Algebraic Structures and Polynomial Identities

Algebraic structures, encompassing groups, rings and various classes of associative and nonassociative algebras, provide a unifying framework for symmetries and operations across mathematics and theoretical physics. Central to this framework is the study of polynomial identities: nontrivial polynomial expressions in noncommuting variables that vanish for all substitutions from a given algebra. Such identities govern the internal constraints of an algebraic system, often reflecting deep connections between its representation theory, combinatorial invariants and geometric or physical applications.

Polynomial identity (PI) theory explores how these identities classify algebras up to structural invariants such as codimension growth—the sequence measuring the dimension of multilinear components of the free algebra modulo its identities—and the associated PI-exponent, which captures the exponential rate of that growth. Extensions of classical PI theory have arisen in contexts where additional operations enrich the algebra: gradings by groups give rise to graded identities and graded exponents; involutions or superinvolutions introduce ∗-identities; derivations lead to differential identities; and even affine group schemes admit a notion of identities by interpreting their coordinate rings as matrix algebras. Across these developments, concrete examples range from matrix and Weyl algebras to enveloping algebras of Lie algebras and quantum deformations, each illustrating how polynomial constraints encode symmetry and cohomological data.

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Technical terms

Algebraic structure: A set equipped with one or more operations (such as addition and multiplication) satisfying specified axioms, including groups, rings and algebras.

Polynomial identity (PI): A nonzero noncommutative polynomial in indeterminates that evaluates to zero under all substitutions of algebra elements, imposing algebraic constraints.

Codimension: For an algebra, the dimension of the space of multilinear polynomials of degree n modulo its identities; its growth rate measures the complexity of identities.

PI-exponent: The limit of the n-th root of the n-th codimension as n tends to infinity, an integer capturing the exponential growth rate of identities.

Graded involution: An involutive antiautomorphism of a graded algebra that preserves or twists the grading, yielding graded ∗-identities.

Differential codimension: The codimension sequence for an algebra endowed with derivations, counting multilinear differential polynomials modulo differential identities.

Affine group scheme: A functor from commutative algebras to groups represented by a finitely generated Hopf algebra, generalising algebraic groups to a scheme-theoretic setting.

References

1. Minimal varieties of PI-algebras with graded involution. Linear Algebra and its Applications (2024).
2. Differential codimensions and exponential growth. Linear Algebra and its Applications (2023).
3. Attaching a matrix algebra to affine group schemes: Polynomial identities. Journal of Algebra (2023).