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Homology of Yang-Baxter modules

Yin Tian, Xiao Wang, Yuxin Zhang

TL;DR

Addresses the one-term Yang-Baxter homology for the family of operators $R_m$ arising from $U_q(sl_m)$ acting on the vector representation $V_m$. The authors construct an operator $\sigma_n$ from the $R_m$-braiding and prove a direct-sum eigenspace decomposition of $V^{\otimes n}$ into $\ker\sigma_n$ and tensor-structured pieces, enabling a splitting of the one-term chain complex as $C^{YB}(M)\cong C^f(M)\otimes B(V_m)$ with $B(V_m)=\bigoplus_n\ker\sigma_n$ being a graded algebra; the finite part $C^f(M)$ is described via the Koszul resolution, allowing explicit computation of $H^{YB}_*(F)$ and, for small $m$, explicit generators of kernels. The paper provides dimension formulas and a graded-algebra structure for $B(V_m)$, including concrete results for $m=2,3$, and illustrates Betti-number computations for certain $V_m$-modules. These results establish a concrete, computable framework for the one-term YB homology of a central quantum-group–related family and set the stage for higher-term homologies and quantum-invariant applications.

Abstract

We study the Yang-Baxter operator for the vector representation $V_m$ of the quantum group $U_q(sl_m)$. We consider the one-term Yang-Baxter homology with coefficients in $V_m$-modules and provide a direct sum decomposition of the one term Yang-Baxter chain complex. The homology is explicitly computed for some specific $V_m$-modules.

Homology of Yang-Baxter modules

TL;DR

Addresses the one-term Yang-Baxter homology for the family of operators arising from acting on the vector representation . The authors construct an operator from the -braiding and prove a direct-sum eigenspace decomposition of into and tensor-structured pieces, enabling a splitting of the one-term chain complex as with being a graded algebra; the finite part is described via the Koszul resolution, allowing explicit computation of and, for small , explicit generators of kernels. The paper provides dimension formulas and a graded-algebra structure for , including concrete results for , and illustrates Betti-number computations for certain -modules. These results establish a concrete, computable framework for the one-term YB homology of a central quantum-group–related family and set the stage for higher-term homologies and quantum-invariant applications.

Abstract

We study the Yang-Baxter operator for the vector representation of the quantum group . We consider the one-term Yang-Baxter homology with coefficients in -modules and provide a direct sum decomposition of the one term Yang-Baxter chain complex. The homology is explicitly computed for some specific -modules.
Paper Structure (6 sections, 20 theorems, 55 equations, 3 figures)

This paper contains 6 sections, 20 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. One term Yang-Baxter homology of $sl_{m}$ operators
  4. One eigenspace decomposition from $\sigma_{n}$
  5. The proof of Theorem \ref{['main']}
  6. The structure of the kernel of $\sigma_{n}$

Key Result

Theorem 1.1

For the Yang-Baxter operator $(V_m, R_m)$, and any $V_m$-module $M$, we have the followings.

Figures (3)

  • Figure 2.1: the wall condition.
  • Figure 2.2: face map $d_{i,n}$.
  • Figure 3.1: $\sigma_{n}$ and ${d_{k}^{n}}$

Theorems & Definitions (62)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Lemma 2.9
  • ...and 52 more