Affine Lie algebraIn mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras.
Virasoro algebraIn mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. The Virasoro algebra is spanned by generators Ln for n ∈ Z and the central charge c. These generators satisfy and The factor of is merely a matter of convention. For a derivation of the algebra as the unique central extension of the Witt algebra, see derivation of the Virasoro algebra.
Simple moduleIn mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R.
Semisimple Lie algebraIn mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra , if nonzero, the following conditions are equivalent: is semisimple; the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate; has no non-zero abelian ideals; has no non-zero solvable ideals; the radical (maximal solvable ideal) of is zero.
Semisimple moduleIn mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient.
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
Module (mathematics)In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.
Projective moduleIn mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains.
Lie algebraIn mathematics, a Lie algebra (pronounced liː ) is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors and is denoted . The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative.
Compact Lie algebraIn the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification.
