On the adjoint representation of a hopf algebra

We consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{{\textrm ad\,fin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{{\textrm ad\,fin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{{\textrm ad\,fin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.

Deligne Categories and the Limit of Categories Rep(GL(m|n))

For each integer $t$ a tensor category $\mathcal{V}_t$ is constructed, such that exact tensor functors $\mathcal{V}_t\rightarrow \mathcal{C}$ classify dualizable $t$-dimensional objects in $\mathcal{C}$ not annihilated by any Schur functor. This means that $\mathcal{V}_t$ is the “abelian envelope” of the Deligne category $\mathcal{D}_t=\operatorname{Rep}(GL_t)$. Any tensor functor $\operatorname{Rep}(GL_t)\longrightarrow \mathcal{C}$ is proved to factor either through $\mathcal{V}_t$ or through one of the classical categories $\operatorname{Rep}(GL(m|n))$ with $m-n=t$. The universal property of $\mathcal{V}_t$ implies that it is equivalent to the categories $\operatorname{Rep}_{\mathcal{D}_{t_1}\otimes \mathcal{D}_{t_2}}(GL(X),\epsilon )$, ($t=t_1+t_2$, $t_1$ not an integer) suggested by Deligne as candidates for the role of abelian envelope.

Handlebody-knot invariants derived from unimodular Hopf algebras

To systematically construct invariants of handlebody-links, we give a new presentation of the braided tensor category [Formula: see text] of handlebody-tangles by generators and relations, and prove that given what we call a quantum-commutative quantum-symmetric algebra A in an arbitrary braided tensor category [Formula: see text], there arises a braided tensor functor [Formula: see text], which gives rise to a desired invariant. Some properties of the invariants and explicit computational results are shown especially when A is a finite-dimensional unimodular Hopf algebra, which is naturally regarded as a quantum-commutative quantum-symmetric algebra in the braided tensor category [Formula: see text] of Yetter–Drinfeld modules.

REGULAR OBJECTS, MULTIPLICATIVE UNITARIES AND CONJUGATION

The notion of left (respectively right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (respectively right) regular object, it can be interpreted as a category of corepresentations (respectively representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.

Morita context functors

Given a Morita context (R, V, W, S), there are functors W⊗() and hom (V, ) from R-mod to; S-mod and a natural transformation λ from the first to the second. This has an epi-mono factorization and the intermediate functor we denote by ()° with natural transformations and . The tensor functor is exact if and only if WR is flat, whilst the hom functor is exact if and only if RV is projective. We begin by determining conditions under which ()° is exact; this is Theorem 1.