Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T15:49:10Z http://mathoverflow.net/feeds/question/93778 http://www.creativecommons.org/licenses/by-nc/2.5/rdf 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/93778/does-there-exist-any-quantum-lie-algebra-embeded-into-the-quantum-enveloping-al/93784#93784 Answer by Jake for Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)? Jake 2012-04-11T17:15:00Z 2012-04-11T17:15:00Z <p>Two seemingly easy to read references are given in the papers 'An introduction to Quantum Lie Algebras' by Delius which is available here: <a href="http://arxiv.org/pdf/q-alg/9605026v1.pdf" rel="nofollow">http://arxiv.org/pdf/q-alg/9605026v1.pdf</a> and 'Quantum Lie Algebras associated to $U_q(\mathfrak{gl}_2)$ and $U_q(\mathfrak{sl}_2)$' available here: <a href="http://arxiv.org/pdf/q-alg/9508013v1.pdf" rel="nofollow">http://arxiv.org/pdf/q-alg/9508013v1.pdf</a></p> <p>They explicitly deal with quantized $\mathfrak{sl}_2$; the first link in terms of $U_h(\mathfrak{sl}_2)$ and the second in terms of $U_q(\mathfrak{sl}_2)$ which it would appear is the case you are interested in.</p> <p>These are from the nineties so I am sure more modern references are available.</p> http://mathoverflow.net/questions/93778/does-there-exist-any-quantum-lie-algebra-embeded-into-the-quantum-enveloping-al/94220#94220 Answer by Réamonn Ó Buachalla for Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)? Réamonn Ó Buachalla 2012-04-16T14:59:46Z 2012-04-16T17:45:00Z <p>You might be interested in the notion of a <strong>braided Lie algebra</strong> 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.</p> <p>The notion of the universal enveloping algebra of a Lie algebra generalizes to this context, and, quoting from Majid's paper <a href="http://arxiv.org/pdf/hep-th/9303148v1.pdf" rel="nofollow">http://arxiv.org/pdf/hep-th/9303148v1.pdf</a>,</p> <blockquote> <p>... the standard quantum deformations $U_q({\frak g})$ are understood as the enveloping algebras of such underlying braided Lie algebras ...</p> </blockquote> <p>The best place to starting learning about these structures is probably Majid's <strong>Quantum Groups Primer</strong> book.</p> <p>The paper <a href="http://arxiv.org/abs/q-alg/9510004%20a" rel="nofollow">arxiv.org/abs/q-alg/9510004</a> mentioned in Jake's answer contains some discussion of these structures.</p> http://mathoverflow.net/questions/93778/does-there-exist-any-quantum-lie-algebra-embeded-into-the-quantum-enveloping-al/94279#94279 Answer by Nicola Ciccoli for Does there exist any "quantum Lie algebra" embeded into the quantum enveloping algebra U_q(g)? Nicola Ciccoli 2012-04-17T08:44:19Z 2012-04-17T09:53:06Z <p>I hope no one gets offended if I summarize a couple of comments adding few details to it. Truly $U_h(\mathfrak g)$ as an associative algebra is not different from $U(\mathfrak g)[[h]]$ (here $\mathfrak g$ is semisimple, everything is char=0). </p> <p>This results is just a rigidity result on the associative algebra $U(\mathfrak g)$ (which reflects rigidity of the Lie algebra $\mathfrak g$). What I wrote inside the bracket seems innocent but it is not: one may say it depends on the fact that the Hochschild cohomology of $U(\mathfrak g)$ is isomorphic to the Chevalley-Eilenberg cohomology of $\mathfrak g$. I find this is nicely explained in <a href="http://people.mpim-bonn.mpg.de/crossi/LectETHbook.pdf" rel="nofollow">http://people.mpim-bonn.mpg.de/crossi/LectETHbook.pdf</a> .</p> <p>When I first saw this result (which is already in the by now classical Chari-Pressley's book) my first impression was "so what's all the fuzz about quantum groups?". The point is that: </p> <ol> <li>They are non trivial deformation of the universal enveloping algebra as a <strong>Hopf algebra</strong>.</li> <li>The isomorphism as associative algebras is neither explicit not canonical. We know it exists from purely cohomological arguments...</li> </ol> <p>How does this connect to the embedding $\mathfrak g\hookrightarrow U(\mathfrak g)$? The canonical way to reconstruct $\mathfrak g$ inside $U(\mathfrak g)$ is to identify it with the set of primitive elements (primitive means $\Delta X=X\otimes 1+1\otimes X$). Therefore this embedding depends on the whole Hopf algebra structure (coproduct to determine primitive elements and product to show that they form a Lie algebra and generate a PBW basis). </p> <p>The set of primitive elements in $U_h(\mathfrak g)$ is trivial and certainly does not allow to reconstruct a PBW basis.</p> <p>One may look for <em>twisted</em> primitive elements with respect to some <em>group-like</em> element different from 1. This is an interesting object, it contains the analogue of simple root vectors, but its algebraic properties are rather weak. Still: starting from twisted primtive elements and performing $q$-commutators, in the context of global quantization $U_q(\mathfrak g)$ it is possible to reconstruct all "root vectors" giving a PBW basis. But there is no obvious algebraic structure even on this set of $q$-root vectors.</p> <p>Of course one may try to understand some kind of embedding $\mathfrak g_h\hookrightarrow U_h(\mathfrak g)$ as a deformation of $\mathfrak g\hookrightarrow U(\mathfrak g)$ as was done in some of the mentioned reference but everything is non canonical and, in my opinion, in the long run it just turns out to be a way of building up an explicit algebra isomorphism; which is known to be technically very complicated.</p> <p>(this comment does not touch on the "braided" side of the story; that, I do not understand)</p>