_{q}(N)$ from the classification of Schmuedgen, K. and Schueler, A. there exist unique Levi-Civita connections.