## at.algebraic topology – Uses for (Framed) E2 algebras twisted by braided monoidal structure

If $$C$$ is a monoidal category (not necessarily a symmetric monoidal category), it’s possible to define the notion of an algebra object $$A$$ in $$C$$, with multiplication operations $$A^{otimes n} (:= Aotimes_C Aotimes_C cdotsotimes_C A)to A.$$

Similarly, if $$C$$ is a braided monoidal category (resp., a ribbon category), one can define a notion of $$E_2$$ DG algebra $$A$$ (resp., framed $$E_2$$ DG algebra $$A$$) “twisted” by $$C$$, consisting of operations $$A^{otimes n}to A$$ compatible with braiding. (Note: I actually don’t know a reference for this, but it follows from standard “homotopy field theory” arguments involving the Ran space.)

In particular, if $$C$$ is a braided monoidal (or ribbon) category coming from an associator on a Lie algebra $$g$$ (with choice of Casimir), there is a whole category of “associator-twisted” $$g$$-equivariant $$E_2$$ (resp., framed $$E_2$$) algebras.

My question is whether algebras of this type have been encountered before. They feel very CFT-ish, and so I’m particularly curious about physics and knot theory applications. In particular, the framed variant should gove some kind of derived 2D TQFT-style invariants.

Any references would be useful. Thanks!

## ct.category theory – Can a braided fusion category have an order-2 Morita equivalence which has no representative that is both connected and isomorphic to its opposite?

Let $$mathcal{B}$$ be a braided fusion category over $$mathbb{C}$$. Let me write $$mathrm{Alg}(mathcal{B})$$ for the set of isomorphism classes of unital associative algebra objects in $$mathcal{B}$$, and $$mathrm{Mor}(mathcal{B}) = mathrm{Alg}(mathcal{B})/sim_M$$ for the set of Morita equivalence classes. These sets are naturally associative (but usually noncommutative) monoids under the tensor product operation. (The formula for multiplication in a tensor product of algebra objects explicitly references the braiding in $$mathcal{B}$$.)

An algebra $$X in mathrm{Alg}(mathcal{B})$$ is called connected if the unit object $$1inmathcal{B}$$ appears with multiplicity $$1$$ in the direct sum decomposition of (the underlying object of) $$mathcal{B}$$. Every simple separable algebra is Morita equivalent to a connected one: given a separable algebra $$X in mathrm{Alg}(mathcal{B})$$, you choose a simple $$X$$-module, and consider its algebra of endomorphisms.

Every algebra object $$X in mathrm{Alg}(mathcal{B})$$ has an opposite algebra $$X^{mathrm{op}}$$ — the formula for multiplication in $$X^{mathrm{op}}$$ uses the braiding in $$mathcal{B}$$. An algebra $$X$$ is called Azumaya if its Morita equivalence class $$(X) in mathrm{Mor}(mathcal{B})$$ is invertible. If $$X$$ is Azumaya, then $$(X)^{-1}$$ is the Morita-equivalence class of $$X^mathrm{op}$$.

I am interested in Azumaya algebras which are Morita self-dual in the sense that $$(X) = (X)^{-1} in mathrm{Mor}(mathcal{B})$$, or in other words there exists a Morita equivalence $$X sim_M X^{mathrm{op}}$$. One way this can occur, of course, is if $$X$$ is isomorphic as an algebra to $$X^{mathrm{op}}$$. Let’s call such an algebra antiautomorphic. Every Morita self-dual algebra is Morita equivalent to an antiautomorphic one: what you do is to look at the category of $$X$$-modules; to take a direct sum of all simple $$X$$-modules, with each isomorphism class appearing with multiplicity one; and take the endomorphism algebra of this sum.

This universal representative, however, will typically fail to be connected. On the other hand, I don’t know very many examples, and in the examples I do know, I can always find an antiautomorphic connected representative of each self-dual Azumaya class.

What is an example of a braided fusion category $$mathcal{B}$$ and an order-2 element in $$mathrm{Mor}(mathcal{B})$$ which has no representative that is simultaneously connected and antiautomorphic?

## qa.quantum algebra – Reference requests : Presentation of the braided dual of \$U_q(frak{sl_2})\$

I am interested in the braided dual of the quantum group $$U_q(frak{sl_2})$$. This is the algebra generated by matric coefficients but where the the multiplications is twisted by an action of the $$R$$-matrix. I have seen (for example in https://arxiv.org/pdf/1908.05233.pdf example 1.23) that it is isomorphic to the algebra generated by elements $$a^1_1, a^1_2, a^2_1$$ and $$a^2_2$$ together with the relations :

$$a^1_2 a^1_1 = a^1_1 a^1_2 + ( 1-q^{-2})a^1_2a^2_2$$
$$a^2_1 a^1_1 = a^1_1 a^2_1 – ( 1-q^{-2})a^2_2a^2_1$$
$$a^2_1 a^1_2 = a^1_2 a^2_1 + ( 1-q^{-2})(a^1_1a^2_2 -a^2_2a^2_2)$$
$$a^2_2a^1_1 = a^1_1a^2_2$$
$$a^2_2a^1_2 = q^2 a^1_2a^2_2$$
$$a^2_2a^2_1 = q^{-2} a^2_1a^2_2$$
$$a^1_1a^2_2 = 1 -q^{-2}a^1_2a^2_1$$

If $$V$$ is the stantard representation of $$U_q(frak{sl_2})$$ and we set $$a^i_j := v^i otimes v_j$$, I can see that those elements indeed generate the whole algebra, but I don’t know if there is more relations needed. According to the litterature this is enough, but I can not find of proof of this.

## reference request – Definition of Braided dual of a Hopf Algebra

I have been reading the paper “Integrating Quantum Groups over Surfaces” by Jordan, Brochier and Ben-Zvi.

At the page 42 they talk about the braided dual algebra $$tilde{H}$$ to a quasi-triangular Hopf algebra $$H$$ but I can’t find a definition of this, even in the references they give in the paper.

Any reference to this would be welcomed.

## Representation theory in braided monoidal categories

The crux of what I wish to know is what results from representation theory, a subject usually framed within the category $$text{Vect}_mathbb{k}$$, follow in more general braided monoidal categories? I would be very satisfied with references to texts that cover this.

I’ll try to be more specific. Let $$mathcal{C}$$ be monoidal, abelian, complete under arbitrary countable biproducts, and enriched over $$text{Vect}_mathbb{C}$$. An algebra in $$mathcal{C}$$ is an object $$A$$ with morphisms $$m:Aotimes Arightarrow A, u:1rightarrow A$$ satisfying suitable axioms, and a left $$A$$-module in $$mathcal{C}$$ is an object $$V$$ with morphism $$a_V:Aotimes Vrightarrow V$$, again satifying some conditions. If $$mathcal{C}$$ has a braiding $$psi$$, we can define commutative algebras as those algebras such that $$mpsi=m$$.

For instance take $$mathcal{C}$$ to be $$Htext{-Mod}$$, the category of (finite-dimensional) modules over quasitriangular Hopf algebra $$H$$ (the quasitriangular structure on $$H$$ makes $$Htext{-Mod}$$ braided monoidal). Recall the following classical (i.e. in $$text{Vect}_mathbb{C}$$) result from representation theory: for commutative algebra $$A$$, every simple finite-dim’l $$A$$-module is $$1$$-dim’l. Is there an analogue of this statement for braided commutative algebras in $$Htext{-Mod}$$?

## ct.category theory – Braided category inside braided 2-category

Let $$mathcal{C}$$ be a semistrict braided monoidal $$2$$-category in the sense of (BN) (so in particular a strict $$2$$-category). Let $$mathcal{C}_1$$ be the category of $$1$$-morphisms (objects) and $$2$$-morphisms (morphisms) in $$mathcal{C}$$. (I hope that this is allowed. I think of $$mathcal{C}_1$$ as a coproduct of all the Hom-categories in $$mathcal{C}$$)

The semistrict monoidal structure ($$boxtimes$$) on $$mathcal{C}$$ induces a monoidal structure ($$otimes$$) on $$mathcal{C}_1$$. Let $$R: boxtimes Rightarrow boxtimes^{op}$$ be the pseudo-natural equivalence corresponding to the braiding in $$mathcal{C}$$. For $$1$$-morphisms $$f:a to c$$ and $$g:b to d$$ we can define $$fwidehat{otimes}g:=R_{c,d}circ (f otimes g)$$ and $$gcheck{otimes}f:=(gotimes f)circ R_{a,b}$$ (analogously on morphisms in $$mathcal{C}_1$$). By pseudo-naturality of $$R$$ we have a natural transformation $$c:widehat{otimes} to check{otimes}^{op}$$ in $$mathcal{C}_1$$ satisfying the hexagon equations of a braiding.

So $$mathcal{C}_1$$ looks very similar to a braided monoidal category, with “the only difference” that $$widehat{otimes} neq check{otimes}$$. Now, assume that $$mathcal{C}$$ had enough structure, so that $$mathcal{C}_1$$ is an abelian category. Is there a way of “modding out” the difference of $$widehat{otimes}$$ and $$check{otimes}$$ in $$mathcal{C}_1$$. Maybe by considering some kind of coequalizer? If so, what are the conditions on $$c$$ for being well-defined on the new category?

(BN) Baez, Neuchl – Higher-Dimensional Algebra I: Braided Monoidal 2-Categories

## at.algebraic topology – What is the etale homotopy type of the Witt group of braided fusion categories?

The Witt group $$mathcal {W}$$ braided merger categories (see also the follow-up paper) can be defined in each field; I am happy to limit myself to the characteristics $$0$$ when it matters.

is $$mathbb k mapsto mathcal W ( mathbb k)$$ an (affine?) algebraic group scheme?

Provided $$mathcal W$$ is so schematic that the following question makes sense. What I really want to know is:

What is the etale homotopy type of $$mathcal {W}$$?