📂Representation Theory

Lie Algebra

Definition¹

A finite-dimensional real (complex) vector space $\mathfrak{g}$ equipped with the following binary operation $[\cdot, \cdot] : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ is called a Lie algebra.

1. $[\cdot, \cdot]$ is bilinear. $$ [ax + by, z] = a[x, z] + b[y, z] $$
2. $[\cdot, \cdot]$ is skew-symmetric. $$ [x, y] = -[y, x] $$
3. $[\cdot, \cdot]$ satisfies the Jacobi identity. $$ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 $$

Explanation

A Lie algebra is denoted in Fraktur as $\mathfrak{g}$. $[\cdot, \cdot]$ is called the bracket. In general, the bracket operation in a Lie algebra need not satisfy associativity. Although associativity does not hold, the Jacobi identity, which can be regarded as a weaker form of it, does hold.

Examples

Associative algebra

Let $A$ be an associative algebra, and let $\mathfrak{g}$ be a subspace of $A$. If the commutator $[X, Y] = XY - YX$ is closed on $\mathfrak{g}$, then $\mathfrak{g}$ forms a Lie algebra.

3-dimensional space and the cross product

Let $\mathfrak{g} = \mathbb{R}^{3}$, and let $[\cdot, \cdot]$ be the cross product in 3-dimensional space.

$$ [\mathbf{x}, \mathbf{y}] = \mathbf{x} \times \mathbf{y} $$

Then $\mathfrak{g}$ is a Lie algebra.