The Orlik-Solomon algebra is a graded algebra associated with a hyperplane arrangement, capturing the combinatorial structure of the arrangement. It is constructed from the intersection lattice of the arrangement and plays a crucial role in the study of the topology of the arrangement's complement.

Orlik-Solomon Algebra

A graded algebra is an algebraic structure that is decomposed into a direct sum of subspaces, each assigned a degree, and the multiplication operation respects this grading by mapping elements of certain degrees to elements of their sum. This structure is particularly useful in areas like algebraic topology and homological algebra, where it helps in organizing and simplifying complex algebraic computations.

Graded Algebra

A hyperplane arrangement is a finite collection of hyperplanes in a vector space, often studied for its combinatorial and topological properties. These arrangements are crucial in understanding geometric structures, optimization problems, and various algebraic and topological phenomena such as partitioning space into regions and studying their intersections.

Hyperplane Arrangement

Combinatorial Structure

An intersection lattice is a mathematical structure that organizes the intersections of a collection of sets, typically subspaces or subgroups, into a lattice based on set inclusion. This structure is crucial for understanding the relationships and hierarchies between different subspaces, often used in fields like combinatorics, algebra, and geometry.

Intersection Lattice

Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations such as stretching and bending, but not tearing or gluing. It provides a foundational framework for understanding concepts of convergence, continuity, and compactness in various mathematical contexts.

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