What is a Borel subalgebra?

I'll give you here a physics motivated definition of Borel subalgebras. I'll start with the case of $SL(2)$, which is the case of interest when treating quantum $SU(2)$, but also generalize the definition to arbitrary Lie groups.

As very well known, the generators of the Lie algebra $\mathfrak {su}(2)$ of the group $SU(2)$ are angular momenta $J _x$, $J_y$, $J_z$. All group elements in a given representation can be generated from the exponentiation of these generators:

$$SU(2) \ni g = e^{i \alpha_x J_x} e^{i \alpha_y J_y} e^{i \alpha_z J_z} \quad \alpha_x, \alpha_y, \alpha_z \in \mathbb{R}$$ However, we also use the raising and lowering operators: $$J_{\pm }= J_x +i J_y $$ These elements are not Hermitian. They are not elements of $\mathfrak{su}(2)$, but they are tracelss, thus together with $J_z$, they generate over the complex numbers the Lie algebra of $\mathfrak{sl}(2, \mathbb{C})$ of traceless complex matrices. Their exponentiation results in the group $SL(2, \mathbb{C})$: $$ SL(2, \mathbb{C}) \ni g = e^{i \beta_- J_-} e^{i \beta_z J_z} e^{i \beta_+ J_+} \quad \beta_-, \beta_z, \beta_+ \in \mathbb{C}$$ The raising and lowering operators appear in the representation theory of angular momenta: $$J_-|j, m\rangle \propto |j, m-1\rangle$$ $$J_+|j, m\rangle \propto |j, m+1\rangle$$ In particular, for the lowest weight, which is sometimes referred to in physics as the vacuum, we have: $$J_-|j, -j\rangle =0 $$ $$J_z|j, -j\rangle =-j| j, -j\rangle $$ These angular momentum representations, are non-unitary representations of $ SL(2, \mathbb{C})$.

Now, we ask ourselves what is the subalgebra of $SL(2, \mathbb{C})$ which stabilizes the vacuum; where "stabilizes" means keep the vacuum in the same one dimensional subspace of the Hilbert space; we see that any complex combination of $J_-$ and $J_z$ does so. It is not hard to see that this combination closes to a Lie algebra, this is the Borel subalgebra $\mathfrak{b}(2)$ of $\mathfrak {sl}(2, \mathbb{C})$. Its defining representation is the algebra of traceless lower triangular matrices of $\mathfrak {sl}(2, \mathbb{C})$. The corresponding Lie group is the Borel subgroup group $B(2) \subset SL(2, \mathbb{C})$: $$ B(2) \ni g = exp(i \beta_-J_- ) exp(i \beta_z J_z ) \quad \beta_-, \beta_z, \in \mathbb{C}$$ (B(2) is the group of lower triangular matrices with unit determinant).

In the language of the theory of spontaneous symmetry breaking, the Borel subgroup is the unbroken group of the vacuum under the action of $SL(2, \mathbb{C})$: $$ B(2) \ni g|j, -j\rangle \propto |j, -j\rangle$$ The Borel group (and its corresponding Lie algebra) are not unique, any conjugate group of the form $gBg^{-1}, \, g \in SL(2, \mathbb{C})$ will stabilize the vector $g| j, -j\rangle $. In particular, if we conjugate by the element $$g = \begin{pmatrix} 0& 1 \\ 1 & 0 \end{pmatrix} $$ we obtain a Borel subalgebra of upper triangular matrices.

The generalization to a general group $G$ is quite straightforward. The maximal subgroup of commuting elements is called the Cartan subgroup. This subgroup can always be represented by diagonal matrices, so do its Lie algebra generators $H_i$. The remaining generators can be divided into positive $E_{\alpha}$ and negative $E_{-\alpha}$ roots generators. The exponentiation of the Cartan, raising, and lowering operators with complex coefficients generates the complexified Lie group $G^c$. (For example, the complexification of $SU(N)$ is $ SL(N, \mathbb{C})$.

An irreducible representation $\lambda$ of $G$ extends to a non-unitary highest weight representation of $G^c$. The highest weight vector is stabilized by the elements: $$ E_{\alpha}| \lambda, \lambda\rangle =0 $$ $$ H_i| \lambda, \lambda\rangle = \lambda_i | \lambda, \lambda\rangle $$ Thus, the algebra generated by $E_{\alpha}$ and $H_i$ stabilizes the vacuum. This is the Borel subalgebra of $\mathfrak{g}^c$. The corresponding group is the Borel subgroup.

The Borel subalgebra can always be represented by lower of upper triangular matrices of zero trace.

Clearly, the Cartan subalgebra is a subalgebra of the Borel subalgebra. Sometimes, mainly in mathematical texts you see an equivalent definition of the Borel subgroup as a maximal solvable subgroup containing a Cartan subgroup, (where a solvable subgroup can be thought as a subgroup representable by triangular matrices). Physically, it is the stability subgroup of $G^c$ of the highest weight (vacuum).

I will first introduce Hopf algebras and the construction of quantum doubles. Then I will discuss how Borel subalgebras emerge in the context of Lie algebras.

Hopf Algebra:

A "quantum group" is in the first place a Hopf algebra that has an additional structure analogous to that of a Lie group. A Hopf algebra $H$ obeys the following axioms:

1. $H$ is a unital algebra over a field $k$.
2. $H$ is a counital algebra $(H,\Delta,\epsilon)$ over $k$.
3. $\Delta,\epsilon$ are algebra homeomorphisms.

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Defining Quantum Double:

Let $H^0$ be the dual algebra to $H$ with opposite comultiplication. Then, canonically associated with $H$ is another Hopf algebra $D(H)$ with the following properties:

1. $D(H)=H\otimes H^0$
2. $H$ and $H^0$ are embedded in $D(H)$ as Hopf subalgebras.
3. $\mathcal{R}\Delta_D(a)=(\sigma\circ\Delta_D)(a)\mathcal{R}$, $\forall a\subset D(H)$, where $\Delta_D$ is comultiplicant in $D(H)$ and $\mathcal{R}$ is the image of the canonical element in $H\otimes H^0$ under the embeding $H\otimes H^0\hookrightarrow D(H)\otimes D(H)$.

This Hopf algebra $D(H)$ is called the quantum double of the algebra $H$. So, if a Hopf algebra can be represented as a quantum double of some of its Hopf subalgebra, the construction gives explicit form of the element $\mathcal{R}$ which defines its quasi-triangular system.

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Boral subalgebra as an essential element of Lie algebra:

The R-matrix of the Quantum Universal Enveloping Algebra (QUEA) $U_q(L)$ of a simple Lie algebra can be constructed and the subalgebra $H$ being QUEA $U_q(B^+)$ (or $U_q(B^-)$) of Lie subalgebras $B^+\subset L$ (or $B_-\subset L$) generated in Chevalley basis by $H_i,X_j^+$ (or $H_i,X_j^-$), $1\leq i,j\leq r=$rank($L$). Such subalgebras $B^\pm$ are called Borel subalgebras. The double $D(B^+)$ or $D(B^-)$ coincides with $U_q(L)$ itself after the natural elimination of the second Cartan subalgebra.

Now, let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra $L$ of rank n and $\mathbb{K}$ be an algebraically closed field of characteristic zero.

Definition: A locally finite Lie algebra is locally solvable if every finite subset of $\mathfrak{g}$ is contained in a solvable subalgebra.

Definition: A Borel subalgebra $B$ of $L$ is defined as a maximal locally solvable algebra. The properties of a Borel subalgebra are as follows:

1. Every Borel subalgebra is self-normalizing.
2. Every Borel subalgebra of $L$ contains the solvable radical $Rad(L)$.
3. Any Borel subalgebra of $L$ contains a Cartan subalgebra of $L$.

Definition: An algebra is called basic if all its finite dimensional irreducible representations are one dimensional.

In this sense, finite dimensional algebras are basic if $\frac{A}{Rad(A)}\cong\mathbb{K}^n$ and universal enveloping algebras of solvable Lie algebras are basic.

Definition: A right coideal subalgebra of the Hopf algebra $U_q(\mathfrak{g})$ is called a Borel subalgebra if it is basic and maximal (with respect to inclusion) among all basic right coideal subalgebras.

This generalizes the characterization of a Borel subalgebra in the maximal solvable Lie algebra.

Borel subalgebras are an essential element in the structure theory of a semisimple Lie algebra $\mathfrak{g}$. The primary motivation for studying Boral algebras of quantum groups is the more general goal to understand the set of all coideal subalgebras. This is considered as the main problem in the area of quantum groups. For Lie algebras, Levi's theorem states that any Lie subalgebra of $\mathfrak{g}$ decomposes into a solvable Lie algebra and a semisimple Lie algebra.