Quantum Matrix Coordinate Algebras

AUTHORS:

  • Travis Scrimshaw (01-2016): initial version
sage.algebras.quantum_matrix_coordinate_algebra.QuantumGL

Quantum coordinate algebra of \(GL(n)\).

The quantum coordinate algebra of \(GL(n)\), or quantum \(GL(n)\) for short and denoted by \(\mathcal{O}_q(GL(n))\), is the quantum coordinate algebra of \(M_R(n, n)\) with the addition of the additional central group-like element \(c\) which satisfies \(c d = d c = 1\), where \(d\) is the quantum determinant.

Quantum \(GL(n)\) is a Hopf algebra where \(\varepsilon(c) = 1\) and the antipode \(S\) is given by the (quantum) matrix inverse. That is to say, we have \(S(c) = c^-1 = d\) and

\[S(x_{ij}) = c * (-q)^{i-j} * \tilde{t}_{ji},\]

where we have the quantum minor

\[\tilde{t}_{ij} = \sum_{\sigma} (-q)^{\ell(\sigma)} x_{1, \sigma(1)} \cdots x_{i-1, \sigma(i-1)} x_{i+1, \sigma(i+1)} \cdots x_{n, \sigma(n)}\]

with the sum over permutations \(\sigma \colon \{1, \ldots, i-1, i+1, \ldots n\} \to \{1, \ldots, j-1, j+1, \ldots, n\}\).

INPUT:

  • n – the integer \(n\)
  • R – (optional) the ring \(R\) if \(q\) is not specified (the default is \(\ZZ\)); otherwise the ring containing \(q\)
  • q – (optional) the variable \(q\); the default is \(q \in R[q, q^{-1}]\)
  • bar – (optional) the involution on the base ring; the default is \(q \mapsto q^{-1}\)

EXAMPLES:

We construct \(\mathcal{O}_q(GL(3))\) and the variables:

sage: O = algebras.QuantumGL(3)
sage: O.inject_variables()
Defining x11, x12, x13, x21, x22, x23, x31, x32, x33, c

We do some basic computations:

sage: x33 * x12
x[1,2]*x[3,3] + (q^-1-q)*x[1,3]*x[3,2]
sage: x23 * x12 * x11
(q^-1)*x[1,1]*x[1,2]*x[2,3] + (q^-2-1)*x[1,1]*x[1,3]*x[2,2]
 + (q^-3-q^-1)*x[1,2]*x[1,3]*x[2,1]
sage: c * O.quantum_determinant()
1

We verify the quantum determinant is in the center and is group-like:

sage: qdet = O.quantum_determinant()
sage: all(qdet * g == g * qdet for g in O.algebra_generators())
True
sage: qdet.coproduct() == tensor([qdet, qdet])
True

We check that the inverse of the quantum determinant is also in the center and group-like:

sage: all(c * g == g * c for g in O.algebra_generators())
True
sage: c.coproduct() == tensor([c, c])
True

Moreover, the antipode interchanges the quantum determinant and its inverse:

sage: c.antipode() == qdet
True
sage: qdet.antipode() == c
True

REFERENCES:

[DD91]R. Dipper and S. Donkin. Quantum \(GL_n\). Proc. London Math. Soc. (3) 63 (1991), no. 1, pp. 165-211.
[Karimipour93]Vahid Karimipour. Representations of the coordinate ring of \(GL_q(n)\). (1993). Arxiv hep-th/9306058.
sage.algebras.quantum_matrix_coordinate_algebra.QuantumMatrixCoordinateAlgebra

A quantum matrix coordinate algebra.

Let \(R\) be a commutative ring. The quantum matrix coordinate algebra of \(M(m, n)\) is the associative algebra over \(R[q, q^{-1}]\) generated by \(x_{ij}\), for \(i = 1, 2, \ldots, m\), \(j = 1, 2, \ldots, n\), and subject to the following relations:

\[\begin{split}\begin{array}{ll} x_{it} x_{ij} = q^{-1} x_{ij} x_{it} & \text{if } j < t, \\ x_{sj} x_{ij} = q^{-1} x_{ij} x_{sj} & \text{if } i < s, \\ x_{st} x_{ij} = x_{ij} x_{st} & \text{if } i < s, j > t, \\ x_{st} x_{ij} = x_{ij} x_{st} + (q^{-1} - q) x_{it} x_{sj} & \text{if } i < s, j < t. \\ \end{array}\end{split}\]

The quantum matrix coordinate algebra is denoted by \(\mathcal{O}_q(M(m, n))\). For \(m = n\), it is also a bialgebra given by

\[\Delta(x_{ij}) = \sum_{k=1}^n x_{ik} \otimes x_{kj}, \varepsilon(x_{ij}) = \delta_{ij}.\]

Moreover, there is a central group-like element called the quantum determinant that is defined by

\[\det_q = \sum_{\sigma \in S_n} (-q)^{\ell(\sigma)} x_{1,\sigma(1)} x_{2,\sigma(2)} \cdots x_{n,\sigma(n)}.\]

The quantum matrix coordinate algebra also has natural inclusions when restricting to submatrices. That is, let \(I \subseteq \{1, 2, \ldots, m\}\) and \(J \subseteq \{1, 2, \ldots, n\}\). Then the subalgebra generated by \(\{ x_{ij} \mid i \in I, j \in J \}\) is naturally isomorphic to \(\mathcal{O}_q(M(|I|, |J|))\).

Note

The \(q\) considered here is \(q^2\) in some references, e.g., [ZZ2005].

INPUT:

  • m – the integer \(m\)
  • n – the integer \(n\)
  • R – (optional) the ring \(R\) if \(q\) is not specified (the default is \(\ZZ\)); otherwise the ring containing \(q\)
  • q – (optional) the variable \(q\); the default is \(q \in R[q, q^{-1}]\)
  • bar – (optional) the involution on the base ring; the default is \(q \mapsto q^{-1}\)

EXAMPLES:

We construct \(\mathcal{O}_q(M(2,3))\) and the variables:

sage: O = algebras.QuantumMatrixCoordinate(2,3)
sage: O.inject_variables()
Defining x11, x12, x13, x21, x22, x23

We do some basic computations:

sage: x21 * x11
(q^-1)*x[1,1]*x[2,1]
sage: x23 * x12 * x11
(q^-1)*x[1,1]*x[1,2]*x[2,3] + (q^-2-1)*x[1,1]*x[1,3]*x[2,2]
 + (q^-3-q^-1)*x[1,2]*x[1,3]*x[2,1]

We construct the maximal quantum minors:

sage: q = O.q()
sage: qm12 = x11*x22 - q*x12*x21
sage: qm13 = x11*x23 - q*x13*x21
sage: qm23 = x12*x23 - q*x13*x22

However, unlike for the quantum determinant, they are not central:

sage: all(qm12 * g == g * qm12 for g in O.algebra_generators())
False
sage: all(qm13 * g == g * qm13 for g in O.algebra_generators())
False
sage: all(qm23 * g == g * qm23 for g in O.algebra_generators())
False

REFERENCES:

sage.algebras.quantum_matrix_coordinate_algebra.QuantumMatrixCoordinateAlgebra_abstract

Abstract base class for quantum coordinate algebras of a set of matrices.