We have known that any finite dim Lie algebra can be embeded into it's enveloping algebra $U(\mathfrak{g})$, my question is: is there any "quantum Lie algebra" embeded into the quantum enveloping algebra $U_q(\mathfrak{g})$?
The related question is, take $sl(2)$ generated by $\{X,Y,H|[XY]=H, [HX]=2X, [HY]=-2Y\}$ for example, consider the representation on polynomial $K[x,y]$, $K[x,y]$ is in fact a module-algebra over $U(sl(2))$, the elment of $sl(2)$ can be represented by $X=x\frac{\partial}{\partial y}, Y=y\frac{\partial}{\partial x}, H=x\frac{\partial_q}{\partial x}-y\frac{\partial_q}{\partial y}$ (see Kassel "Quantum groups" (GTM155), pp. 109). In fact, $\{x\frac{\partial}{\partial y}, y\frac{\partial}{\partial x}, x\frac{\partial_q}{\partial x}-y\frac{\partial_q}{\partial y}\}$ generated a three dim Lie subalgbebra (isomorphic to $sl(2)$ under the above correspondence) of derivation algebra of $K[x,y]$.
Similariy, Is there quantum Lie algebra contained in $U_q(sl(2))$? In fact, by Kassel "Quantum groups" (GTM155), pp. 146-149, there is an action of $U_q(sl(2))$ on quantum plane $K_q[x,y], E=x\frac{\partial_q}{\partial y}, E=y\frac{\partial_q}{\partial x}, K=\sigma_x\sigma_y^{-1}, K^{-1}=\sigma_y\sigma_x^{-1}$ , so is there any finite dim quantum Lie algebra generated by $E,F,K,K^{-1}$, or does the operators $x\frac{\partial_q}{\partial y}, y\frac{\partial_q}{\partial x}, \sigma_x, \sigma_y^{-1}, \sigma_y, \sigma_x^{-1}$ generate a Lie subalgebra of of derivation algebra of $K_q[x,y]$?
Best Answer
You might be interested in the notion of a braided Lie algebra due to Majid. Roughly speaking this is a coalgebra ${L}$ in a braided category (ie $L$ is an object in a braided category category, with morphisms $\Delta:L \otimes L \to L$, and $\epsilon:L \to C$ satisfying the natural generalization of the axioms of a coalgebra), and in addition a morphism $$ [ , ]:L \otimes L \to L, $$ satisfying a "braided version" of the axioms of a Lie algebra.
The notion of the universal enveloping algebra of a Lie algebra generalizes to this context, and, quoting from Majid's paper http://arxiv.org/pdf/hep-th/9303148v1.pdf,
The best place to starting learning about these structures is probably Majid's Quantum Groups Primer book.
The paper arxiv.org/abs/q-alg/9510004 mentioned in Jake's answer contains some discussion of these structures.