User tzhang - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:32:56Z http://mathoverflow.net/feeds/user/22829 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101911/why-should-morphisms-between-two-graded-vector-spaces-preserve-grading Why should morphisms between two graded vector spaces preserve grading? tzhang 2012-07-11T02:55:14Z 2012-07-14T06:40:37Z <p>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?</p> http://mathoverflow.net/questions/93778/does-there-exist-any-quantum-lie-algebra-embeded-into-the-quantum-enveloping-al Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)? tzhang 2012-04-11T15:51:32Z 2012-04-17T09:53:06Z <p>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 $U_q(\mathfrak{g})$?</p> <p>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 $K[x,y]$.</p> <p>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 $K_q[x,y]$?</p> http://mathoverflow.net/questions/101911/why-should-morphisms-between-two-graded-vector-spaces-preserve-grading/101975#101975 Comment by tzhang tzhang 2012-07-14T07:39:47Z 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. http://mathoverflow.net/questions/101911/why-should-morphisms-between-two-graded-vector-spaces-preserve-grading/101944#101944 Comment by tzhang tzhang 2012-07-14T07:14:18Z 2012-07-14T07:14:18Z Yes, all $h$ allowed. By the way you mean it is a category with HOM!? But Mariano Su&#225;rez-Alvarez and Yuan suggest it gives wrong notion of isomorphism. What's wrong with it? That is my questions. http://mathoverflow.net/questions/93778/does-there-exist-any-quantum-lie-algebra-embeded-into-the-quantum-enveloping-al Comment by tzhang tzhang 2012-04-12T05:33:13Z 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 <a href="http://mathoverflow.net/questions/55647/relationship-between-different-quantum-deformations" rel="nofollow" title="relationship between different quantum deformations">mathoverflow.net/questions/55647/&hellip;</a> http://mathoverflow.net/questions/93778/does-there-exist-any-quantum-lie-algebra-embeded-into-the-quantum-enveloping-al/93784#93784 Comment by tzhang tzhang 2012-04-12T05:30:30Z 2012-04-12T05:30:30Z Thank you for the references you have given, but it seems that the axioms of &quot;Quantum Lie Algebras&quot; there are not very algebraic explictly. I have found that V. Lyubashenko, A.Sudbery &quot;Quantum Lie algebras of type A_n&quot; <a href="http://arxiv.org/abs/q-alg/9510004" rel="nofollow">arxiv.org/abs/q-alg/9510004</a> and &quot;Quantum deformations of simple Lie algebras&quot; <a href="http://cms.math.ca/10.4153/CMB-1997-017-6" rel="nofollow">cms.math.ca/10.4153/CMB-1997-017-6</a> more usefull to the problems. Thank you all the same!