\nopagenumbers
\def\sm{\setminus}
\def\ra{\rightarrow}
\def\eqv{\Leftrightarrow}
\def\ot{\otimes}
\def\TM{M_1\ot\dots\ot M_n}
\def\TMs{M_1^*\ot\dots\ot M_n^*}
\def\gf{\varphi}
\def\gre{\varepsilon}
\def\gk{\kappa}
\def\gd{\delta}
\def\gi{\iota}
\def\F{{\cal F}}
\def\G{{\cal G}}
\def\F{{\cal F}}
\def\C{{\bf C}}
\def\N{{\bf N}}
\def\Z{{\bf Z}}
\def\Re{\mathop{\rm Re}\nolimits}
\def\exp{\mathop{\rm exp}\nolimits}
\def\ker{\mathop{\rm ker}\nolimits}
\def\id{\mathop{\rm id}\nolimits}
\font\smallrm=cmr7
\centerline{\bf Varchenko's View of Quantum Groups}
\medskip
\centerline{\smallrm typeset by S.Duzhin, Nov. 30, 1991}
\bigskip
{\bf Main theorem.} {\sl The diagram
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\smash{
    \mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow
  \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{V(\TM)_{m-2k} &\mapright{S}
        &V(\TMs)_{m-2k}\cr
\mapdown{\gf_C}&&\mapdown{\gf_C}\cr
H_k(P_{n,k}(z)\sm U_{n,k}(z);\F(z))_{ant}
&\mapright{I}
&H_k(P_{n,k}(z), U_{n,k}(z);\F(z))_{ant} }
$$
is commutative.  The isomorphisms $\gf_C$ are horizontal with
respect to the Gauss-Manin connection and send the $R$-matrix
representation of the pure braid group in the upper line
to its monodromy representation in the lower line.}
\medskip
{\bf Remark.} The upper line is the `contravariant operator'
between `vacuum spaces'--- classical objects in the theory of
quantum groups.
The lower line features their geometric interpretation discovered by
Varchenko.
\medskip
{\bf Data that define the diagram.}

The {\sl upper line\/} is defined by the choice of
a natural number $n$, a non-negative integer $k$,
a non-zero complex number $q=\exp(2\pi i/\gk)$, $\gk\in\C^*$
(parameter of the quantum group $U_q sl_2$), and a set $m_1,\dots,m_n$
of arbitrary complex numbers (parameters of Verma modules).

To define the {\sl lower line\/}, one has to fix, moreover,
a set $z_1,\dots,z_n$ of different complex numbers.

The {\sl isomorphisms\/} $\gf_C$ are defined by a comb $C$
with boundary $z_1,\dots,z_n$, i.e. a collection on $n$ non-intersecting
curves in $\C$ such that the $j$-th curve coincides with
$\Re(z)=j$ in a neighbourhood of infinity and ends at $z_j$.
\medskip
{\bf Explanation of the upper line.}

$M=\TM$ is the tensor product (over $\C$) of Verma modules $M_i=M(m_i)$.
$M^*=\TMs$ is the tensor product of corresponding dual modules.
$M_i^*$ is defined as the space of linear functionals on $M_i$ that take
non-zero values only on a finite number of the standard
basis elements. Module structure is given by
$h\psi(\cdot)=\psi(h\cdot)$, $e\psi(\cdot)=\psi(f\cdot)$,
$f\psi(\cdot)=\psi(e\cdot)$.

$V(M)_{m-2k}=(M)_{m-2k} \cap\bigcap_{l=1}^k \ker[e^l]_q$ is the
{\sl vacuum subspace\/} of level $k$.  Here $(M)_{m-2k}$ is the eigenspace
of $h$ with eigenvalue $m-2k$ (note: the entire $M$ is the direct sum
of $M_{m-2k}$ over all non-negative integers $k$) and the operator
$[e^l]_q$ is defined as $(1/c_{l,q})e^l$, if
$c_{l,q}=(q^{1/2}+q^{-1/2})(q+1+q^{-1})\dots
(q^{(l-1)/2}+\dots q^{-(l-1)/2}) \ne 0$, otherwise one has to
consider the limit of an appropriate family of operators
$[e^l]_{\tilde q}$ as $\tilde q\ra q$.
It may happen that, for some $v\in M$, $ev=0$ but $[e^l]_qv\ne0$,
although $[e^l]_q$ looks like a multiple of $e^l$.
If $q$ is not a root of 1, then $V(M)_{m-2k}=(M)_{m-2k}\cap \ker e$.
If $q$ is a primitive root of $q^p=1$, then
$V(M)_{m-2k} = (M)_{m-2k}\cap\ker e\cap\ker[e^p]_q$.

For the dual module, the vacuum subspace is simply
$V(M^*)_{m-2k} = (M^*)_{m-2k}\cap\ker e$.

The operator $S:\TM\ra \TMs$ takes $x_1\ot\dots\ot x_n$ into
$B(x_1,\cdot)\ot\dots\ot B(x_n,\cdot)$ where the bilinear form $B$
is symmetric and satisfies $B(v,v)=1$, $B(fx,y)=B(x,ey)$.

\medskip
{\bf Explanation of the lower line.}

$z=(z_1,\dots,z_n)$ is a point in the complex space $\C^n$.
$P_{n,k}$ is the fibre over $z$ of the projection
$\C^{n+k}\ra\C^n: (t_1,\dots,t_{n+k}) \mapsto (t_1,\dots,t_n)$.
$U_{n,k} = P_{n,k}\cap U_{n+k}$ where $U_{n+k}$ is the union of
all diagonal hyperplanes in $\C^{n+k}$.

$\F$ is the local system of coefficients defined by the function
$$F(t_1,\dots,t_k)=
\Pi(z_i-z_j)^{m_im_j/2\gk}\Pi(z_i-t_j)^{-m_i/\gk}\Pi(t_i-t_j)^{2/\gk}.$$
Subscript $ant$ denotes the totally antisymmetric part of the homology
spaces with respect to the natural action of the permutation group of
coordinates in $P_{n,k}=\C^k$.  Operator $I$ of the lower line is
the natural operator in homology arising from the identity chain mapping.

\eject
\centerline{\bf Help for beginners}
\medskip

{\bf Hopf algebras.}

A Hopf algebra (=bialgebra) over a basic ring $K$ is a $K$-module
$A$ equipped with two coherent sructures: that of an associative
algebra with unity and that of an associative coalgebra with counity.
The coalgebra structure consists of comultiplication
$\gd:A\ra A\ot A$ and counity $\gre:A\ra K$. To obtain the axioms
that govern $\gd$ and $\gre$, one has to write the standard
algebra axioms in the form of commutative diagrams involving
multiplication $\mu:A\ot A\ra A$ and unity $\gi:K\ra A$ and take the
dual diagrams. The algebra and coalgebra structures of a Hopf algebra
must agree in the sense that the following 4 conditions hold:

(1) $\gre(1)=1$,

(2) $\gd(1)=1\ot 1$,

(3) $\gre(ab)=\gre(a)\gre(b)$,

(4) $\gd(ab)=\gd(a)\gd(b)$.

\noindent
Observing that

(1) \& (3) $\eqv$ $\gre$ is an algebra homomorphism,

(2) \& (4) $\eqv$ $\gd$ is an algebra homomorphism,

(1) \& (2) $\eqv$ $\gi$ is a coalgebra homomorphism,

(3) \& (4) $\eqv$ $\mu$ is a coalgebra homomorphism,

\noindent
we can state the definition of a Hopf algebra in either of
the two forms:

(A) $\gre$ and $\gd$ are algebra homomorphisms, or

(B) $\gi$ and $\mu$ are coalgebra homomorphisms.
\medskip
{\sl Examples.}

\noindent
In the space of functions on a finite group, there are two
natural Hopf algebra structures:

(1) The commutative functional algebra with comultiplication
$\gd(f)(x,y)=f(xy)$.

(2) The (convolution) group algebra  with $\gd(g)=g\ot g$ on generators.

The universal enveloping algebra $U(\G)$ of a Lie algebra $\G$ is
a Hopf algebra with $\gre(1)=1$, $\gre(x)=0$, $\gd(1)=0$,
$\gd(x)= x\ot 1 + 1\ot x$ for $x\in \G$.

\medskip
{\bf Quantum groups and Verma modules.}
The quantum group $A=U_q sl_2$ is an infinite dimensional associative
\C--algebra whose elements are formal power series in 3 non-commutative
variables $e$, $f$ and $h$ subject to relations
$[h,e]=2e$, $[h,f]=-2f$ and $[e,f]=(h)_q$, where
$(a)_q=q^{a/2}-q^{-a/2}$ and $q^b=\exp(2\pi ib/\gk)$
(both definitions equally apply to formal series and complex numbers).
$A$ can be viewed as a deformation of the universal enveloping algebra
$U(sl_2)$, the latter appearing when the series $(h)_q$ is replaced
by its first term.

The coalgebra structure in $A$ is defined by counity
$\gre(1)=1,\,\gre(x)=0,\,x=e,f,h$ and comultiplication
$\gd(h)=h\ot 1 +1\ot h$,
$\gd(e)=e\ot q^{h/4} + q^{-h/4}\ot e$,
$\gd(f)=f\ot q^{h/4} + q^{-h/4}\ot f$.

A Verma module over $A$ with highest weight $m\in\C$ is the
$A$--module $M(m)$ with one generator $v$ satisfying $ev=0$, $hv=mv$.
(Choosing $m\in\N$ and taking $U(sl_2)$ instead of $A$ gives the usual
irreducible finite dimensional representations of $sl_2$.)
Vectors $v_k=f^k v$, $k=0,1,2,\dots$ form a \C--basis of $M(m)$ satisfying
$fv_k=v_{k+1}$, $hv_k=(m-2k)v_k$, $ev_k=(m-k+1)_q(k)_q/(1)_qv_{k-1}$.

If $M_1,M_2$  are modules over the quantum group $A$,
then $M_1\ot_{\C}M_2$ also has the structure of an $A$--module:
the action of an element $a\in A$ is the natural action of
$\gd(a)\in A\ot A$.  Modules $M_1\ot M_2$ and $M_2\ot M_1$ are
isomorphic. The isomorphism is established by $PR$ where $P$ is the
transposition of factors, while $R$ is the operator of multiplication
by the so called $R$--matrix
$$R= q^{h\ot h/4}\sum_{k\ge0}q^{-k(k+1)/2}(1)^k_q/(k)_q!
q^{kh/4}e^k\ot q^{-kh/4}f^k,$$
where $(k)_q!=(1)_q\cdot\dots\cdot(k)_q$.

The $R$-matrix representation of the pure braid group on $n$ strings
in the tensor product of $n$ $A$--modules is defined by the action
of the $j$-th braid generator $\sigma_j$ as $PR$ on
the $j$-th and $(j+1)$-th factors.


\eject
{\bf Homology with local coefficients.}
A {\sl local system} of abelian groups on a topological space $X$ consists
in assigning a group $G_x$ to every point $x\in X$ and an isomorphism
$\tau_s: G_x\ra G_y$ to every path in $X$ beginning at $x$
and ending at $y$ in such a way that (i) $\tau_s$ depends only on the
homotopy class of $s$, (ii) $\tau_s=\id$, if $s$ is a constant path,
and (iii) $\tau_{s's''}=\tau_{s'}\tau_{s''}$ whenever both sides make
sense.  A section of the local system $\G$ over a subset $Y\subset X$
is a family $\{\,g_y\in G_y \mid y\in Y\,\}$ of elements going into
one another under isomorphisms over the paths contained entirely in $Y$.
This shows that the notion of the local system is equivalent to that
of a locally constant sheaf.

Let $F$ be a multivalued complex analytic function in a domain
$U\subset \C^n$. Suppose that (i) any two branches of $F$ differ by
a non-zero multiplicative constant, (ii) $F$ has no zeros in $U$.
Then $F$ defines a local system of additive groups $\C$ in $U$.

To define {\sl homology} with coefficients in a local system,
one has to modify the standard definition of singular chains
by considering formal combinations $\sum g_i\gf_i$, where
$\gf:Q^k\ra X$ is a singular $k$--cube and $g_i$ is
a section of the local system $\gf^*(\G)$ over $Q^k$.
In other words, the homology with local coefficients is
nothing but the homology with coefficients in the corresponding
locally constant sheaf.

{\sl Examples\/}.

(i) Take $X=S^1$ and fix a point $\star\in S^1$.
Let $G_x=\Z$ for every $x\in S^1$ and $\tau_s=\id$ or $-\id$
according to the parity of the number of times the path $s$
goes through $\star$.
The homology is $H_0(S^1;\G)=\Z_2$, $H_1(S^1;\G)=0$.

(ii) If $\F$ is the system of local coefficients in $\C\sm0$
defined by the function $F(z)=z^{1/2}$, then
$H_0(\C\sm0;\F)=0$, $H_1(\C\sm0;\F)=0$.

{\sl Relative homology with local coefficients\/}.
If $Y\subset X$, where $X$ is equipped with a local system $\G$,
then one can restrict $\G$ to $Y$ and obtain a pair of chain
complexes with local coefficients $C_k(Y;\F)\subset C_k(X;\F)$
whose quotient in a standard way defines the relative homology
with local coefficients.

This construction is valid only if the local system is defined
on the entire space $X$.  However, if we consider the local system
$\F$ associated with an analytic function $F$ as above, we can
define the relative homology of the pair $\bar U,Z$, where
$\bar U= U\cup Z,\,Z=F^{-1}(0)$, with coefficients in $\F$, as the
homology of the corresponding sheaf extended to $\bar U$ in a natural
way.  Note that (i) the extended sheaf is no longer locally constant
and (ii) the relative homology modulo $Z$ coincides with
absolute homology of $\bar U$, since all the chains of $Z$ appear
with zero coefficients.

{\sl Example\/}.
If $\F$ is defined by the function $F(z)=z^{1/2}$, then
$H_0(\C,0;\F)=0$, $H_1(\C,0;\F)=0$.


\end