Aristophanes Dimakis, Folkert Muller-Hoissen
If the bimodule of 1-forms of a differential calculus over an associative algebra is the direct sum of 1-dimensional bimodules, a relation with automorphisms of the algebra shows up. This happens for some familiar quantum space calculi.
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Differential calculi of Poincare-Birkhoff-Witt type on universal enveloping algebras of Lie algebras g are defined. This definition turns out to be independent of the basis chosen in g. The role of automorphisms of g is explained. It is proved that no differential calculus of Poincare-Birkhoff-Witt type exists on semi-simple Lie algebras. Examples are given, namely gl_n, Abelian Lie algebras, the Heisenberg algebra, the Witt and the Virasoro algebra. Completely treated are the 2-dimensional solvable Lie algebra, and the 3-dimensional Heisenberg algebra.
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 ope…
A variety of three-dimensional left-covariant differential calculi on the quantum group $SU_q(2)$ is considered using an approach based on global $ U(1) $ -covariance. Explicit representations of possible $q $-Lie algebras are constructed in terms of differential operators. A gauge covariant differential algebra is uniquely determined. The non-standard Leibnitz rule is obtained for a corresponding $q$-Lie algebra.
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 we…
All bicovariant first order differential calculi on the quantum group GLq(3,C) are determined. There are two distinct one-parameter families of calculi. In terms of a suitable basis of 1-forms the commutation relations can be expressed with the help of the R-matrix of GLq(3,C). Some calculi induce bicovariant differential calculi on SLq(3,C) and on real forms of GLq(3,C). For generic deformation parameter q there are six calculi on SLq(3,C), on SUq(3) there are only two. The classical limit q-->1 of bicovariant calculi on SLq(3,C) is not the ordinary calculus on SL(3,C). One obtains a deformat…
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…
We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of the coordinate algebra of the quantum SL(2) group relative to twisting automorphisms acting by rescaling the standard generators a,b,c,d. We discover a family of automorphisms for which the "twisted" Hochschild dimension coincides with the classical dimension of SL(2, C), thus avoiding the "dimension drop" in Hochschild homology seen for many quantum deformations. Strikingly, the simplest such automorphism is the canonical modular automorphism arising from the Haar functional. In addition,…
We investigate covariant first order differential calculi on the quantum complex projective spaces CP_q^{N-1} which are quantum homogeneous spaces for the quantum group SU_q(N). Hereby, one more well-studied example of covariant first order differential calculus on a quantum homogeneous space is given. Since the complex projective spaces are subalgebras of the quantum spheres S_q^{2N-1} introduced by Vaksman and Soibelman, we get also an example of the relations between covariant differential calculus on two closely related quantum spaces. Two approaches are combined in obtaining covariant fir…
Using the Hecke $\hat R$-matrix, we give a definition of the lattice $(l,q)$-deformed $n$-component boson and Grassmann fields. Here $l$ is a deformation parameter for the commutation relations of "values" of these fields in two arbitrary lattice sites and $q$ is a deformation parameter for $n$-component $q$-boson or $q$-Grassmann variable. In framework of the Wess-Zumino approach to the noncommutative differential calculus the commutation relations between differentials and derivatives of these fields are determined. The $SL_q(n,C)$-invariant generalization of the Berezin integration for the …
In one of our recent papers, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were defined and studied, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and $F[D]$ is the polynomial algebra of a commutative derivation subalgebra $D$ of $A$. In the present paper, a class of the above associative and Lie algebras $A[D]$ with $F$ being a field of characteristic 0, $D$ consisting of locally finite but not locally nilpotent derivations of $A$, are studied. The isomorphism classes and automorphism groups of these as…
The derivation algebras, automorphism groups and second cohomology groups of the generalized loop Schrödinger-Virasoro algebras are completely determined in this paper.
Let L be a positive definite even lattice and V_L^+ be the fixed points of the lattice VOA V_L associated to L under an automorphism of V_L lifting the -1 isometry of L. For any positive rank, the full automotphism group of V_L^+ is determined if L does not have vectors of norm 2 or 4. For any L of rank 2, a set of generators and the full automorphism group of V_L^+ are determined.
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.
The conjugacy classes of the involutive automorphisms of the affine Kac-Moody algebras \Cli\ for $\ell\geq 2$ are determined using the matrix formulation of automorphisms of an affine Kac-Moody algebra.
We determine the automorphism groups of the cyclic orbifold vertex operator algebras associated with coinvariant lattices of isometries of the Leech lattice in the conjugacy classes $4C,6E,6G,8E$ and $10F$. As a consequence, we have determined the automorphism groups of all the $10$ vertex operator algebras in [Hö], which are useful to analyze holomorphic vertex operator algebras of central charge $24$.
Let L be a positive definite even lattice of rank one and V_L^+ be the fixed points of the lattice VOA V_L associated to L under an automorphism of V_L lifting the -1$ isometry of L. A set of generators and the full automorphism group of V_L^+ are determined.
In this paper, we completely determine the group of algebra automorphisms for the two-parameter Hopf algebra ${\check U}_{r,s}^{\geq 0}({\mathfrak sl_{3}})$. As a result, the group of Hopf algebra automorphisms is determined for $\V$ as well. We further characterize all the derivations of the subalgebra $U^{+}_{r,s}({\mathfrak sl_{3})}$, and calculate its first degree Hochschild cohomology group.
Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.
Let X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra $H=kM\cobicross k(G)$ is itself a bicrossproduct $kX\cobicross k(Y)$ associated to a group YX, where $Y=G\times M^{op}$. This provides a class of bicrossproduct Hopf algebras which are quasitriangular. We also construct a subgroup $Y^θX^θ$ associated to every order-reversing automorphism $θ$ of X. The corresponding Hopf algebra $kX^θ\cobicross k(Y^θ)$ has the same coalgebra as H. Using related results, we classify the first order bicovariant differential calculi on H…
The automorphism group of the vertex operator algebra $V_L^+$ is studied by using its action on isomorphism classes of irreducible $V_L^+$-modules. In particular, the shape of the automorphism group of $V_L^+$ is determined when $L$ is isomorphic to an even unimodular lattice without roots, $\sqrt2R$ for an irreducible root lattice $R$ of type $ADE$ and the Barnes-Wall lattice of rank 16.
Let $\mathbb{F}$ be a field of characteristic 0, $G$ an additive subgroup of $\mathbb{F}$, $α\in \mathbb{F}$ satisfying $α\notin G, 2α\in G$. We define a class of infinite-dimensional Lie algebras which are called generalized Schrödinger-Virasoro algebras and use $\mathfrak{gsv}[G,α]$ to denote the one corresponding to $G$ and $α$. In this paper the automorphism group and irreducibility of Verma modules for $\mathfrak{gsv}[G,α]$ are completely determined.
An algebraic analysis framework for quantum calculus is proposed. The quantum derivative operator $D_{τ,σ}$ is based on two commuting bijections $τ$ and $σ$ defined on an arbitrary set $M$ equipped with a tension structure determined by a single tension function $θ$, i.e. a 1-dimensional case is analyzed here. The well known cases, i.e. $h$- and $q$-calculi together with their symmetric versions, can be obtained owing to special choice of mappings $τ$ and $σ$.
Automorphisms of algebras $R$ from a very large axiomatic class of quantum nilpotent algebras are studied using techniques from noncommutative unique factorization domains and quantum cluster algebras. First, the Nakayama automorphism of $R$ (associated to its structure as a twisted Calabi-Yau algebra) is determined and shown to be given by conjugation by a normal element, namely, the product of the homogeneous prime elements of $R$ (there are finitely many up to associates). Second, in the case when $R$ is connected graded, the unipotent automorphisms of $R$ are classified up to minor excepti…
We recall the emergence of a generalized gauge theory from a noncommutative Riemannian spin manifold, viz. a real spectral triple $(A,H,D;J)$. This includes a gauge group determined by the unitaries in the $*$-algebra $A$ and gauge fields arising from a so-called perturbation semigroup which is associated to $A$. Our main new result is the interpretation of this generalized gauge theory in terms of an upper semi-continuous $C^*$-bundle on a (Hausdorff) base space $X$. The gauge group acts by vertical automorphisms on this $C^*$-bundle and can (under some mild conditions) be identified with the…
We construct noncommutative `Riemannian manifold' structures on dual quasitriangular Hopf algebras such as $C_q[SU_2]$ with its standard bicovariant differential calculus, using the quantum frame bundle formalism introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the $n$-bein by the Maurer-Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras $C[G]$ with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group $C[S_3]$ is…
The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the a…
We study super-braided Hopf algebras $Λ$ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules $Λ^1$ over a Hopf algebra $A$ which are quotients of the augmentation ideal $A^+$ under right multiplication and the adjoint coaction. Here super-bosonisation $Ω=A\ltimesΛ$ provides a bicovariant differential graded algebra on $A$. We introduce $Λ_{max}$ providing the maximal prolongation, while the canonical braided-exterior algebra $Λ_{min}=B_-(Λ^1)$ provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator $\sharp$ by sup…
The restricted $S$-matrix of $V^G$ is determined for any regular vertex operator algebra $V$ and finite automorphism group $G$ of $V.$ As an application, the $S$-matrices for cyclic permutation orbifolds of prime orders are computed.
Let $V$ be a simple vertex operator superalgebra and $G$ a finite automorphism group of $V$ containing the canonical automorphism $σ$ such that $V^G$ is regular. It is proved that every irreducible $V^G$-module occurs in an irreducible $g$-twisted $V$-module for some $g\in G$ and the irreducible $V^G$-modules are classified. Moreover, the quantum dimensions of irreducible $V^G$-modules are determined, a global dimension formula for $V$ in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition, the $S$-matrix of $V^G$ is computed
For the algebra $A$ in the title, its prime, primitive and maximal spectra are classified. The group of automorphisms of $A$ is determined. The simple unfaithful $A$-modules and the simple weight $A$-modules are classified.
Asymmetric graphs with quantum symmetry
本文首次构造出经典自同构平凡但量子自同构非平凡的无限图序列，并证明每个有限经典群都可实现为某有限图的量子自同构群，从而否定“量子排除群”的存在。
Quantum Symmetries of Vertex-Transitive Graphs on 12 Vertices
本文扩展了图的量子自同构群研究，确定了所有12顶点顶点传递图的量子对称性，并明确给出了大多数图的量子自同构群，填补了该领域顶点数研究的空白。
We show that many noetherian Hopf algebras A have a rigid dualising complex R with R isomorphic to ^νA^1 [d]. Here, d is the injective dimension of the algebra and νis a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite dimensional theory which is hereby generalised we call νthe Nakayama automorphism of A. We prove that ν= S^2\XXi, where S is the antipode of A and \XXi is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are s…
Complete Structural Analysis of $q$-Heisenberg Algebras: Homology, Rigidity, Automorphisms, and Deformations
本文研究了$q$-Heisenberg代数的结构性质，包括同调、刚性、自同构与变形，揭示其在量子变形和表示论中的重要贡献。
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