lattices

Verify lattice conventions and bilinear form relations with exact arithmetic.

1|Updated Feb 24, 2026

Search Tags:#sagemath #lattices #bilinear-forms #gram-matrix #cartan-matrix #exact-arithmetic #mathematical-conventions

Ready to run

npx skills add https://github.com/dzackgarza/ai --skill lattices

💬 Prompt for ChatGPT / Claude Chat▼

Please help me install this Agent Skill.
Skill: lattices
Source: https://github.com/dzackgarza/ai/tree/main/opencode/skills/lattices
Command: npx skills add https://github.com/dzackgarza/ai --skill lattices

SYSTEM DOCUMENTATION & REQUIREMENTS

What problem does it solve?

Clarifies and standardizes mathematical conventions for lattices, bilinear forms, Gram and Cartan matrices, and related invariants to aid researchers in reasoning, communicating, and implementing computations.

Core Features & Use Cases

Establishes precise terminology for bilinear forms vs inner products, Gram vs Cartan matrices, and related constructs.
Guides exact arithmetic practices (working over Z, Q, and rings) and safe implementation in software like SageMath.
Use Case: A researcher validates a lattice basis by checking B(v,v) definiteness and transformation relations between Gram and Cartan matrices.

Quick Start

Provide a lattice basis and request a formal verification of the associated Gram and Cartan conventions.

Frequently Asked Questions about lattices

High-intent search queries and answers about installing and using this skill.

FAQPage Schema

How do I verify Gram matrix conventions for a lattice basis using exact arithmetic?▼

Verify Gram matrix conventions by providing a lattice basis to check B(v,v) definiteness and transformation relations. The Skill automates this verification using exact arithmetic over Z and Q.

What is the difference between a Gram matrix and a Cartan matrix in bilinear forms?▼

The difference between a Gram matrix and a Cartan matrix lies in their associated bilinear form conventions. The Skill establishes precise terminology to distinguish these constructs and standardizes their relationships for mathematical research.

How do I validate lattice basis transformations in SageMath?▼

Validate lattice basis transformations in SageMath by applying exact arithmetic over Z and Q. The Skill guides safe implementation practices and checks transformation relations between Gram and Cartan matrices.

Can I use exact arithmetic over minimal fields for bilinear form computations?▼

Yes, you can use exact arithmetic over minimal fields for bilinear form computations. The Skill requires reasoning with exact arithmetic over Z, Q, and minimal fields to verify lattice and bilinear form relations accurately.

When do I need to standardize bilinear form terminology for mathematical implementation?▼

Standardize bilinear form terminology for mathematical implementation when communicating or reasoning about lattice invariants. The Skill clarifies conventions for bilinear forms versus inner products to aid researchers in precise computation.

What are the limitations of verifying lattice conventions without exact arithmetic?▼

Verifying lattice conventions without exact arithmetic risks numerical instability and incorrect bilinear form relations. The Skill enforces exact arithmetic over Z and Q to ensure safe implementation and mathematically rigorous results.