nLab quantum linear group


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

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

p ijq ij=q 2,q ij=q ji 1,i<j,andq ii=p ii,foralli. 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) 𝒪(M P,Q(F,n)):=FT j i,i,j=1,,n/I\mathcal{O}(M_{P,Q}(F,n)):= F \langle T^i_j , i,j = 1,\ldots, n\rangle/I, where II is the ideal spanned by the relations

T i kT j k=q ijT j kT i k, i<j T i kT i l=p klT i lT i k, k<l q ijT j kT i l=p klT i lT j k, i<j,k<l T i kT j lq ijq kl 1T j lT i k=(q ijp ij 1)T j kT i l, i<j,k<l\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 }

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

In these conventions, the 1-parametric version 𝒪(M q(F))\mathcal{O}(M_q(F)) is obtained as a special case when P=QP = Q and q ij=qq_{ij} = q for i<ji \lt j and q ij=q 1q_{ij} = q^{-1} for i>ji \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 𝒪(GL q(n))\mathcal{O}(GL_q(n)), 𝒪(SL q(n))\mathcal{O}(SL_q(n)) and variants thereof.


  • related nnLab pages: quantum group, quantum Gauss decomposition, quantized function algebra, general linear group
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Last revised on January 2, 2021 at 09:41:06. See the history of this page for a list of all contributions to it.