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In mathematics, the special linear Lie algebra of order over a field, denoted or (,), is the Lie algebra of all the matrices (with entries in ) with trace zero and with the Lie bracket [,]:= given by the commutator. This algebra is well studied and understood, and is often used as a model for the study of other Lie algebras.
In mathematics, the special linear group SL(n, R) of degree n over a commutative ring R is the set of n × n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.
Download as PDF; Printable version; ... It is called the general linear Lie algebra. When F is the real numbers, ... the special linear group (,) consists of ...
The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear representation of PSL(2, R). This can alternatively be described as the action of PSL(2, R) on the space of quadratic forms on R 2. The result is the following representation:
Let be a Lie algebra and let be a vector space. We let () denote the space of endomorphisms of , that is, the space of all linear maps of to itself. Here, the associative algebra () is turned into a Lie algebra with bracket given by the commutator: [,] = for all s,t in ().
In positive characteristic, there can be many different connected subgroups of a group G with the same Lie algebra (again, the torus G = (G m) 2 provides examples). For these reasons, although the Lie algebra of an algebraic group is important, the structure theory of algebraic groups requires more global tools.
In other words, a linear Lie algebra is the image of a Lie algebra representation. Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of g {\displaystyle {\mathfrak {g}}} (in fact, on a finite-dimensional vector space by Ado's theorem if g {\displaystyle {\mathfrak {g}}} is itself finite ...
In the theory of Lie algebras, an sl 2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra sl 2. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to their nilpotent orbits.
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