**Quantum linear semigroups** are bialgebras which are deformations of bialgebras of coordinate functions on the groups of $n\times n$-matrices for some $n$. They belong to the class of matrix bialgebras. The usual notation for the one-parametric version is $\mathcal{O}(M_q(n))$ or sometimes simply $\mathcal{M}_q(n)$.

Suppose we are given $n\times n$ matrices $P = (p_{ij})$ and $Q = (q_{ij})$ with invertible entries in the ground field $F$, for which there exist $q$ such that

$p_{ij} q_{ij} = q^2, q_{ij} = q^{-1}_{ji},\,\,\,\,\,i \lt j,\,\,and \,\,\,\,\,\,
q_{ii} = p_{ii},\,\,\,\,\,\,\, for\,\,\, all\,\,\,\, i.$

The multiparametric quantized matrix bialgebra (synonym: multiparametric quantum linear semigroup) $\mathcal{O}(M_{P,Q}(F,n)):= F \langle T^i_j , i,j = 1,\ldots, n\rangle/I$, where $I$ is the ideal spanned by the relations

$\array{
T^k_i T^k_j = q_{ij} T^k_j T^k_i, & i \lt j \\
T^k_i T^l_i = p_{kl} T^l_i T^k_i, & k \lt l \\
q_{ij} T^k_j T^l_i = p_{kl} T^l_i T^k_j, & i\lt j,\,\,\,\,k\lt l \\
T^k_i T^l_j - q_{ij} q^{-1}_{kl} T^l_j T^k_i
= (q_{ij}-p_{ij}^{-1}) T^k_j T^l_i,& i\lt j,\,\,\,\,k\lt l
}$

$\mathcal{M} = \mathcal{O}(M_{P,Q}(F,n))$ is a bialgebra with respect to the “matrix” comultiplication which is the unique algebra homomorphism $\Delta : \mathcal{M} \to\mathcal{M} \otimes\mathcal{M}$ extending the formulas which are written in the matrix form as $\Delta T^i_j = \sum T^i_k \otimes T^k_j$ with counit $\epsilon T^i_j = \delta^i_j$ (Kronecker delta). This means that it is a matrix bialgebra with basis $\{T^i_j\}_{i,j=1,\ldots,n}$, in fact a free matrix bialgebra over $F$.

In these conventions, the 1-parametric version $\mathcal{O}(M_q(F))$ is obtained as a special case when $P = Q$ and $q_{ij} = q$ for $i \lt j$ and $q_{ij} = q^{-1}$ for $i \gt j$.

**Quantum linear groups** are Hopf algebras which are quantum deformations of Hopf algebras of coordinate functions on the general linear group or special linear group. There exist one parametric and many parametric versions as well as super analogues. They belong to the class of matrix Hopf algebras.

The usual notation for one-parametric versions is $\mathcal{O}(GL_q(n))$, $\mathcal{O}(SL_q(n))$ and variants thereof.

- related $n$Lab pages: quantum group, quantum Gauss decomposition, quantized function algebra, general linear group
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