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quantum-group

Exact symbolic verification of R-matrix and Yang–Baxter identities for U_q(sl_2) and GL_q(2|1).

quantum-group builds the explicit matrices behind concrete quantum-group calculations with SymPy, forms the difference between the two sides of an identity, and checks that every entry of this residual simplifies exactly to zero. It covers finite-dimensional U_q(sl_2) representations and the graded GL_q(2|1) R-matrix of Çelik and Çelik (2021). It verifies identities on concrete representations; it is not a general quantum-group computer algebra system and not a theorem prover.

Paper

Taha Berk Terekli (2026), quantum-group: Exact symbolic verification of R-matrix and Yang–Baxter identities for U_q(sl_2) and GL_q(2|1). arXiv:2610.07867, DOI 10.48550/arXiv.2610.07867. The paper's mathematical background and worked examples complement this site; see Citation for how to cite it.

python3 -m pip install quantum-group

Main features

  • q-integers, q-factorials and q-binomials as Laurent polynomials, and classical limits.
  • Explicit E, F, K, K_inv matrices for the modules V_n of U_q(sl_2).
  • Defining relations checked as exact residual matrices.
  • Coproduct, counit and antipode, with generator-level Hopf identities.
  • Tensor products, Clebsch–Gordan highest-weight vectors.
  • The fundamental R-matrix on V_1 ⊗ V_1: QYBE, braid and Hecke relations, coproduct intertwining, in both the historical and the coproduct-compatible convention (see below).
  • The 9x9 GL_q(2|1) R-matrix, super-permutation, the 27x27 graded Yang–Baxter residual, and R_ij placements on V^{⊗n} via graded swaps.
  • The combinatorial crystal graph B(n) and simple weight/crystal plots.

Where to go next

Task Page
Install the package and run a first calculation Getting started
Coproduct, basis and grading conventions; what is out of scope Conventions and scope
Reproduce the paper's listings and verification claims Reproducing the paper
Look up a function API reference
Report a bug or ask a question GitHub issues