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

$newcommand{C}{mathcal{C}}$ $newcommand{g}{mathfrak{g}}$
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} $?