Mubarakzyanov's classification of low-dimensional real Lie algebras, published in Russian in 1963. It complements the article on Lie algebra in the area of abstract...

the first table for real Lie algebras. Classification of low-dimensional real Lie algebras Simple Lie group#Full classification Fulton, William; Harris...

groups Classification of low-dimensional real Lie algebras Classification of finite simple groups Jacobson, Nathan (1971). Exceptional Lie Algebras. CRC...

over any field. Some Lie algebras of low dimension are described here. See the classification of low-dimensional real Lie algebras for further examples...

semisimple rings Classification of Clifford algebras Classification of low-dimensional real Lie algebras Classification of Simple Lie algebras and groups Classification...

Octonions Lie algebras Jordan algebras Alternative algebras Flexible algebras Power-associative algebras The definition of an associative K-algebra with unit...

orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different n are the compact real forms of two of the four...

mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation...

(finite-dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the...

isomorphic to the Lie algebra of SO(3), the rotation group. The Bianchi types refer to the classification of three-dimensional Lie algebras by the Italian...

is a quotient of the tensor algebra of V enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally...

infinite-dimensional topological group.) Once this result is established, we see that H {\displaystyle H} is a one-dimensional Lie group central extension of G...

spacetime is 26-dimensional, while in superstring theory it is 10-dimensional, and in M-theory it is 11-dimensional. In order to describe real physical phenomena...

conjugates of one another. In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another...

role of quantum Borel part of a pointed Hopf algebra such as a quantum groups and their well known finite-dimensional truncations. Nichols algebras can...

Calabi–Yau manifold of (complex) dimension n {\displaystyle n} is sometimes defined as a compact n {\displaystyle n} -dimensional Kähler manifold M {\displaystyle...

of the Leech lattice, quotiented out by its center. There are only three finite-dimensional associative division algebras over the reals — the real numbers...

(or reductive) Lie group, can be done for all. Therefore, once the role of some low-dimensional Lie groups such as GL(2) in the theory of modular forms...

neighborhood that is homeomorphic to an open subset of n {\displaystyle n} -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but...

algebras) Hurwitz's theorem (normed division algebras) Jacobson–Morozov theorem (Lie algebra) Levi's theorem (Lie groups) Lie's theorem (Lie algebra)...

there are isomorphisms between low-dimensional spin groups and certain classical Lie groups, owing to low-dimensional isomorphisms between the root systems...

as commutative von Neumann algebras do. Together with Francis Murray, he produced a classification of von Neumann algebras. The direct integral construction...

can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which...

not only of the n-dimensional Lorentz group, but also of a Lie algebra called the n-dimensional Clifford algebra. The most commonly used basis of the complex...

manifolds need to be odd-dimensional. The cotangent bundle T*N of any n-dimensional manifold N is itself a manifold (of dimension 2n) and supports naturally...

of complex semisimple Lie groups and Lie algebras. McKay conjecture: in a group G {\displaystyle G} , the number of irreducible complex characters of...

corresponding to twists along directions of a four-dimensional maximal torus in the twelve-dimensional Standard Model Lie group, SU(3)×SU(2)×U(1). In grand unified...

manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension 2 n {\displaystyle 2n} , so its cohomology groups lie in degrees...

surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds. K3 surfaces have been applied to Kac–Moody algebras, mirror...

as zero-dimensional points but as one-dimensional extended objects called strings. The physics of strings can be studied in various numbers of dimensions...