nLab quantum linear group

Idea

Quantum linear semigroups are bialgebras which are deformations of bialgebras of coordinate functions on the groups of $n\times n$ invertible matrices for some $n \in \mathbb{N}$. 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 entries

• quantum group

• quantum Gauss decomposition

• quantized function algebra

• general linear group

• quantum determinant

Literature

• Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.
• Yu. I. Manin, Multiparametric quantum deformation of the general linear supergroup, Comm. Math. Phys. 123 (1989) 163–175.
• B. Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.
• E. E. Demidov, Multiparameter quantum deformations of the group $GL(n)$, (Russian) Uspehi Mat. Nauk 46 (1991), no. 4 (280) 147–148; translation in Russian Math. Surveys 46 (1991) no. 4, 169–171.
• M. Hashimoto, T. Hayashi, Quantum multilinear algebra, Tohoku Math. J. 44 (1992) 471–521 doi
• Michael Artin, W. Schelter, John Tate, Quantum deformations of $GL_n$, Commun. Pure Appl. Math. XLIV, 879–895 (1991)
• Zoran Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications 61, pp. 265–298, Warszawa 2003.
• Z. Škoda, Every quantum minor generates an Ore set, International Math. Res. Notices 2008, rnn063-8; math.QA/0604610
• Karel Casteels, Siân Fryer, From restricted permutations to Grassmann necklaces and back again, Algebras and Representation Theory 20, 895–921 (2017). doi arxiv/1511.06664
• Hechun Zhang, R.B.Zhang, Dual canonical bases for the quantum special linear group and invariant subalgebras, Letters in Mathematical Physics (2005) 73:165–181 doi; Dual canonical bases for the quantum general linear supergroup, J. Algebra 304 (2006) 1026–1058 doi
• Hans Plesner Jakobsen, Hechun Zhang, The center of the quantized matrix algebra, J. Algebra 196:2 (1997) 458-474 doi
• K. R. Goodearl, E.S. Letzter, Prime factor algebras of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121 (1994) 1017–1025 doi
• S. Launois, T. H. Lenagan, B. M. Nolan, Total positivity is a quantum phenomenon: the Grassmannian case, Memoirs of the Amer. Math. Soc. 1448 (2023) 123 p.