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\title{Quantum Groups Notes\\February 27, 2006\\Dan An}
\author{Gabriel C. Drummond-Cole}
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I wanted more people to come because what I am talking on is very easy.  So first the main theme of today's talk is quantization, which is a loosely defined term.  You have a strict structure, your structure is deformed, with a parameter $\bar{h}$ such that $\bar{h}=0$ gives you the original structure.  Most of the time when you do a deformation of the structure you lose commutativity, taking commutative numbers to noncommutative numbers.  So quantum groups mean you have a Lie group and you are doing some kind of deformation.  So how do we deform a Lie group? Maybe why as well?

To answer the why case first, you have these, in classical mechanics you have a bunch of particles, and with the Hamiltonian equations, certain integrable systems will let you solve this system.  People wanted to solve this system.  People wanted to find a quantum analogue where nice things would still work.

Okay, there are essentially two viewpoints on this.  The first is to work with matrix groups, the so-called classical groups, $SL,\ SO,\ U,\ Sp,$ and these can be written as matrix groups.  The things that are commutative are the matrix elements , the real numbers.  Take matrix elements and deform them so that they are not commutative.

The second viewpoint is that to study Lie groups is really hard so you want to study Lie algebras, and then the universal enveloping algebra.  Once you deform the Lie algebra, it no longer becomes a normal Lie algebra.  In all of these deformations, you need to think of what this will live inside of.  At the enveloping algebra level you need to keep some structures and eliminate some.

Rather than doing it at this second level, we will do it at the first level, because that is where this thing started.

These matrix groups are varieties, so for example $SL_2(k),$ with $k$ an algebraically closed field.  This is $k[X_{11},X_{12},X_{21},X_{22}]/\langle X_{11}X_{22}-X_{21},X_{12}-1\rangle.$

We want to deform this coordinate ring so that it is not commutative.  So the first viewpoint is to deform rings.  Since we're discussing deformations, we want to retain some structures, so we want to keep associativity, and we have to be really careful about what to keep and what to lose.  So let's see what structures transfer, and so on.  Let me give you a general situation.  Here $G$ is a Lie group with $\mathscr{O}(G)$ the coordinate ring or ring of regular functions.  Grothiendieck wanted to work with sheafs on the varieties.  Now $\mathscr{O}(G)$ is an algebra already, but $G$ has a multiplication.  This gives a comultiplication $\mathscr{O}(G)\to \mathscr{O}(G\times G)\cong \mathscr{O}(G)\otimes \mathscr{O}(G).$  So $f$ goes to $f\circ\mu$ where $\mu$ is the multiplication of the Lie group.

There is a counit $\epsilon:\mathscr{O}(G)\to k$ which takes $f$ to $f(1)$ and an antipode $S:\mathscr{O}(G)\to \mathscr{O}(G).$ which sends $f\mapsto f\circ i$ where $i$ is the Lie group inversion.

This gives a richer structure, and they combine to form a structure called a Hopf algebra.  

$\{\mathscr{O}(G),\mu,1,\Delta,\epsilon,S\}$ is a Hopf algebra.   This is an algebra and coalgebra, associative and coassociative, and these are compatible meaning that $\Delta$ and $\epsilon$ are homomorphisms of algebras and $\mu$ and $1$ are homomorphisms of coalgebras.  For this to be a Hopf algebra you need the antipode $S$ which takes $A$ to $A$ and makes the diagram commute:
$$
\xymatrix{
A\otimes A\ar[r]^{S\otimes id}&A\otimes A\ar[d]_\mu\\
A\ar[u]^\Delta\ar[r]^{i\circ \epsilon}&A}
$$

This approach is nice because you feel like you will be constructing the deformation.  So example one is, this is Manin's quantum plane.  This just takes $k[x_1,x_2]=\mathscr{O}(k^2)$ and define 
$\mathscr{O}_q(k^2)=k\langle x_1,x_2|x_1x_2=qx_2x_1\rangle.$  So $q$ is $e^{\pm \bar{h}}.$

The second example is, you want to deform $O_q(SL_2(k)),$ but too many things are happening at once.  So first we want to go to $M_2(k).$  We want to recognize it with matrix multiplication and quantize it.  

So we write down the structure of this and see what we can do about it.  First look at the comultiplication, the functions are generated by the $x_{ij}.$  So we send $x_{ij}$ to $\sum x_{ik}\otimes x_{kj}.$  The counit becoumes $x_{ij}\to \delta_{ij}.$  The antipode is defined as $S(X_{11})=X_{22}$ and $S(X_{12})=-X_{21}$ along with the other expected relations.  This also has an extra structure $\mathscr{O}(k^2)\to \mathscr{O}(M^2(k))\otimes \mathscr{O}(k^2)$ which sends $x_i$ to $X_{i1}\otimes x_1+X_{i2}\otimes x_2.$  You have a similar thing for the right action.  

Since we have all the structures written down we can try to deform them.  We want this to go along with the choice of deformations of the plane.  You want to retain the structure of the Hopf algebra, and you want to retain everything else except commutativity.

First let's work with $\mathscr{O}_q(k^2)$ where $x_1x_2=qx_2x_1.$  Then applying a matrix to both sides of this, we get $(X_{11}\otimes x_1+X_{12}\otimes x_2)(X_{21}\otimes x_1+X_{22}\otimes x_2)=q(X_{21}\otimes x_1+X_{22}\otimes x_2)(X_{11}\otimes x_1+X_{12}\otimes x_2).$  With the appropriate expansion, omitting the calculations, you get $X_{12}X_{22}=qX_{22}X_{12},\\ X_{11}X_{21}=qX_{21}X_{11}$ and \\
$X_{12}X_{21}+qX_{11}X_{22}=qX_{22}X_{11}+q^2X_{21}X_{12}.$

If you write everything down for the right action as well you get 
$ab=qba,\ ac=qca,\ bc=cb,\ bd=qdb,\ cd= qdc,\ ad-da = q-q^{-1}bc.$

To go on from here you need to define the quantum determinant.  We want $\wedge(k^2)\to \mathscr{O}(M_2(k))\otimes\wedge (k^2)$ where $x_1\wedge x_2\mapsto \det (X_{ij})\otimes (x_1\wedge x_2).$

So we need $\wedge_q(k^2)=k\langle \xi_1,\xi_2|\xi_i^2=0,\ \xi_1\xi_2=-q\xi_2\xi_1\rangle.$

Under this map, we get 
$(X_{11}\otimes \xi_1+X_{12}\otimes \xi_2)\wedge(X_{21}\otimes \xi_1+X_{22}\otimes \xi_2)=$``det''$\otimes \xi_1\wedge \xi_2.$ and at the end of the day we get $D_q=X_{11}X_{22}-qX_{12}X_{21}$ so $\mathscr{O}_q(SL_2(k))=\mathscr{O}_q(M_2(k))/\langle D_q-1\rangle.$

So eventually they began with Lie algebras, here you have $\mathbf{U}(\la{sl}_2(k))$ which is generated by $H,\ E,\ F$ subject to the usual relations $[H,E]=2E, [H,F]=-2F$ and $[E,F]=H.$  So the enveloping algebra is the algebra generated by $H,\ E,\ F$ subject to $HE-EH=2E,\ HF-FH=2F,\ EF-FE=H.$

So I could not find in the literature how this went, you have $\mathbf{U}_q(\la{sl}_2(k))$ is generated by $E,F,$ and $K=e^{\bar{h}H}$ as well as $K^{-1}.$  Then $KE=q^2EK,\ KF=q^-2FK,\ EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.$  This quantized one has no good relationship with $SL_2(k)$ by the exponential map.  

You like to work with Lie algebras because you can go back to the Lie group; why would we do this without the exponential?  Well, $\mathbf{U}(\la{sl}_2(k))$ is isomorphic to $C^{-\infty}_{\{e\}}(SL_2(k)$ (distributions supported by the identity).  Since $\mathscr{O}(SL_2(k))$ is a subset, you have it in here and you want to preserve this so $\mathbf{U}(\la{sl}_2(k))$ acts on $\mathscr{O}(\la{SL}_2(k)).$  When you try to study the representations, there is a relation.  That's what justifies working with the Lie algebras.

[Next Monday Dan will speak on more of the same.]



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