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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 $sl(2)={X,Y,H|[XY]=H, sl(2)$ generated by ${X,Y,H|[XY]=H, [HX]=2X, [HY]=2Y}$ 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}$. (see Kassel "Quantum groups" (GTM155),pp109) In fact,${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)$under the above correspendence) of derivation algebra of$K[x,y]$. Similariy, Is there quantum Lie algebra contained in$U_q(sl(2))$? In fact, by Kassel "Quantum groups" (GTM155),pp146--149, there is an action of$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]$? 2 added 38 characters in body We have known that any finite dim Lie algebra can be embeded into it's enveloping algebra$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)={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}$. (see Kassel "Quantum groups" (GTM155),pp109) In fact,${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)$under the above correspendence) of derivation algebra of$K[x,y]$. Similariy, Is there quantum Lie algebra contained in$U_q(sl(2))$? In fact, by Kassel "Quantum groups" (GTM155),pp146--149, there is an action of$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}$, or is does the operator operators$x\frac{\partial_q}{\partial yy}, y\frac{\partial_q}{\partial x}, \sigma_x,\sigma_y$generated sigma_x, \sigma_y^{-1}, \sigma_y, \sigma_x^{-1}$ generate a Lie subalgebra of of derivation algebra of $K_q[x,y]$?

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# Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)?

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 $sl(2)={X,Y,H|[XY]=H, [HX]=2X, [HY]=2Y}$ 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),pp109) 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 correspendence) 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),pp146--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 is the operator $x\frac{\partial_q}{\partial y, y\frac{\partial_q}{\partial x}, \sigma_x,\sigma_y$ generated a Lie subalgebra of of derivation algebra of $K_q[x,y]$?