Searcharxiv⌕ Search

arXiv subjects

Thomas Kerler

Publications and source records attributed to Thomas Kerler.

17 recordsLinked to original sources
arXiv2025

Hopf Ideals, Integrality, and Automorphisms of Quantum Groups at Roots of 1

We consider skew-commutative subalgebras in Drinfeld-Jimbo quantum groups at a root of unity $\zeta$ generated by primitive power elements. We classify the centrality and commutativity of these skew-polynomial algebras depending on the Lie type and the order of $\zeta$ modulo 8. We describe Hopf ideals in the quantum group induced from ideals in these subalgebras, including the non-commutative cases. Among these, we construct and analyze a family of Hopf ideals that depend on the choice of an element in the Weyl group. We show that they arise naturally both in the construction of (partial) $R$-matrices and as vanishing ideals of Bruhat subgroups. Specialization to the maximal element yields a rigorous construction of restricted quantum groups as pre-triangular Hopf algebras, independent of any choices. Our treatment also includes even orders of $\zeta$, non-simply laced Lie types, and minimal ground rings. Consequently, we extend some results of De Concini-Kac-Procesi, whose work focuses on odd orders of $\zeta$, which forces the subalgebra to be strictly central, and complex ground fields. This includes the identification of the subalgebras for Lie types $\mathsf{A}_n$ and $\mathsf{B}_2$ with the coordinate rings of associated algebraic groups in the commutative cases, even if $\zeta$ has even order. Our descriptions are computationally explicit and do not utilize Poisson structures. As technical preparations, we discuss PBW bases over minimal rings, dependencies on choices of convex orderings, as well as various new constructions of, and relations among, automorphisms on quantum groups. The latter include formulae for the Garside element in the Lustzig-Artin group action and the family of Che-transformations.

math.QA↗
arXiv2014

Random Walk Invariants of String Links from R-Matrices

We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting properties of these representations of the string link monoids are discussed.

math.GT↗
arXiv2013

The Lawrence-Krammer-Bigelow Representations of the Braid Groups via Quantum SL_2

We construct representations of the braid groups B_n on n strands on free Z[q,q^-1,s,s^-1]-modules W_{n,l} using generic Verma modules for an integral version of quantum sl_2. We prove that the W_{n,2} are isomorphic to the faithful Lawrence Krammer Bigelow representations of B_n after appropriate identification of parameters of Laurent polynomial rings by constructing explicit integral bases and isomorphism. We also prove that the B_n-representations W_{n,l} are irreducible over the fractional field Q (q,s).

math.GT↗
arXiv2013

Integrality and Gauge Dependence of Hennings TQFTs

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbf{k}$, starting from a finite Hopf algebra over $\mathbf{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds. We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotoic integers for surfaces with boundary. We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element. These results are combined and applied to show that the Hennings invariant associated to quantum-$sl_2$ takes values in the cyclotomic integers. Using prior results of Chen et al we infer integrality also of the Witten-Reshetikhin-Turaev $SO(3)$ invariant for rational homology spheres. As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras.

math.GT↗
arXiv2007

Higher Rank TQFT Representations of SL(2,Z) are Reducible

In this article we give examples which show that the TQFT representations of the mapping class groups derived from quantum SU(N) for N>2 are generically decomposable. One general decomposition of the representations is induced by the symmetry which exchanges SU(N) representation labels by their conjugates. The respective summands of a given parity are typically still reducible into many further components. Specifically, we give an explicit basis for an irreducible direct summand in the SL(2,Z) representation obtained from quantum PSU(3) when the order of the root of unity is a prime r=2 mod 3. We show that this summand is isomorphic to the respective PSU(2) representation.

math.GT↗
arXiv2002

Resolutions of p-Modular TQFT's and Representations of Symmetric Groups

We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to the p-modular representation theory of the symmetric groups S_n. We construct resolutions of simple p-modular TQFT's and S_n-modules over Z/pZ using powers of Lefschetz operators. These yield expansions of the irreducible p-modular S_n-characters in terms of the ordianry ones as well as expressions for the Alexander Polynomial of a 3-manifold evaluated at a p-th root of unity in terms of traces in the irreducible modular TQFT's over covering cobordisms. Together with identifications with the constant orders of quantum TQFT's this results, e.g., in non-trivial criteria for a group to be a 3-manifold group.

math.GT↗
arXiv2002

Homology TQFT's and the Alexander-Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory

We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the intersection homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra ${\cal N}=\Z/2\ltimes \ext *\R^2$ on the other side. We find that both TQFT's are $SL(2,\R)$-equivariant functors and also as such isomorphic. The $SL(2,\R)$-action in the Hennings construction comes from the natural action on $\cal N$ and in the case of the Frohman-Nicas theory from the Hard-Lefschetz decomposition of the U(1)-moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of $SL(2,\Z)$ and $Sp(2g,\Z)$, thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg-Witten theories, Casson type theories for homology circles á la Donaldson, higher rank gauge theories following Frohman and Nicas, and the $\Z/r$ reductions of Reshetikhin-Turaev theories over the cyclotomic integers $\Z[ζ_r]$. We also conjecture that the Hennings TQFT for quantum-${\mathfrak sl}_2$ is the product of the Reshetikhin-Turaev TQFT and such a homological TQFT.

math.GT↗
arXiv2001

p-Modular TQFT's and Torsion

We give an overview over several constructions of TQFT's over finite fields and cyclotomic integers and their applications to characterizing 3-manifolds and their fundamental groups.

math.GT↗
arXiv2001

Towards an Algebraic Characterization of 3-dimensional Cobordisms

The goal of this paper is to find a close to isomorphic presentation of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3-dimensional topology. The first is the category \Cob of connected surfaces with one boundary component and 3-dimensional relative cobordisms, the second is a category \Tgl of tangles with relations, and the third is a natural algebraic category \Alg freely generated by a Hopf algebra object. From previous work we know that {\Tgl} and \Cob are equivalent. We use this fact and the idea of Heegaard splittings to construct a surjective functor from \Alg onto {\Cob}. We also find a map that associates to the generators of the mapping class group in \Cob preimages in \Alg. The single block relations in the mapping class group are verified for these expressions. We propose to find a version of \Alg with possibly additional relations to obatin isomorphic algebraic presentations of the mapping class groups and eventually of \Cob.

math.GT↗
arXiv1998

Equivalence of a Bridged Link Calculus and Kirby's Calculus on Non-Simply Connected 3-Manifolds

We recall an extension of Kirby's Calculus on non-simply connected 3-manifolds given in [FR], and the surgery calculus of bridged links from [Ke], which involves only local moves. We give a short combinatorial proof that the two calculi are equivalent, and thus describe the same classes of 3-manifolds. This makes the proofs for the validity of surgery calculi in [FR] and [Ke] interchangeable.

math.GT↗
arXiv1998

Bridged Links and Tangle Presentations of Cobordism Categories

We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is valid also on non-simply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3-manifold by doing surgery on any other 3-manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerf-theoretical derivation of presentations of 2+1-dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the Matveev-Polyak presentations of the mapping class group, and, furthermore, find systematically surgery presentations of general mapping tori. We discuss a natural extension of the Reshetikhin Turaev invariant to the calculus of bridged links. Invariance follows now - similar as for knot invariants - from simple identifications of the elementary moves with elementary categorial relations for invariances or cointegrals, respectively. Hence, we avoid the lengthy computations and the unnatural Fenn-Rourke reduction of the original proofs. Moreover, we are able to start from a much weaker ``modularity''-condition, which implies the one of Turaev. Generalizations of the presentation to cobordisms of surfaces with boundaries are outlined.

math.GT↗
arXiv1997

Integrals for braided Hopf algebras

Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object Int H is invertible. The fully braided version of Radford's formula for the fourth power of the antipode is obtained. Connections of integration with cross-product and transmutation are studied. The results apply to topological Hopf algebras, e.g. a torus with a hole, which do not have additive structure.

q-alg↗
arXiv1996

Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family

We study the relations between the invariants $τ_{RT}$, $τ_{HKR}$, and $τ_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko, respectively. In particular, we discuss explicitly how $τ_L$ specializes to $τ_{RT}$ for semisimple categories and to $τ_{HKR}$ for Tannakian categories. We give arguments for that $τ_L$ is the most general invariant that stems from an extended TQFT. We introduce a canonical, central element, {\sf Q}, for a quasi-triangular Hopf algebra, $\A$, that allows us to apply the Hennings algorithm directly, in order to compute $τ_{RT}$, which is originally obtained from the semisimple trace-subquotient of $\A-mod$. Moreover, we generalize Hennings' rules to the context of cobordisms, in order to obtain a TQFT for connected surfaces compatible with $τ_{HKR}\,$. As an application we show that, for lens spaces and $\A=U_q(sl_2)\,$, the ratio of $τ_{HKR}$ and $τ_{RT}$ is the order of the first homology group. In the course of this paper we also outline the topology and the algebra that enter invariance proofs, which contain no reference to 2-handle slides, but to other moves that are local. Finally, we give a list of open questions regarding cellular invariants, as defined by Turaev-Viro, Kuperberg, and others, their relations among each other, and their relations to the surgical invariants from above.

q-alg↗
arXiv1996

On the Connectivity of Cobordisms and Half-Projective TQFT's

We consider a generalization of the axioms of a TQFT, so called half-projective TQFT's, with an anomaly, $x^μ$, in the composition law. $μ$ is a coboundary on the cobordism categories with non-negative, integer values. The element $x$ of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be 0. This modification makes it possible to extend quantum-invariants, which vanish on $S^1\times S^2$, to non-trivial TQFT's. (A TQFT in the sense of Atiyah with this property has to be trivial all together). Under a few natural assumptions the notion of a half-projective TQFT is shown to be the only possible generalization. Based on separate work with Lyubashenko on connected TQFT's, we construct a large class of half-projective TQFT's with $x=0$. Their invariants vanish on $S^1\times S^2$, and they coincide with the Hennings invariant for non-semisimple Hopf algebras. Several toplogical tools that are relevant for vanishing properties of such TQFT's are developed. They are concerned with connectivity properties of cobordisms, as for example maximal non-separating surfaces. We introduce in particular the notions of ``interior'' homotopy and homology groups, and of coordinate graphs, which are functions on cobordisms with values in the morphisms of a graph category. For applications we will prove that half-projective TQFT's with $x=0$ vanish on cobordisms with infinite interior homology, and we argue that the order of divergence of the TQFT on a cobordism in the ``classical limit'' can be estimated by the rank of its maximal free interior group.

q-alg↗
arXiv1994

On braided tensor categories

We investigate invertible elements and gradings in braided tensor categories. This leads us to the definition of theta-, product-, subgrading and orbitcategories in order to construct new families of BTC's from given ones. We use the representation theory of Hecke algebras in order to relate the fusionring of a BTC generated by an object $X$ with a two component decomposition of its tensorsquare to the fusionring of quantum groups of type $A$ at roots of unity. We find the condition of `local isomorphie' on a special fusionring morphism implying that a BTC is obtained from the above constructions applied to the semisimplified representation category of a quantum group. This family of BTC's contains new series of twisted categories that do not stem from known Hopf algebras. Using the language of incidence graphs and the balancing structure on a BTC we also find strong constraints on the fusionring morphism. For Temperley Lieb type categories these are sufficient to show local isomorphie. Thus we obtain a classification for the subclass of Temperley Lieb type categories.

hep-th↗
arXiv1994

Mapping Class Group Actions on Quantum Doubles

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for the ${\cal S}^{\pm 1}$-matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projective $SL(2, Z)$-action on the center of $U_q(sl_2)$ for $q$ an $l=2m+1$-st root of unity. It appears that the $3m+1$-dimensional representation decomposes into an $m+1$-dimensional finite representation and a $2m$-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of $SL(2, Z)$ and the finite, $m$-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category of $U_q(sl_2)\,$.

hep-th↗