User tzhang - MathOverflow

Why should morphisms between two graded vector spaces preserve grading?

2012-07-11T02:55:14Z

Denote the category of $G$-graded vector spaces by SVect, where $G$ is an abelian group. Then morphisms $f:V\to W$ (where $V$ and $W$ are two $G$-graded vector spaces) should be $f(V_g) \subseteq W_g$ for any $g\in G$ in many references. Why don't we take morphisms as other homomorphisms between $V$ and $W$ such as $f(V_{g}) \subseteq W_{g+h}$ for any $g\in G$? In other words, why we use $Hom_{G}(V,W)=\{f|f(V_g) \subseteq W_g\}$ other than $HOM(V,W)=\{f|f(V_g) \subseteq W_{g+h}\forall h\in G\}$ where $HOM(V,W)$ is usually called internal hom in References. Does not SVect with $HOM(V,W)$ as morphisms form a category?

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

2012-04-11T15:51:32Z

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),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 $K_q[x,y]$?

Comment by tzhang

2012-07-14T07:39:47Z

Yes, it seems if we want an isomorphism between $Hom(U\otimes V, W)$ and $Hom(U, Hom(V,W))$ like in the ordinary case, so $Hom(V,W)$ should be an object in this category and we must replace with HOM.

Comment by tzhang

2012-07-14T07:14:18Z

Yes, all $h$ allowed. By the way you mean it is a category with HOM!? But Mariano Suárez-Alvarez and Yuan suggest it gives wrong notion of isomorphism. What's wrong with it? That is my questions.

Comment by tzhang

2012-04-12T05:33:13Z

U(g) and Uq(g) are different as algebras and coalgebras, the relationship between “different” quantum deformations are given in mathoverflow.net/questions/55647/…

Comment by tzhang

2012-04-12T05:30:30Z

Thank you for the references you have given, but it seems that the axioms of "Quantum Lie Algebras" there are not very algebraic explictly. I have found that V. Lyubashenko, A.Sudbery "Quantum Lie algebras of type A_n" arxiv.org/abs/q-alg/9510004 and "Quantum deformations of simple Lie algebras" cms.math.ca/10.4153/CMB-1997-017-6 more usefull to the problems. Thank you all the same!