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Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...

are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" An, Bn, Cn and...

Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups Mumford, David (1970), Abelian varieties, Oxford University Press...

theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically...

These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such...

unital, but Lie algebras never are. The nonassociative algebra structure of A may be studied by associating it with other associative algebras which are...

in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group...

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called...

basis (a sort of integral form but over finite fields) for all the complex simple Lie algebras (or rather of their universal enveloping algebras), which can...

classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that...

representation theory of compact connected Lie groups, or the finite-dimensional representation theory of complex semisimple Lie algebras. For k of characteristic...

dimensions". arXiv:math/0509708. Gilmore, Robert (1974). Lie Groups, Lie Algebras and some of their Applications. John Wiley & Sons. pp. 52, 201−205. MR 1275599...

algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras...

diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but...

from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups. The...

R), consists of the real n×n matrices with positive determinant. This is also a Lie group of dimension n2; it has the same Lie algebra as GL(n, R). The...

{\displaystyle k[X_{1},\ldots ,X_{n}].} An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by the defining...

property often allows a variety of different formulations of the algebra to be given, as noted in the examples below. Poisson algebras occur in various settings...

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

mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over...