Verma module

We can explain the idea of a Verma module as follows. Let \mathfrak{g} be a semisimple Lie algebra (over \mathbb{C}, for simplicity). Let \mathfrak{h} be a fixed Cartan subalgebra of \mathfrak{g} and let R be the associated root system. Let R^+ be a fixed set of positive roots. For each \alpha\in R^+, choose a nonzero element X_\alpha for the corresponding root space \mathfrak{g}_\alpha and a nonzero element Y_\alpha in the root space \mathfrak{g}_{-\alpha}. We think of the X_\alpha's as "raising operators" and the Y_\alpha's as "lowering operators." Now let \lambda\in\mathfrak{h}^* be an arbitrary linear functional, not necessarily dominant or integral. Our goal is to construct a representation W_\lambda of \mathfrak{g} with highest weight \lambda that is generated by a single nonzero vector v with weight \lambda. The Verma module is one particular such highest-weight module, one that is maximal in the sense that every other highest-weight module with highest weight \lambda is a quotient of the Verma module. It will turn out that Verma modules are always infinite dimensional; if \lambda is dominant integral, however, one can construct a finite-dimensional quotient module of the Verma module. Thus, Verma modules play an important role in the classification of finite-dimensional representations of \mathfrak{g}. Specifically, they are an important tool in the hard part of the theorem of the highest weight, namely showing that every dominant integral element actually arises as the highest weight of a finite-dimensional irreducible representation of \mathfrak{g}. We now attempt to understand intuitively what the Verma module with highest weight \lambda should look like. Since v is to be a highest weight vector with weight \lambda, we certainly want :H\cdot v=\lambda(H)v,\quad H\in\mathfrak{h} and :X_\alpha\cdot v=0,\quad\alpha\in R^+. Then W_\lambda should be spanned by elements obtained by lowering v by the action of the Y_\alpha's: :Y_{\alpha_{i_1}}\cdots Y_{\alpha_{i_M}}\cdot v. We now impose only those relations among vectors of the above form required by the commutation relations among the Y's. In particular, the Verma module is always infinite-dimensional. The weights of the Verma module with highest weight \lambda will consist of all elements \mu that can be obtained from \lambda by subtracting integer combinations of positive roots. The figure shows the weights of a Verma module for \mathfrak{sl}(3;\mathbb C). A simple re-ordering argument shows that there is only one possible way the full Lie algebra \mathfrak{g} can act on this space. Specifically, if Z is any element of \mathfrak{g}, then by the easy part of the Poincaré–Birkhoff–Witt theorem, we can rewrite :ZY_{\alpha_{i_1}}\cdots Y_{\alpha_{i_M}} as a linear combination of products of Lie algebra elements with the raising operators X_\alpha acting first, the elements of the Cartan subalgebra, and last the lowering operators Y_\alpha. Applying this sum of terms to v, any term with a raising operator is zero, any factors in the Cartan act as scalars, and thus we end up with an element of the original form. To understand the structure of the Verma module a bit better, we may choose an ordering of the positive roots as \alpha_1,\ldots\alpha_n and we denote the corresponding lowering operators by Y_1,\ldots Y_n. Then by a simple re-ordering argument, every element of the above form can be rewritten as a linear combination of elements with the Y's in a specific order: :Y_1^{k_1}\cdots Y_n^{k_n}v, where the k_j's are non-negative integers. Actually, it turns out that such vectors form a basis for the Verma module. Although this description of the Verma module gives an intuitive idea of what W_\lambda looks like, it still remains to give a rigorous construction of it. In any case, the Verma module gives—for any \lambda, not necessarily dominant or integral—a representation with highest weight \lambda. The price we pay for this relatively simple construction is that W_\lambda is always infinite dimensional. In the case where \lambda is dominant and integral, one can construct a finite-dimensional, irreducible quotient of the Verma module. ==The case of sl(2; C)==

Let {X,Y,H} be the usual basis for \mathrm{sl}(2;\mathbb{C}): : X = \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} \qquad Y = \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix} \qquad H = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} ~, with the Cartan subalgebra being the span of H. Let \lambda be defined by \lambda(H)=m for an arbitrary complex number m. Then the Verma module with highest weight \lambda is spanned by linearly independent vectors v_0,v_1,v_2,\dots and the action of the basis elements is as follows: :Y\cdot v_j=v_{j+1};\quad X\cdot v_j=j(m-(j-1))v_{j-1};\quad H\cdot v_j=(m-2j)v_j. (This means in particular that H\cdot v_0=mv_0 and that X\cdot v_0=0.) These formulas are motivated by the way the basis elements act in the finite-dimensional representations of \mathrm{sl}(2;\mathbb{C}), except that we no longer require that the "chain" of eigenvectors for H has to terminate. In this construction, m is an arbitrary complex number, not necessarily real or positive or an integer. Nevertheless, the case where m is a non-negative integer is special. In that case, the span of the vectors v_{m+1},v_{m+2},\ldots is easily seen to be invariant—because X\cdot v_{m+1}=0. The quotient module is then the finite-dimensional irreducible representation of \mathrm{sl}(2;\mathbb{C}) of dimension m+1. ==Definition of Verma modules==

There are two standard constructions of the Verma module, both of which involve the concept of universal enveloping algebra. We continue the notation of the previous section: \mathfrak{g} is a complex semisimple Lie algebra, \mathfrak{h} is a fixed Cartan subalgebra, R is the associated root system with a fixed set R^+ of positive roots. For each \alpha\in R^+, we choose nonzero elements X_\alpha\in\mathfrak{g}_\alpha and Y_\alpha\in\mathfrak{g}_{-\alpha}. As a quotient of the enveloping algebra The first construction of the Verma module is a quotient of the universal enveloping algebra U(\mathfrak{g}) of \mathfrak{g}. Since the Verma module is supposed to be a \mathfrak{g}-module, it will also be a U(\mathfrak{g})-module, by the universal property of the enveloping algebra. Thus, if we have a Verma module W_\lambda with highest weight vector v, there will be a linear map \Phi from U(\mathfrak{g}) into W_\lambda given by :\Phi(x)=x\cdot v,\quad x\in U(\mathfrak{g}). Since W_\lambda is supposed to be generated by v, the map \Phi should be surjective. Since v is supposed to be a highest weight vector, the kernel of \Phi should include all the root vectors X_\alpha for \alpha in R^+. Since, also, v is supposed to be a weight vector with weight \lambda, the kernel of \Phi should include all vectors of the form :H-\lambda(H)1,\quad H\in\mathfrak{h}. Finally, the kernel of \Phi should be a left ideal in U(\mathfrak{g}); after all, if x\cdot v=0 then (yx)\cdot v=y\cdot (x\cdot v)=0 for all y\in U(\mathfrak{g}). The previous discussion motivates the following construction of Verma module. We define W_\lambda as the quotient vector space :W_\lambda=U(\mathfrak{g})/I_\lambda, where I_\lambda is the left ideal generated by all elements of the form :X_\alpha,\quad\alpha\in R^+, and :H-\lambda(H)1,\quad H\in\mathfrak{h}. Because I_\lambda is a left ideal, the natural left action of U(\mathfrak{g}) on itself carries over to the quotient. Thus, W_\lambda is a U(\mathfrak{g})-module and therefore also a \mathfrak{g}-module. By extension of scalars The "extension of scalars" procedure is a method for changing a left module V over one algebra A_1 (not necessarily commutative) into a left module over a larger algebra A_2 that contains A_1 as a subalgebra. We can think of A_2 as a right A_1-module, where A_1 acts on A_2 by multiplication on the right. Since V is a left A_1-module and A_2 is a right A_1-module, we can form the tensor product of the two over the algebra A_1: :A_2\otimes_{A_1}V. Now, since A_2 is a left A_2-module over itself, the above tensor product carries a left module structure over the larger algebra A_2, uniquely determined by the requirement that :a_1\cdot (a_2\otimes v)=(a_1a_2)\otimes v for all a_1 and a_2 in A_2. Thus, starting from the left A_1-module V, we have produced a left A_2-module A_2\otimes_{A_1}V. We now apply this construction in the setting of a semisimple Lie algebra. We let \mathfrak{b} be the subalgebra of \mathfrak{g} spanned by \mathfrak{h} and the root vectors X_\alpha with \alpha\in R^+. (Thus, \mathfrak{b} is a "Borel subalgebra" of \mathfrak{g}.) We can form a left module F_\lambda over the universal enveloping algebra U(\mathfrak{b}) as follows: • F_\lambda is the one-dimensional vector space spanned by a single vector v together with a \mathfrak{b}-module structure such that \mathfrak{h} acts as multiplication by \lambda and the positive root spaces act trivially: :\quad H\cdot v=\lambda(H)v,\quad H\in\mathfrak{h};\quad X_\alpha\cdot v=0,\quad \alpha\in R^+. The motivation for this formula is that it describes how U(\mathfrak{b}) is supposed to act on the highest weight vector in a Verma module. Now, it follows from the Poincaré–Birkhoff–Witt theorem that U(\mathfrak{b}) is a subalgebra of U(\mathfrak{g}). Thus, we may apply the extension of scalars technique to convert F_\lambda from a left U(\mathfrak{b})-module into a left U(\mathfrak{g})-module W_\lambda as follow: : W_\lambda := U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} F_\lambda. Since W_\lambda is a left U(\mathfrak{g})-module, it is, in particular, a module (representation) for \mathfrak{g}. The structure of the Verma module Whichever construction of the Verma module is used, one has to prove that it is nontrivial, i.e., not the zero module. Actually, it is possible to use the Poincaré–Birkhoff–Witt theorem to show that the underlying vector space of W_\lambda is isomorphic to : U(\mathfrak{g}_-) where \mathfrak{g}_- is the Lie subalgebra generated by the negative root spaces of \mathfrak{g} (that is, the Y_\alpha's). ==Basic properties==

Verma modules, considered as \mathfrak{g}-modules, are highest weight modules, i.e. they are generated by a highest weight vector. This highest weight vector is 1\otimes 1 (the first 1 is the unit in \mathcal{U}(\mathfrak{g}) and the second is the unit in the field F, considered as the \mathfrak{b}-module F_\lambda) and it has weight \lambda. Multiplicities Verma modules are weight modules, i.e. W_\lambda is a direct sum of all its weight spaces. Each weight space in W_\lambda is finite-dimensional and the dimension of the \mu-weight space W_\mu is the number of ways of expressing \lambda-\mu as a sum of positive roots (this is closely related to the so-called Kostant partition function). This assertion follows from the earlier claim that the Verma module is isomorphic as a vector space to U(\mathfrak{g}_-), along with the Poincaré–Birkhoff–Witt theorem for U(\mathfrak{g}_-). Universal property Verma modules have a very important property: If V is any representation generated by a highest weight vector of weight \lambda, there is a surjective \mathfrak{g}-homomorphism W_\lambda\to V. That is, all representations with highest weight \lambda that are generated by the highest weight vector (so called highest weight modules) are quotients of W_\lambda. Irreducible quotient module W_\lambda contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight \lambda. If the highest weight \lambda is dominant and integral, one then proves that this irreducible quotient is actually finite dimensional. As an example, consider the case \mathfrak g = \operatorname{sl}(2;\mathbb C) discussed above. If the highest weight m is "dominant integral"—meaning simply that it is a non-negative integer—then Xv_{m+1}=0 and the span of the elements v_{m+1},v_{m+2},\ldots is invariant. The quotient representation is then irreducible with dimension m+1. The quotient representation is spanned by linearly independent vectors v_0,v_1,\ldots,v_m. The action of \operatorname{sl}(2;\mathbb C) is the same as in the Verma module, except that Yv_m=0 in the quotient, as compared to Yv_m=v_{m+1} in the Verma module. The Verma module W_\lambda itself is irreducible if and only if \lambda is antidominant. Consequently, when \lambda is integral, W_\lambda is irreducible if and only if none of the coordinates of \lambda in the basis of fundamental weights is from the set \{0,1,2,\ldots\}, while in general, this condition is necessary but insufficient for W_\lambda to be irreducible. Other properties The Verma module W_\lambda is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight \tilde\lambda. In other word, there exist an element w of the Weyl group W such that :\lambda=w\cdot\tilde\lambda where \cdot is the affine action of the Weyl group. The Verma module W_\lambda is called singular, if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight \tilde\lambda so that \tilde\lambda+\delta is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights). ==Homomorphisms of Verma modules==

For any two weights \lambda, \mu a non-trivial homomorphism :W_\mu\rightarrow W_\lambda may exist only if \mu and \lambda are linked with an affine action of the Weyl group W of the Lie algebra \mathfrak{g}. This follows easily from the Harish-Chandra theorem on infinitesimal central characters. Each homomorphism of Verma modules is injective and the dimension :\dim(\operatorname{Hom}(W_\mu, W_\lambda))\leq 1 for any \mu, \lambda. So, there exists a nonzero W_\mu\rightarrow W_\lambda if and only if W_\mu is isomorphic to a (unique) submodule of W_\lambda. The full classification of Verma module homomorphisms was done by Bernstein–Gelfand–Gelfand and Verma and can be summed up in the following statement: There exists a nonzero homomorphism W_\mu\rightarrow W_\lambda if and only if there exists a sequence of weights ::\mu=\nu_0\leq\nu_1\leq\ldots\leq\nu_k=\lambda such that \nu_{i-1}+\delta=s_{\gamma_i}(\nu_i+\delta) for some positive roots \gamma_i (and s_{\gamma_i} is the corresponding root reflection and \delta is the sum of all fundamental weights) and for each 1\leq i\leq k, (\nu_i+\delta)(H_{\gamma_i}) is a natural number (H_{\gamma_i} is the coroot associated to the root \gamma_i). If the Verma modules M_\mu and M_\lambda are regular, then there exists a unique dominant weight \tilde\lambda and unique elements w, w′ of the Weyl group W such that :\mu=w'\cdot\tilde\lambda and :\lambda=w\cdot\tilde\lambda, where \cdot is the affine action of the Weyl group. If the weights are further integral, then there exists a nonzero homomorphism :W_\mu\to W_\lambda if and only if :w \leq w' in the Bruhat ordering of the Weyl group. ==Jordan–Hölder series==

Let :0\subset A\subset B\subset W_\lambda be a sequence of \mathfrak{g}-modules so that the quotient B/A is irreducible with highest weight μ. Then there exists a nonzero homomorphism W_\mu\to W_\lambda. An easy consequence of this is, that for any highest weight modules V_\mu, V_\lambda such that :V_\mu\subset V_\lambda there exists a nonzero homomorphism W_\mu\to W_\lambda. ==Bernstein–Gelfand–Gelfand resolution==

Let V_\lambda be a finite-dimensional irreducible representation of the Lie algebra \mathfrak{g} with highest weight λ. We know from the section about homomorphisms of Verma modules that there exists a homomorphism :W_{w'\cdot\lambda}\to W_{w\cdot\lambda} if and only if :w\leq w' in the Bruhat ordering of the Weyl group. The following theorem describes a projective resolution of V_\lambda in terms of Verma modules (it was proved by Bernstein–Gelfand–Gelfand in 1975) : There exists an exact sequence of \mathfrak{g}-homomorphisms : 0\to \oplus_{w\in W,\,\, \ell(w)=n} W_{w\cdot \lambda}\to \cdots \to \oplus_{w\in W,\,\, \ell(w)=2} W_{w\cdot \lambda}\to \oplus_{w\in W,\,\, \ell(w)=1} W_{w\cdot \lambda}\to W_\lambda\to V_\lambda\to 0 where n is the length of the largest element of the Weyl group. A similar resolution exists for generalized Verma modules as well. It is denoted shortly as the BGG resolution. ==See also==