Category: Theorems about algebras

Quillen's lemma
In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field k is algebraic over k. In contrast to a version
Ado's theorem
In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.
Lie's theorem
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if is a finite-dimensional representation of a solvable L
Poincaré–Birkhoff–Witt theorem
In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a
Theorem of the highest weight
In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra . There is a closely related theorem
Artin approximation theorem
In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theory which implies that formal power series with coefficients in a field k are well-approximat
Artin–Tate lemma
In algebra, the Artin–Tate lemma, named after Emil Artin and John Tate, states: Let A be a commutative Noetherian ring and commutative algebras over A. If C is of finite type over A and if C is finite
Frobenius theorem (real division algebras)
In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the r
Hurwitz's theorem (composition algebras)
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras end
Milnor–Moore theorem
In algebra, the Milnor–Moore theorem, introduced by John W. Milnor and John C. Moore classifies an important class of Hopf algebras, of the sort that often show up as cohomology rings in algebraic top
Zariski's lemma
In algebra, Zariski's lemma, proved by Oscar Zariski, states that, if a field K is finitely generated as an associative algebra over another field k, then K is a finite field extension of k (that is,
Brauer–Nesbitt theorem
In mathematics, the Brauer–Nesbitt theorem can refer to several different theorems proved by Richard Brauer and Cecil J. Nesbitt in the representation theory of finite groups. In modular representatio
Serre's theorem on a semisimple Lie algebra
In abstract algebra, specifically the theory of Lie algebras, Serre's theorem states: given a (finite reduced) root system , there exists a finite-dimensional semisimple Lie algebra whose root system
Lie's third theorem
In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra over the real numbers is associated to a Lie group . The theorem is part of the Lie group–Lie alg