VI. Modules for quantum groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 28 October 2012

The isomorphism theorem in (1.1) is found (with proof) in [Dri1990] p. 330-331. The proof of this theorem uses several cohomological facts,

$$H^2(𝔤,𝔘𝔤)=0,H^1(𝔥,𝔘𝔤/{\left(𝔘𝔤\right)}^{𝔥})=0,\text{and}{H}^1(𝔤,𝔘𝔤\otimes 𝔘𝔤)=0\text{.}$$

The correspondence theorem in (1.3) is also found in [Dri1990] p.331. All of the results in section 2 can be found, with detailed proofs, in [Jan1995] Chapt. 5.

Finite dimensional ${𝔘}_h𝔤\text{-modules}$

As algebras, ${𝔘}_h𝔤\cong 𝔘𝔤\left[\left[h\right]\right]$

The algebra $𝔘𝔤\left[\left[h\right]\right]$ is just the enveloping algebra of the Lie algebra $𝔤$ except over the ring $ℂ\left[\left[h\right]\right]$ (and then $h\text{-adically}$ completed) instead over the field $ℂ\text{.}$ It acts exactly like the algebra $𝔘𝔤,$ the only difference is that we have extended coefficients.

The following theorem says that the algebra ${𝔘}_h𝔤$ and the algebra $𝔘𝔤\left[\left[h\right]\right]$ are exactly the same! In fact we have already seen that this must be so, since $𝔘𝔤$ has no deformations as an algebra (Chapt. III Theorem (2.6)). One might ask: If ${𝔘}_h𝔤$ and $𝔘𝔤\left[\left[h\right]\right]$ are the same then what is the big deal about quantum groups? The answer is: They are the same as algebras but they are not the same when you look at them as Hopf algebras.

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let ${𝔘}_h𝔤$ be the Drinfel'd-Jimbo quantum group corresponding to $𝔤\text{.}$ Then there is an isomorphism of algebras

$$\begin{array}{c}\phi :{𝔘}_h𝔤⟶𝔘𝔤\left[\left[h\right]\right],\text{such that}\\ \\ \text{(a)}\phi ={\text{id}}_{𝔘𝔤}\text{(mod}h\text{),}\text{and}\\ \text{(b)}\phi {\mid }_{𝔥}={\text{id}}_{𝔥},\end{array}$$

where, in the second condition, $𝔥=ℂ\text{-span}\{H_1,\dots ,H_r\}\subseteq {𝔘}_h𝔤\text{.}$

Definition of weight spaces in a ${𝔘}_h𝔤$ module

A finite dimensional ${𝔘}_h𝔤\text{-module}$ is a ${𝔘}_h𝔤\text{-module}$ that is a finitely generated free module as a $ℂ\left[\left[h\right]\right]\text{-module.}$ If $M$ is a finite dimensional ${𝔘}_h𝔤\text{-module}$ and $\mu \in {𝔥}^{*}$ define the $\mu \text{-weight}$ space of $M$ to be the subspace

$$M_{\mu }=\{m\in M\mid am=\mu \left(a\right)m,\text{for all}a\in 𝔥\}\text{.}$$

The dimension of the weight space $M_{\mu }$ is the number of elements in a basis for it, as a $ℂ\left[\left[h\right]\right]\text{-module.}$

Classification of modules for ${𝔘}_h𝔤$

Theorem (1.1) says that ${𝔘}_h𝔤$ and $𝔘𝔤\left[\left[h\right]\right]$ are the same as algebras. Since the category of finite dimensional modules for an algebra depends only on its algebra structure it follows immediately that the category of finite dimensional modules for ${𝔘}_h𝔤$ is the same as the category of modules for $𝔘𝔤\left[\left[h\right]\right]\text{.}$

There is a one to one correspondence between the isomorphism classes of finite dimensional ${𝔘}_h𝔤\text{-modules}$ and the isomorphism classes of finite dimensional $𝔘𝔤\text{-modules}$ given by

$$\begin{array}{ccc}{𝔘}_h𝔤\text{-modules}& \stackrel{1-1}{⟷}& 𝔘𝔤\text{-modules}\\ \\ M& ⟷& M/hM\\ V\left[\left[h\right]\right]& ⟷& V\end{array}$$

where the ${𝔘}_h𝔤$ module structure on $V\left[\left[h\right]\right]$ is defined by the composition

$${𝔘}_h𝔤\stackrel{\sim }{⟶}𝔘𝔤\left[\left[h\right]\right]⟶\text{End}\left(V\left[\left[h\right]\right]\right)\text{.}$$

It follows from condition (b) of Theorem (1.1) that, under the correspondence in the Theorem above, weight spaces of ${𝔘}_h𝔤\text{-modules}$ are taken to weight spaces of $𝔘𝔤\text{-modules}$ and their dimension remains the same. Furthermore, irreducible finite dimensional $𝔘𝔤\text{-modules}$ correspond taken to indecomposable ${𝔘}_h𝔤\text{-modules}$ and vice versa. (Note that $hV\left[\left[h\right]\right]$ is always a ${𝔘}_h𝔤\text{-submodule}$ of the ${𝔘}_h𝔤\text{-module}$ $V\left[\left[h\right]\right]\text{.)}$

The previous theorem combined with Chapt. II Theorem (2.5) gives the following corollary.

Let $P^{+}=\sum _{i=1}^r\mathbb{N}{\omega }_i$ be the set of dominant integral weights for $𝔤,$ as in (2.4). For every $\lambda \in P^{+}$ there is a unique (up to isomorphism) finite dimensional indecomposable ${𝔘}_h𝔤\text{-module}$ $L\left(\lambda \right)$ corresponding to $\lambda \text{.}$

Finite dimensional $U_q𝔤\text{-modules}$

The category of finite dimensional modules for the rational form $U_q𝔤$ of the quantum group is slightly different from the category of finite dimensional modules for ${𝔘}_h𝔤\text{.}$ The construction of the finite dimensional irreducible modules for $U_q𝔤$ is similar to the construction of these modules in the case of the Lie algebra $𝔤\text{.}$ Let us describe how this is done.

Construction of the Verma module $M\left(\lambda \right)$ and the simple module $L\left(\lambda \right)$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let $U_q𝔤$ be the corresponding rational form of the quantum group over a field $k$ and with $q\in k\text{.}$ We shall assume that

$$\text{char}k\ne 2,3\text{and}q\text{is not a root of unity in}k\text{.}$$

Let $\lambda \in P$ be an element of the weight lattice for $𝔤\text{.}$ The Verma module $M\left(\lambda \right)$ is be the $U_q𝔤\text{-module}$ generated by a single vector $v_{\lambda }$ where the action of $U_q𝔤$ satisfies the relations

$$E_i{v}_{\lambda }=0,\text{and}{K}_i{v}_{\lambda }=q^{(\lambda ,{\alpha }_i)}{v}_{\lambda },\text{for all}1\le i\le r\text{.}$$

The map

$$\begin{array}{ccc}{U}_q{𝔫}^{-}& ⟶& M\left(\lambda \right)\\ y& ⟼& y{v}_{\lambda }\end{array}$$

is a vector space isomorphism.

The module $M\left(\lambda \right)$ has a unique maximal proper submodule. For each $\lambda \in P$ define

$$L\left(\lambda \right)=\frac{M\left(\lambda \right)}{N}$$

where $N$ is the maximal proper submodule of the Verma module $M\left(\lambda \right)\text{.}$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let $U_q𝔤$ be the corresponding rational form of the quantum group over a field $k$ with $q\in k\text{.}$ Assume that char $k\ne 2,3$ and that $q$ is not a root of unity in $k\text{.}$ Let $\lambda \in P$ be an element of the weight lattice of $𝔤$ and let $L\left(\lambda \right)$ be the $U_q𝔤\text{-module}$ defined above.

1. The module $L\left(\lambda \right)$ is a simple $U_q𝔤\text{-module.}$
2. The module $L\left(\lambda \right)$ is finite dimensional if and only if $\lambda$ is a dominant integral weight.

Twisting $L\left(\lambda \right)$ to get $L(\lambda ,\sigma )$

Let $Q$ be the root lattice corresponding to $𝔤$ as give in II (2.6) and let $\sigma :Q\to \{\pm 1\}$ be a group homomorphism. The homomorphism $\sigma$ induces an automorphism $\sigma :U_q𝔤\to U_q𝔤$ of $U_q𝔤$ defined by

$$\begin{array}{cccc}\sigma :& U_q𝔤& ⟶& U_q𝔤\\ \\ & E_i& ⟼& \sigma \left({\alpha }_i\right)E_i\\ & F_i& ⟼& F_i\\ \\ & K_i^{\pm 1}& ⟼& \sigma (\pm {\alpha }_i)K_i^{\pm 1},\end{array}$$

where ${\alpha }_1,\dots ,{\alpha }_r$ are the simple roots for $𝔤\text{.}$ Let $\lambda \in P$ be an element of the weight lattice and let $L\left(\lambda \right)$ be the irreducible $U_q𝔤\text{-module}$ defined in (2.1). Define a $U_q𝔤\text{-module}$ $L(\lambda ,\sigma )$ by defining

1. $L(\lambda ,\sigma )=L\left(\lambda \right)$ as vector spaces,
2. $U_q𝔤$ acts on $L(\lambda ,\sigma )$ by the formulas $$u✶m-\sigma \left(u\right)m,\text{for all}u\in U_q𝔤,m\in L\left(\lambda \right),$$ where $\sigma$ is the automorphism of $U_q𝔤$ defined above.

Classification of finite dimensional irreducible modules for $U_q𝔤$

Let $𝔤$ be a finite dimensional complex simple Lie algebra and let $U_q𝔤$ be the rational form of the quantum group over a field $k\text{.}$ Assume that $\text{char}k\ne 2,3$ and $q\in k$ is not a root of unity in $k\text{.}$ Let $P^{+}$ be the set of dominant integral weights for $𝔤$ and let $Q$ be the root lattice for $𝔤$ (see II (2.6)).

1. Let $\lambda \in P^{+}$ and let $\sigma :Q\to \{\pm 1\}$ be a group homomorphism. The modules $L(\lambda ,\sigma )$ defined in (2.2) are all finite dimensional irreducible $U_q𝔤\text{-modules.}$
2. Every finite dimensional $U_q𝔤\text{-module}$ is isomorphic to $L(\lambda ,\sigma )$ for some $\lambda \in P^{+}$ and some group homomorphism $\sigma :Q\to \{\pm 1\}\text{.}$

Weight spaces for $U_q𝔤\text{-modules}$

Retain the notations and assumptions from (2.3) and let $(,)$ be the inner product on ${𝔥}_{ℝ}^{*}$ defined in II (2.7). Let $M$ be a finite dimensional $U_q𝔤\text{-module.}$ Let $\sigma :Q\to \{\pm 1\}$ be a group homomorphism and let $\lambda \in P\text{.}$ The $(\lambda ,\sigma )\text{-weight space}$ of $M$ is the vector space

$$M_{(\lambda ,\sigma )}=\{m\in M\mid K_{i}m=\sigma \left({\alpha }_i\right)q^{(\lambda ,{\alpha }_i)}m\text{for all}1\le i\le r\text{.}\}$$

The following proposition if analogous to Chapt. II Proposition (2.4).

Every finite dimensional $U_q𝔤\text{-module}$ is a direct sum of its weight spaces.

The following theorem says that the dimensions of the weight spaces of irreducible $U_q𝔤\text{-modules}$ coincide with the dimensions of the weight space of corresponding irreducible modules for the Lie algebra $𝔤\text{.}$

Let $\lambda \in P^{+}$ be a dominant integral weight and let $\sigma$ be a group homomorphism $\sigma :Q\to \{\pm 1\}\text{.}$ Let $V^{\lambda }$ be the simple $𝔤\text{-module}$ indexed by the $\lambda$ and let $L(\lambda ,\sigma )$ be the irreducible $U_{𝔤}\text{-module}$ indexed by the pair $(\lambda ,\sigma )\text{.}$ Then, for all $\mu \in P$ and all group homomorphisms $\tau :Q\to \{\pm 1\},$

$${\text{dim}}_k\left(L{(\lambda ,\sigma )}_{\mu ,\sigma }\right)={\text{dim}}_{ℂ}\left({\left(V^{\lambda }\right)}_{\mu }\right)\text{and}{\text{dim}}_k\left(L{(\lambda ,\sigma )}_{\mu ,\tau }\right)=0,\text{if}\sigma \ne \tau \text{.}$$

Notes and References

This is an excerpt from a paper entitled Quantum groups: A survey of definitions, motivations and results by Arun Ram. Research and writing supported in part by an Australian Research Council fellowship and a National Science Foundation grant DMS-9622985.

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