Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting operators, i.e.:

It is the universal $C^*$-algebra generated by matrices $u = (u_{ij})$ of generators, such that each $u_{ij}$ is a projection and on each row and column of $u$ these projections are orthogonal and sum up to $1$.

From the point of view of generators and relations the braid group is not that far away from the permutation group. Although it is hard to find faithful representations as was done by Bigelow in this paper. An interesting theorem by Artin also shows that the braid group $B_n$ embeds into Aut($F_n$), i.e. the automorphisms of the free group, as those elements that fix $x_1\cdot \dots \cdot x_n$ (see also this question).

Is there a definition of a quantum braid group? Or is this too difficult due to the non-finiteness of $B_n$? Is there a way to define what a free quantum group would be? (to obtain quantum-$B_n$ as a fixed point algebra)

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting operators, i.e.:

It is the universal $C^*$-algebra generated by matrices $u = (u_{ij})$ of generators, such that each $u_{ij}$ is a projection and on each row and column of $u$ these projections are orthogonal and sum up to $1$.

From the point of view of generators and relations the braid group is not that far away from the permutation group. Although it is hard to find faithful representations as was done by Bigelow in this paper. An interesting theorem by Artin also shows that the braid group $B_n$ embeds into Aut($F_n$), i.e. the automorphisms of the free group, as those elements that fix $x_1\cdot \dots \cdot x_n$ (see also this question).

Is there a definition of a quantum braid group? Or is this too difficult due to the non-finiteness of $B_n$? Is there a way to define what a free quantum group would be? (to obtain quantum-$B_n$ as a fixed point algebra)

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting operators, i.e.:

It is the universal $C^*$-algebra generated by matrices $u = (u_{ij})$ of generators, such that each $u_{ij}$ is a projection and on each row and column of $u$ these projections are orthogonal and sum up to $1$.

From the point of view of generators and relations the braid group is not that far away from the permutation group. Although it is hard to find faithful representations as was done by Bigelow in this paper. An interesting theorem by Artin also shows that the braid group $B_n$ embeds into Aut($F_n$), i.e. the automorphisms of the free group, as those elements that fix $x_1\cdot \dots \cdot x_n$ (see also this question).

Is there a definition of a quantum braid group? Or is this too difficult due to the non-finiteness of $B_n$? Is there a way to define what a free quantum group would be? (to obtain quantum-$B_n$ as a fixed point algebra)

Recently I learned the definition of the quantum permutation group $A_s(n)$, which starts from the fundamental representation by permutation matrices and exchanges the entries by noncommuting operators, i.e.:

It is the universal $C^*$-algebra generated by matrices $u = (u_{ij})$ of generators, such that each $u_{ij}$ is a projection and on each row and column of $u$ these projections are orthogonal and sum up to $1$.

From the point of view of generators and relations the braid group is not that far away from the permutation group. Although it is hard to find faithful representations as was done by Bigelow in this paper. An interesting theorem by Artin also shows that the braid group $B_n$ embeds into Aut($F_n$), i.e. the automorphisms of the free group, as those elements that fix $x_1\cdot \dots \cdot x_n$ (see also this question).

Is there a definition of a quantum braid group? Or is this too difficult due to the non-finiteness of $B_n$? Is there a way to define what a free quantum group would be? (to obtain quantum-$B_n$ as a fixed point algebra)