Let s, t be two positive integers and k be an algebraically closed field with char (k)∤st. We show that the Drinfeld double D(⋀st,t*cop) of generalized Taft–Hopf algebra ⋀st,t*cop has ribbon elements ...if and only if t is odd. Moreover, if s is even and t is odd, then D(⋀st,t*cop) has two ribbon elements, and if both s and t are odd, then D(⋀st,t*cop) has only one ribbon element. Finally, we compute explicitly all ribbon elements of D(⋀st,t*cop).
The second author constructed a topological ribbon Hopf algebra from the unrolled quantum group associated with the super Lie algebra sl(2|1). We generalize this fact to the context of unrolled ...quantum groups and construct the associated topological ribbon Hopf algebras. Then we use such an algebra, the discrete Fourier transforms, a symmetrized graded integral and a modified trace to define a modified graded Hennings invariant of 3-manifolds endowed with a cohomology class and which contains a ribbon graph. Finally, we use the notion of a modified integral to extend this invariant to manifolds without ribbon graphs inside and show that it recovers the invariant of 6.
It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, ...we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.
On Bar-Natan–van der Veen’s perturbed Gaussians Becerra, Jorge
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas,
04/2024, Volume:
118, Issue:
2
Journal Article
Peer reviewed
Open access
We elucidate further properties of the novel family of polynomial time knot polynomials devised by Bar-Natan and van der Veen based on the Gaussian calculus of generating series for noncommutative ...algebras. These polynomials determine all coloured Jones polynomials and the simplest of these is expected to coincide with the one-variable 2-loop polynomial. We prove a conjecture stating that half of these polynomials vanish and give concrete formulas for three of these knot polynomial invariants. We also study the behaviour of these polynomials under the connected sum of knots.
For any finite-dimensional factorizable ribbon Hopf algebra H and any ribbon automorphism of H, we establish the existence of the following structure: an H-bimodule Fω and a bimodule morphism Zω from ...Lyubashenkoʼs Hopf algebra object K for the bimodule category to Fω. This morphism is invariant under the natural action of the mapping class group of the one-punctured torus on the space of bimodule morphisms from K to Fω. We further show that the bimodule Fω can be endowed with a natural structure of a commutative symmetric Frobenius algebra in the monoidal category of H-bimodules, and that it is a special Frobenius algebra iff H is semisimple.
The bimodules K and Fω can both be characterized as coends of suitable bifunctors. The morphism Zω is obtained by applying a monodromy operation to the coproduct of Fω; a similar construction for the product of Fω exists as well.
Our results are motivated by the quest to understand the bulk state space and the bulk partition function in two-dimensional conformal field theories with chiral algebras that are not necessarily semisimple.
Quasi-triangular Hopf algebras were introduced by Drinfel’d in his construction of solutions to the Yang–Baxter Equation. This algebra is built upon Uh(sl2), the quantized universal enveloping ...algebra of the Lie algebra sl2. In this paper, combinatorial structure in Uh(sl2) is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case n=1. We apply this formalism to the quasi-triangular Hopf algebras and obtain a constructive account not only for the derivation of the Drinfel’d’s sR-matrix, but also for the arguably mysterious ribbon elements of Uh(sl2). Finally, we extend these techniques to the higher-dimensional algebras Uh(sln+1). While these explicit algebraic results are well known, our contribution is in our formalism and perspective: our emphasis is on the combinatorial structure of these algebras and how that structure may guide algebraic constructions.
Quasi-triangular Hopf algebras were introduced by Drinfel’d in his construction of solutions to the Yang–Baxter Equation. This algebra is built upon
U
h
(
sl
2
)
, the quantized universal enveloping ...algebra of the Lie algebra
sl
2
. In this paper, combinatorial structure in
U
h
(
sl
2
)
is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case
n
=
1
. We apply this formalism to the quasi-triangular Hopf algebras and obtain a constructive account not only for the derivation of the Drinfel’d’s
sR
-matrix, but also for the arguably mysterious ribbon elements of
U
h
(
sl
2
)
. Finally, we extend these techniques to the higher-dimensional algebras
U
h
(
sl
n
+
1
)
. While these explicit algebraic results are well known, our contribution is in our formalism and perspective: our emphasis is on the combinatorial structure of these algebras and how that structure may guide algebraic constructions.
We construct convex PBW-type Lyndon bases for two-parameter quantum groups
U
r
,
s
(
s
o
2
n
+
1
)
with detailed commutation relations. It turns out that under a certain condition, the restricted ...two-parameter quantum group
u
r
,
s
(
s
o
2
n
+
1
)
(
r
,
s
are roots of unity) is a Drinfel’d double. All Hopf isomorphisms of
u
r
,
s
(
s
o
2
n
+
1
)
, as well as
u
r
,
s
(
s
l
n
)
, are determined. Finally, necessary and sufficient conditions for
u
r
,
s
(
s
o
2
n
+
1
)
to be a ribbon Hopf algebra are singled out by describing the left and right integrals.
The modularity of a ribbon Hopf algebra is characterized by the Drinfeld map. An elementary approach to Etingof and Gelaki's (1998, Math. Res. Lett.5, 119–-197) result on the dimensions of ...irreducible modules is given by deducing the necessary identities involving the matrix (Sij) from the well-known orthogonal relations of Hopf algebra characters.
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