Twisted Hopf Algebra Research Articles

Invariants from the Sweedler power maps on integrals

For a finite-dimensional Hopf algebra A with a nonzero left integral Λ, we investigate a relationship between Pn(Λ) and PnJ(Λ), where Pn and PnJ are respectively the n-th Sweedler power maps of A and the twisted Hopf algebra AJ. We use this relation to give several invariants of the representation category Rep(A) considered as a tensor category. As applications, we distinguish the representation categories of 12-dimensional pointed nonsemisimple Hopf algebras. Also, these invariants are sufficient to distinguish the representation categories Rep(K8), Rep(kQ8) and Rep(kD4), although they have been completely distinguished by their Frobenius-Schur indicators. We further reveal a relationship between the right integrals λ in A⁎ and λJ in (AJ)⁎. This can be used to give a uniform proof of the remarkable result which says that the n-th indicator νn(A) is a gauge invariant of A for any n∈Z. We also use the expression for λJ to give an alternative proof of the known result that the Killing form of the Hopf algebra A is invariant under twisting. As a result, the dimension of the Killing radical of A is a gauge invariant of A.

Twisting of affine algebraic groups, II

Abstract We use [11] to study the algebra structure of twisted cotriangular Hopf algebras ${}_J\mathcal{O}(G)_{J}$, where $J$ is a Hopf $2$-cocycle for a connected nilpotent algebraic group $G$ over $\mathbb{C}$. In particular, we show that ${}_J\mathcal{O}(G)_{J}$ is an affine Noetherian domain with Gelfand–Kirillov dimension $\dim (G)$, and that if $G$ is unipotent and $J$ is supported on $G$, then ${}_J\mathcal{O}(G)_{J}\cong U({\mathfrak{g}})$ as algebras, where ${\mathfrak{g}}={\textrm{Lie}}(G)$. We also determine the finite dimensional irreducible representations of ${}_J\mathcal{O}(G)_{J}$, by analyzing twisted function algebras on $(H,H)$-double cosets of the support $H\subset G$ of $J$. Finally, we work out several examples to illustrate our results.

Calabi–Yau property under monoidal Morita–Takeuchi equivalence

Let $H$ and $L$ be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if $H$ is a twisted Calabi-Yau (CY) Hopf algebra, then $L$ is a twisted CY algebra when it is homologically smooth. Especially, if $H$ is a Noetherian twisted CY Hopf algebra and $L$ has finite global dimension, then $L$ is a twisted CY algebra.

HOMOLOGICAL DIMENSION OF SMASH PRODUCT OVER QUASITRIANGULAR WEAK HOPF ALGEBRA

Let (H, R) be a quasitriangular weak Hopf algebra, and A a quantum commutative weak H-module algebra. We establish the relationship of homological dimensions between weak smash product algebra A#H and A under some conditions. As an application, we consider the case of twisted weak Hopf algebra.

Towers of graded superalgebras categorify the twisted Heisenberg double

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double, introduced in [9], and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilCoxeter graded superalgebras satisfy our axioms. In the latter case, one obtains a categorification of the quantum Weyl algebra.

Twisted Heisenberg Doubles

We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted Hopf pairing. We state a Stone--von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction.

An Unfolded Quantization for Twisted Hopf Algebras

In this talk I discuss a recently developed "Unfolded Quantization Framework". It allows to introduce a Hamiltonian Second Quantization based on a Hopf algebra endowed with a coproduct satisfying, for the Hamiltonian, the physical requirement of being a primitive element. The scheme can be applied to theories deformed via a Drinfel'd twist. I discuss in particular two cases: the abelian twist deformation of a rotationally invariant nonrelativistic Quantum Mechanics (the twist induces a standard noncommutativity) and the Jordanian twist of the harmonic oscillator. In the latter case the twist induces a Snyder non-commutativity for the space-coordinates, with a pseudo-Hermitian deformed Hamiltonian.The "Unfolded Quantization Framework" unambiguously fixes the non-additive effective interactions in the multi-particle sector of the deformed quantum theory. The statistics of the particles is preserved even in the presence of a deformation.

Noncommutative geometry in string and twisted Hopf algebra of diffeomorphism

We discuss the Hopf algebra structure in string theory and present the twist quantization as a unified formulation of the world sheet quantization of the string and the symmetry of the target spacetime. Applying it to the case with a nonzero B-field background, we explain a method to decompose the twist into two successive twists. There are two different possibilities of decomposition: The first is a natural decomposition from the viewpoint of the twist quantization, leading to a new type of twisted Poincare symmetry. The second decomposition reveals the relation of our formulation to the twisted Poincare symmetry on the Moyal type noncommutative space.

Hall basis of twisted Lie algebras

In this paper we define a minimal generating system for the free twisted Lie algebra. This gives a correct formulation and a proof to an old statement of Barratt. To this aim we use properties of the Lyndon words and of the Klyachko idempotent which generalize to twisted Hopf algebras some similar results well known in the classical case.

Constructing Quasitriangular Multiplier Hopf Algebras By Twisted Tensor Coproducts

Let A and B be multiplier Hopf algebras, and let R ∈ M(B ⊗ A) be an anti-copairing multiplier, i.e, the inverse of R is a skew-copairing multiplier in the sense of Delvaux [5]. Then one can construct a twisted tensor coproduct multiplier Hopf algebra A ⊗ R B. Using this, we establish the correspondence between the existence of quasitriangular structures in A ⊗ R B and the existence of such structures in the factors A and B. We illustrate our theory with a profusion of examples which cannot be obtained by using classical Hopf algebras. Also, we study the class of minimal quasitriangular multiplier Hopf algebras and show that every minimal quasitriangular Hopf algebra is a quotient of a Drinfel’d double for some algebraic quantum group.

Twist quantization of string andBfield background

In a previous paper, we investigated the Hopf algebra structure in string theory and gave a unified formulation of the quantization of the string and the space-time symmetry. In this paper, this formulation is applied to the case with a nonzero B-field background, and the twist of the Poincare symmetry is studied. The Drinfeld twist accompanied by the B-field background gives an alternative quantization scheme, which requires a new normal ordering. In order to obtain a physical interpretation of this twisted Hopf algebra structure, we propose a method to decompose the twist into two successive twists and we give two different possibilities of decomposition. The first is a natural decomposition from the viewpoint of the twist quantization, leading to a new type of twisted Poincare symmetry. The second decomposition reveals the relation of our formulation to the twisted Poincare symmetry on the Moyal type noncommutative space.

Wigner oscillators, twisted Hopf algebras, and second quantization

By correctly identifying the role of the central extension in the centrally extended Heisenberg algebra h, we show that it is indeed possible to construct a Hopf algebraic structure on the corresponding enveloping algebra U(h) and eventually deform it through the Drinfeld twist. This Hopf algebraic structure and its deformed version UF(h) are shown to be induced from a more “fundamental” Hopf algebra obtained from the Schrödinger field/oscillator algebra and its deformed version provided that the fields/oscillators are regarded as odd elements of a given superalgebra. We also discuss the possible implications in the context of quantum statistics.

Noether analysis of the twisted Hopf symmetries of canonical noncommutative spacetimes

We study the twisted Hopf-algebra symmetries of observer-independent canonical spacetime noncommutativity, for which the commutators of the spacetime coordinates take the form [x{sup {mu}},x{sup {nu}}]=i{theta}{sup {mu}}{sup {nu}} with observer-independent (and coordinate-independent) {theta}{sup {mu}}{sup {nu}}. We find that it is necessary to introduce nontrivial commutators between transformation parameters and spacetime coordinates, and that the form of these commutators implies that all symmetry transformations must include a translation component. We show that with our noncommutative transformation parameters the Noether analysis of the symmetries is straightforward, and we compare our canonical-noncommutativity results with the structure of the conserved charges and the ''no-pure-boost'' requirement derived in a previous study of {kappa}-Minkowski noncommutativity. We also verify that, while at intermediate stages of the analysis we do find terms that depend on the ordering convention adopted in setting up the Weyl map, the final result for the conserved charges is reassuringly independent of the choice of Weyl map and (the corresponding choice of) star product.

κ-deformed spacetime from twist

We twist the Hopf algebra of igl(n,R) to obtain the κ-deformed spacetime coordinates. Coproducts of the twisted Hopf algebras are explicitly given. The κ-deformed spacetime obtained this way satisfies the same commutation relation as that of the conventional κ-Minkowski spacetime, but its Hopf algebra structure is different from the well-known κ-deformed Poincaré algebra in that it has larger symmetry algebra than the κ-Minkowski case. There are some physical models which consider this symmetry [R. Percacci, Phys. Lett. B 144 (1984) 37; R. Percacci, Geometry of Nonlinear Field Theories, World Scientific, Singapore, 1986; L. Smolin, Nucl. Phys. B 132 (1978) 138; R. Floreanini, R. Percacci, Class. Quantum Grav. 7 (1990) 975]. Incidentally, we obtain the canonical (θ-deformed) non-commutative spacetime from canonically twisted igl(n,R) Hopf algebra.

Deformed Gauge Theories

Gauge theories are studied on a space of functions with the Moyal-Weyl product. The development of these ideas follows the differential geometry of the usual gauge theories, but several changes are forced upon us. The Leibniz rule has to be changed such that the theory is now based on a twisted Hopf algebra. Nevertheless, this twisted symmetry structure leads to conservation laws. The symmetry has to be extended from Lie algebra valued to enveloping algebra valued and new vector potentials have to be introduced. As usual, field equations are subjected to consistency conditions that restrict the possible models. Some examples are studied.

Trees, set compositions and the twisted descent algebra

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). Among others, a number of enveloping algebra structures are introduced and studied in detail. For example, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.

LORENTZ INVARIANT AND SUPERSYMMETRIC INTERPRETATION OF NONCOMMUTATIVE QUANTUM FIELD THEORY

In this paper, using a Hopf-algebraic method, we construct deformed Poincaré SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed super-Poincaré algebra as our fundamental symmetry, we can see the consistency between the algebra and non(anti)commutative relation among (super)coordinates and interpret that symmetry of non(anti)commutative QFT is in fact twisted one. The key point is validity of our new twist element that guarantees non(anti)commutativity of space. It is checked in this paper for [Formula: see text] case. We also comment on the possibility of noncommutative central charge coordinate. Finally, because our twist operation does not break the original algebra, we can claim that (twisted) SUSY is not broken in contrast to the string inspired [Formula: see text] SUSY in [Formula: see text] non(anti)commutative superspace.

The module structure of the Solomon–Tits algebra of the symmetric group

Let ( W , S ) be a finite Coxeter system. Tits defined an associative product on the set Σ of simplices of the associated Coxeter complex. The corresponding semigroup algebra is the Solomon–Tits algebra of W. It contains the Solomon algebra of W as the algebra of invariants with respect to the natural action of W on Σ. For the symmetric group S n , there is a 1–1 correspondence between Σ and the set of all set compositions (or ordered set partitions) of { 1 , … , n } . The product on Σ has a simple combinatorial description in terms of set compositions. We study in detail the representation theory of the Solomon–Tits algebra of S n over an arbitrary field, and show how our results relate to the corresponding results on the Solomon algebra of S n . This includes the construction of irreducible and principal indecomposable modules, a description of the Cartan invariants, of the Ext-quiver, and of the descending Loewy series. Our approach builds on a (twisted) Hopf algebra structure on the direct sum of all Solomon–Tits algebras.

Skew differential operator algebras of twisted Hopf algebras

By introducing a twisted Hopf algebra we unify several important objects of study. Skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied. This generalizes results in the Weyl algebra. Applying this investigation to the twisted Ringel–Hall algebra we get, in particular, a natural realization of the non-positive part of a quantized generalized Kac–Moody algebra, by identifying the canonical generators with some linear, skew differential operators. This also induces some algebras which are quantum-group like.

Twisted Hopf Algebras, Ringel–Hall Algebras, and Green's Categories: With an appendix by the referee

The concept and some basic properties of a twisted Hopf algebra are introduced and investigated. Its unique difference from a Hopf algebra is that the comultiplication δ: A→A⊗A is an algebra homomorphism, not for the componentwise multiplication on A⊗A, but for the twisted multiplication on A⊗A given by Lusztig's rule.Also, it is proved that any object A in Green's category has a twisted Hopf algebra structure, any morphism between objects is a twisted Hopf algebra homomorphism, the antipode s of A is self-adjoint under the Lusztig form (−,−) on A, and the Green polynomials Ma,b(t) share a so-called cyclic-symmetry.As examples, the twisted Ringel–Hall algebras, Ringel's twisted composition algebras, Lusztig's free algebras ′F and non-degenerate algebras F, the positive part U+ of the Drinfeld–Jimbo quantized enveloping algebras U, and Rosso's quantum shuffle algebra T(V) all are twisted Hopf algebras. The antipode and its inverse for a twisted Ringel–Hall are explicitly given, and all δ-primitive elements are determined.