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Quantum super Manin matrices

Naihuan Jing, Yinlong Liu, Jian Zhang

TL;DR

This work develops the theory of $q$-super Manin matrices by constructing the associated bialgebra $\,\mathcal{M}_{m|n}^q$ that governs endomorphisms of the quantum symmetric superalgebra on a $\,\mathbb{Z}_2$-graded space. It defines the quantum Berezinian $Ber_q(M)$ for such matrices and provides explicit, factorized expressions using both $M$ and $M^{-1}$, alongside a suite of fundamental identities. Central results include a Jacobi-type ratio theorem, a Schur-complement identity, and a suite of minor/quotient identities expressed via quasideterminants, uniting $q$-deformations with superalgebraic structure. Collectively, these contributions extend classical linear algebra into the noncommutative, quantum-super context, with potential applications to quantum groups and integrable systems.

Abstract

Manin matrices are quantum linear transformations of general quantum spaces. In this paper, we study the $q$-analogue of super Manin matrices and obtain several quantum versions of classical identities, such as Jacobi's ratio theorem, Schur's complement theorem, Cayley's complementary theorem, Muir's law, Sylvester's theorem, MacMahon Master Theorem and Newton's identities.

Quantum super Manin matrices

TL;DR

This work develops the theory of -super Manin matrices by constructing the associated bialgebra that governs endomorphisms of the quantum symmetric superalgebra on a -graded space. It defines the quantum Berezinian for such matrices and provides explicit, factorized expressions using both and , alongside a suite of fundamental identities. Central results include a Jacobi-type ratio theorem, a Schur-complement identity, and a suite of minor/quotient identities expressed via quasideterminants, uniting -deformations with superalgebraic structure. Collectively, these contributions extend classical linear algebra into the noncommutative, quantum-super context, with potential applications to quantum groups and integrable systems.

Abstract

Manin matrices are quantum linear transformations of general quantum spaces. In this paper, we study the -analogue of super Manin matrices and obtain several quantum versions of classical identities, such as Jacobi's ratio theorem, Schur's complement theorem, Cayley's complementary theorem, Muir's law, Sylvester's theorem, MacMahon Master Theorem and Newton's identities.

Paper Structure

This paper contains 4 sections, 19 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Quantum super Manin matrices
  3. Quantum Berezinian
  4. Minor identities for Berezinians

Key Result

Proposition 2.1

The quadratic relations M-rel of the bialgebra $\mathcal{M}_{m|n}$ are equivalent to one of the following three conditions: where $M_1=\sum_{i,j}(-1)^{\bar{i}\bar{j}+\bar{j}} M_{ij}\otimes E_{ij} \otimes 1$, $M_2=\sum_{i,j}(-1)^{\bar{i}\bar{j}+\bar{j}} M_{ij} \otimes 1 \otimes E_{ij}$.

Theorems & Definitions (32)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5: Newton's identities
  • proof
  • Proposition 2.6
  • proof
  • ...and 22 more