What are quantum symmetric matrices and quantum exterior matrices?

Question

What are quantum symmetric matrices and quantum exterior matrices? I searched google but didn't find the definitions. Thank you very much.

Edit: "algebras of quantum symmetric and quantum exterior matrices" is used in Remark 1.13 of the paper.

Answer 1

As mentioned in Casteels comment, the quantum symmetric algebra is a well-known construction for a $k$-vector space $V$: $$ S_q(V):=k\langle v_1, \ldots, v_n\mid v_iv_j=q_{ij}v_jv_i\rangle. $$ (see e.g http://arxiv.org/abs/1111.5243). Here, the scalars $q_{ij}$ need to satisfy $q_{ii}=1$ and $q_{ji}=q_{ij}^{-1}$. This is sometimes also referred to as the quantum $n$-space (see e.g. http://arxiv.org/pdf/math/9905055v1.pdf).

One can also consider a quantum exterior algebra: $$ \Lambda_q(V):= k\langle v_1,\ldots,v_n\mid v_iv_j+q_{ij}v_jv_i\rangle.$$ (see http://arxiv.org/abs/1111.5243, p.4). Note that this algebra will turn out to be finite-dimensional in characteristic $0$ at least, and $v_i^2=0$ is a consequence from the relations.

These example can be understood in the more general context of Nichols algebras.