The Lie algebra gl ∞ and its representations

For every n ∈ N, we can define a Lie algebra gl_n of n×n-matrices over C. One can wonder how this can be generalized to the “n = ∞ case”, i. e., to infinite matrices.

Obviously, not every pair of infinite matrices has a reasonable commutator (because not any such pair can be multiplied), but there are certain restrictions on infinite matrices which allow us to multiply them and form their commutators. These restrictions can be used to define various Lie algebras consisting of infinite matrices. We will be concerned with some such Lie algebras; the first of them is gl_∞:

Definition 3.5.1. We define gl_∞ to be the vector space of infinite matrices whose rows and columns are labeled by integers (not only positive integers!) such that only finitely many entries of the matrix are nonzero. This vector space gl_∞ is an associative algebra without unit (by matrix multiplication); we can thus make gl_∞ into a Lie algebra by the commutator in this associative algebra.

We will study the representations of this gl_∞. The theory of these representations will extend the well-known (Schur-Weyl) theory of representations of gl_n.

Definition 3.5.2. Thevector representation V ofgl_∞ is defined as the vector space C^(Z) =

(x_i)_i∈

Z | x_i ∈C; only finitely many x_i are nonzero . The Lie algebra gl_∞ acts on the vector representation V in the obvious way: namely, for any a ∈ gl_∞ and v ∈ V, we let a * v be the product of the matrix a with the column vector v.

Here, every element (x_i)_i∈

Z of V is identified with the column vector



















 ...

x−2

x−1

x₀ x₁ x2

...



















 .

For every j ∈Z, let vj be the vector (δi,j)_i∈

Z ∈V. Then, (vj)_j∈

Z is a basis of the vector space V.

Convention 3.5.3. When we draw infinite matrices whose rows and columns are labeled by integers, the index of the rows is supposed to increase as we go from left to right, and the index of the columns is supposed to increase as we go from top to bottom.

Remark 3.5.4. In Definition 3.5.2, we used the following (very simple) fact: For every a ∈ gl_∞ and every v ∈ V, the product av of the matrix a with the column vectorv is a well-defined element ofV. This fact can be generalized: Ifais an infinite matrix (whose rows and columns are labeled by integers) such that every column of a has only finitely many nonzero entries, andv is an element of V, then the product av is a well-defined element of V. However, this does no longer hold if we drop

the condition that every column of a have only finitely many nonzero entries. (For example, if a would be the matrix whose all entries equal 1, then the product av0

would not be an element ofV, but rather the element



We can consider the representation ∧ⁱV of gl_∞ for every i∈N. More generally, we have the so-called Schur modules:

Definition 3.5.5. If π ∈ IrrS_n, then we can define a representation S_π(V) of gl_∞ by S_π(V) = Hom_Sn(π, V^⊗n) (where S_n acts on V^⊗n by permuting the tensorands).

This S_π(V) is called the π-th Schur module of V.

This definition mimics the well-known definition (or, more precisely, one of the defi-nitions) of the Schur modules of a finite-dimensional vector space.

Proposition 3.5.6. For every π ∈ IrrS_n, the representation S_π(V) of gl_∞ is irre-ducible.

Proof of Proposition 3.5.6. The following is not a self-contained proof; it is just a way to reduce Proposition 3.5.6 to the similar fact about finite-dimensional vector spaces (which is a well-known fact in the representation theory ofgl_m).

For every vector subspaceW ⊆V, we can canonically identify S_π(W) with a vector subspace of S_π(V).

For every subset I of Z, let W_I be the subset of V generated by all v_i with i ∈ I.

Clearly, whenever two subsets I and J of Z satisfy I ⊆ J, we have W_I ⊆ W_J. Also, whenever I is a finite subset of Z, the vector spaceW_I is finite-dimensional.

For every tensor u∈ V^⊗n, there exists a finite subset I of Z such that u ∈(W_I)^⊗n. V). Thus, we can write the tensoru∈V^⊗n as aC-linear combination of finitely many tensors of the form v_i1⊗v_i2⊗...⊗v_in with (i₁, i₂, ..., i_n)∈Zⁿ. LetIbe the union of the sets {i1, i₂, ..., i_n} over all the tensors which appear in this linear combination. Since only finitely many tensors appear in this linear combination, the set I is finite. Every tensor v_i1 ⊗v_i2 ⊗...⊗v_in which appears in this linear combination satisfies {i₁, i₂, ..., i_n} ⊆I (by the construction of I) and thus vi₁⊗vi₂⊗...⊗vi_n∈(WI)^⊗n. Thus,umust lie in (WI)^⊗n, too (becauseuis the value of this linear combination). Hence, we have found a finite subsetI ofZsuch thatu∈(WI)^⊗n. Qed.

For every w ∈S_π(V), there exists some finite subset I of Z such that w∈S_π(W_I).

102 Denote this subset I byI(w). Thus, w∈S_π W_I(w)

for every w∈S_π(V).

Let wand w⁰ be two vectors in S_π(V) such that w6= 0. We are going to prove that w⁰ ∈ U(gl_∞)w. Once this is proven, it will be obvious that S_π(V) is irreducible, and we will be done.

There exists a finite subset I of Z such that w ∈ S_π(W_I) and w⁰ ∈ S_π(W_I). ¹⁰³ Consider thisI.

Since I is finite, the vector space W_I is finite-dimensional. Thus, by the analogue of Proposition 3.5.6 for representations of gl_m, the representation S_π(W_I) of the Lie algebragl(W_I) is irreducible. Hence, w⁰ ∈U(gl(W_I))w.

Now, we have a canonical injective Lie algebra homomorphism gl(W_I)→gl_∞ ¹⁰⁴. Thus, we can view gl(W_I) as a Lie subalgebra of gl_∞ in a canonical way. Moreover, the classical action gl(W_I)×S_π(W_I) → S_π(W_I) of the Lie algebra gl(W_I) on the Schur module S_π(W_I) can be viewed as the restriction of the action gl_∞×S_π(V) → S_π(V) to gl(W_I)×S_π(W_I). Hence, U(gl(W_I))w ⊆ U(gl_∞)w. Since we know that w⁰ ∈ U(gl(W_I))w, we thus conclude w⁰ ∈ U(gl_∞)w. This completes the proof of Proposition 3.5.6.

On the other hand, we can define so-calledhighest-weight representations. Before we do so, let us makegl_∞ into a graded Lie algebra:

Definition 3.5.7. For everyi∈Z, letglⁱ_∞ be the subspace of gl_∞ which consists of matrices which have nonzero entries only on the i-th diagonal. (The i-th diagonal consists of the entries in the (α, β)-th places with β−α=i.)

Then, gl_∞ = L

i∈Z

glⁱ_∞, and this makes gl_∞ into a Z-graded Lie algebra. Note that gl⁰_∞ is abelian. Let gl_∞ = n−⊕h⊕n+ be the triangular decomposition of gl_∞, so

102Proof. Letw∈Sπ(V). Then, w∈Sπ(V) = HomS_n(π, V^⊗n). But since πis a finite-dimensional vector space, the image w(π) must be finite-dimensional. Hence, w(π) is a finite-dimensional vector subspace of V^⊗n. Thus, w(π) is generated by some elements u1, u2, ..., uk ∈ V^⊗n. Let I be the union

k

S

j=1

I(uj). Then,Iis finite (because for everyj ∈ {1,2, ..., k}, the setI(uj) is finite) and satisfiesI(u_j)⊆I for everyj∈ {1,2, ..., k}.

Recall that every u ∈ V^⊗n satisfies u ∈ WI(u)

⊗n

. Thus, every j ∈ {1,2, ..., k} satisfies uj∈ W_I(uj)⊗n

⊆(WI)^⊗n(sinceI(uj)⊆Iand thusW_I(uj)⊆WI). In other words, allkelements u1, u2, ..., uklie in the vector space (WI)^⊗n. Since the elementsu1, u2, ..., ukgenerate the subspace w(π), this yields that w(π) ⊆ (WI)^⊗n. Hence, the map w : π → V^⊗n factors through a map π→(W_I)^⊗n. In other words,w∈Hom_Sn(π, V^⊗n) is contained in Hom_Sn

π,(W_I)^⊗n

=S_π(W_I), qed.

103Proof. LetI=I(w)∪I(w⁰). Then,Iis a finite subset ofZ(sinceI(w) andI(w⁰) are finite subsets of Z), and I(w)⊆I and I(w⁰)⊆I. We have w ∈Sπ W_I(w)

⊆Sπ(WI) (since I(w)⊆I and thusWI(w)⊆WI) and similarlyw⁰ ∈Sπ(WI). Thus, there exists a finite subset I ofZsuch that w∈Sπ(WI) andw⁰∈Sπ(WI), qed.

104Here is how it is defined: For every linear mapA∈gl(WI), we define a linear map A⁰ ∈gl(V) by setting

A⁰vi=

Av_i, ifi∈I;

0, ifi /∈I for alli∈Z. This linear map A⁰ is represented (with respect to the basis (v_i)_i∈

Z of V) by an infinite matrix whose rows and columns are labeled by integers. This matrix lies ingl_∞.

Thus, we have assigned to every A ∈ gl(WI) a matrix in gl_∞. This defines an injective Lie algebra homomorphismgl(WI)→gl_∞.

that the subspace n− = L

i<0

glⁱ_∞ is the space of all strictly lower-triangular matrices in gl_∞, the subspace h = gl⁰_∞ is the space of all diagonal matrices in gl_∞, and the subspace n+ =L

i>0

glⁱ_∞ is the space of all strictly upper-triangular matrices in gl_∞. Definition 3.5.8. For everyi, j ∈Z, letE_i,j be the matrix (with rows and columns labeled by integers) whose (i, j)-th entry is 1 and whose all other entries are 0. Then, (Ei,j)_(i,j)∈

Z² is a basis of the vector space gl_∞.

Definition 3.5.9. For everyλ∈h^∗, letM_λbe the highest-weight Verma moduleM_λ⁺ (as defined in Definition 2.5.14). Let J_λ = Ker (·,·) ⊆ M_λ be the maximal proper graded submodule. Let Lλ be the quotient moduleMλJλ =M_λ⁺J_λ⁺ =L⁺_λ; then, L_λ is irreducible (as we know).

Definition 3.5.10. We can define an antilinear R-antiinvolution †: gl_∞→ gl_∞ on gl_∞ by setting

E_i,j^† =E_j,i for all (i, j)∈Z².

(Thus, † : gl_∞ → gl_∞ is the operator which transposes a matrix and then applies complex conjugation to each of its entries.) Thus we can speak of Hermitian and unitary gl_∞-modules.

A very important remark:

For the Lie algebra gl_n, the highest-weight modules are the Schur modules up to tensoring with a power of the determinant module. (More precisely: For gl_n, every finite-dimensional irreducible representation and any unitary irreducible representation is of the form Sπ(Vn)⊗(∧ⁿ(V_n^∗))^⊗j for some partition π and some j ∈N, where Vn is the gl_n-moduleCⁿ.)

Nothing like this is true for gl_∞. Instead, exterior powers of V and highest-weight representations live “in different worlds”. This is because V is composed of infinite-dimensional vectors which have “no top or bottom”;V has no highest or lowest weight and does not lie in categoryO⁺ orO⁻.

This is important, because many beautiful properties of representations of gl_n come from the equality of the highest-weight and Schur module representations.

A way to marry these two worlds is by considering so-called semiinfinite wedges.

3.5.1. Semiinfinite wedges

Let us first give an informal definition of semiinfinite wedges and the semiinfinite wedge space ∧

∞

2 V (we will later define these things formally):

An elementary semiinfinite wedge will mean a formal infinite “wedge product” v_i0∧ v_i1 ∧v_i2 ∧...with (i₀, i₁, i₂, ...) being a sequence of integers satisfyingi₀ > i₁ > i₂ > ...

and ik+1 =ik−1 for all sufficiently large k. (At the moment, we consider this wedge product v_i0∧v_i1 ∧v_i2 ∧...just as a fancy symbol for the sequence (i₀, i₁, i₂, ...).)

The semiinfinite wedge space ∧

∞

2 V is defined as the free vector space with basis given by elementary semiinfinite wedges.

Note that, despite the notation ∧

∞

2 V, the semiinfinite wedge space is not a functor in the vector space V. We could replace our definition of ∧

∞

2 V by a somewhat more functorial one, which doesn’t use the basis (v_i)_i∈

ZofV anymore. But it would still need a topology on V (which makes V locally linearly compact), and some working with formal Laurent series. It proceeds through the semiinfinite Grassmannian, and will not be done in these lectures.¹⁰⁵ For us, the definition using the basis will be enough.

The space ∧

∞

2 V is countably dimensional. More precisely, we can write∧

∞ 2 V as

∧

∞

2 V =M

m∈Z

∧

∞

2 ^,mV, where

∧

∞

2 ^,mV = span{v_i0 ∧v_i1 ∧v_i2 ∧... | i_k+k =m for sufficiently large k}.

The space∧

∞

2 ^,mV has basis{v_i0 ∧v_i1 ∧v_i2 ∧... | i_k+k =m for sufficiently largek}, which is easily seen to be countable. We will see later that this basis can be naturally labeled by partitions (of all integers, not just ofm).

3.5.2. The action of gl_∞ on ∧

∞ 2 V

For every m ∈ Z, we want to define an action of the Lie algebra gl_∞ on the space

∧

∞

2 ^,mV which is given “by the usual Leibniz rule”, i. e., satisfies the equation a *(v_i0 ∧v_i1 ∧v_i2 ∧...) = X

k≥0

v_i0 ∧v_i1 ∧...∧v_ik−1 ∧(a * v_ik)∧v_ik+1∧v_ik+2∧...

for all a ∈ gl_∞ and all elementary semiinfinite wedges v_i0 ∧v_i1 ∧v_i2 ∧... (where, of course, a * v_ik is the same as av_ik due to our definition of the action of gl_∞ on V).

Of course, it is not immediately clear how to interpret the infinite wedge products v_i0∧v_i1∧...∧v_ik−1∧(a * v_ik)∧v_ik+1∧v_ik+2∧...on the right hand side of this equation, since they are (in general) not elementary semiinfinite wedges anymore. We must find a reasonable definition for such wedge products. What properties should a wedge product

105Some pointers to the more functorial definition:

Consider the fieldC((t)) of formal Laurent series overCas aC-vector space.

Let Gr =

U vector subspace ofC((t)) |

U ⊇tⁿC[[t]] and dim (U(tⁿC[[t]]))<∞

for some sufficiently highn

. For every U ∈Gr, define an integer sdimU by sdimU = dim (U(tⁿC[[t]]))−nfor anyn∈Z

satisfying U ⊇tⁿC[[t]]. Note that this integer does not depend onn as long as nis sufficiently high to satisfyU ⊇tⁿC[[t]].

This Grassmannian Gr is the disjoint union `Gr_n.

There is something called a determinant line bundle on Gr. The space of semiinfinite wedges is then defined as the space of regular sections of this line bundle (in the sense of algebraic geometry).

See the book by Pressley and Segal about loop groups for explanations of these matters.

(infinite as it is) satisfy? It should be multilinear¹⁰⁶ and antisymmetric¹⁰⁷. These properties make it possible to compute any wedge product of the formb₀∧b₁∧b₂∧...

with b₀, b₁, b₂, ... being vectors inV which satisfy

b_i =v_m−i for sufficiently largei.

In fact, whenever we are given such vectors b₀, b₁, b₂, ..., we can compute the wedge product b₀∧b₁∧b₂∧... by the following procedure:

• Find an integerM ∈Nsuch that every i≥M satisfiesb_i =vm−i. (ThisM exists by the condition thatb_i =v_m−i for sufficiently large i.)

• Expand each of the vectors b₀, b₁, ..., b_M−1 as a C-linear combination of the basis vectorsv_`.

• Using these expansions and the multilinearity of the wedge product, reduce the computation ofb₀∧b₁∧b₂∧...to the computation of finitely many wedge products of basis vectors.

• Each wedge product of basis vectors can now be computed as follows: If two of the basis vectors are equal, then it must be 0 (by antisymmetry of the wedge product). If not, reorder the basis vectors in such a way that their indices decrease (this is possible, because “most” of these basis vectors are already in order, and only the first few must be reordered). Due to the antisymmetry of the wedge product, the wedge product of the basis vectors before reordering must be (−1)^π times the wedge product of the basis vectors after reordering, where π is the permutation which corresponds to our reordering. But the wedge product of the basis vectors after reordering is an elementary semiinfinite wedge, and thus we know how to compute it.

This procedure is not exactly a formal definition, and it is not immediately clear that the value of b₀∧b₁∧b₂ ∧...that it computes is independent of, e. g., the choice of M. In the following subsection (Subsection 3.5.3), we will give a formal version of this definition.

3.5.3. The gl_∞-module ∧

∞

2 V: a formal definition

Before we formally define the value of b₀ ∧b₁ ∧b₂∧..., let us start from scratch and repeat the definitions of ∧

∞

2 V and ∧

∞

2 ^,mV in a cleaner fashion than how we defined them above.

106i. e., it should satisfy

b0∧b1∧...∧b_k−1∧(λb+λ⁰b⁰)∧bk+1∧bk+2∧...

=λb₀∧b₁∧...∧b_k−1∧b∧b_k+1∧b_k+2∧...+λ⁰b₀∧b₁∧...∧b_k−1∧b⁰∧b_k+1∧b_k+2∧...

for allk∈N,b0, b1, b2, ...∈V,b, b⁰∈V andλ, λ⁰∈Cfor which the right hand side is well-defined

107i. e., a well-defined wedge productb0∧b1∧b2∧...should be 0 whenever two of thebk are equal

Warning 3.5.11. Some of the nomenclature defined in the following (particularly, the notions of “m-degression” and “straying m-degression”) is mine (=Darij’s). I don’t know whether there are established names for these things.

First, we introduce the notion ofm-degressionsand formalize the definitions of∧

∞ 2 V and ∧

∞ 2 ^,mV.

Definition 3.5.12. Let m ∈ Z. An m-degression will mean a strictly decreasing sequence (i₀, i₁, i₂, ...) of integers such that every sufficiently high k ∈ N satisfies i_k +k = m. It is clear that any m-degression (i₀, i₁, i₂, ...) automatically satisfies i_k−i_k+1 = 1 for all sufficiently highk.

For anym-degression (i₀, i₁, i₂, ...), we introduce a new symbolv_i0 ∧v_i1 ∧v_i2 ∧....

This symbol is, for the time being, devoid of any meaning. The symbol v_i0 ∧v_i1 ∧ v_i2 ∧...will be called an elementary semiinfinite wedge.

Definition 3.5.13. (a) Let ∧

∞

2 V denote the free C-vector space with basis (v_i0 ∧v_i1 ∧v_i2 ∧...)_{m∈Z; (i}

0,i1,i2,...) is anm-degression. We will refer to ∧

∞

2 V as the semi-infinite wedge space.

(b) For everym ∈Z, define a C-vector subspace ∧

∞

2 ^,mV of ∧

∞ 2 V by

∧

∞

2 ^,mV = span{v_i0 ∧v_i1 ∧v_i2 ∧... | (i₀, i₁, i₂, ...) is an m-degression}. Clearly, ∧

∞

2 ^,mV has basis (v_i0 ∧v_i1 ∧v_i2 ∧...)_(i

0,i1,i2,...) is anm-degression. Obviously, ∧

∞

2 V = L

m∈Z

∧

∞ 2 ^,mV.

Now, let us introduce the (more flexible) notion of straying m-degressions. This no-tion is obtained from the nono-tion ofm-degressions by dropping the “strictly decreasing”

condition:

Definition 3.5.14. Let m ∈ Z. A straying m-degression will mean a sequence (i₀, i₁, i₂, ...) of integers such that every sufficiently highk ∈N satisfiesi_k+k=m.

As a consequence, a strayingm-degression is strictly decreasing from some point on-wards, but needs not be strictly decreasing from the beginning (it can “stray”, whence the name). A strictly decreasing straying m-degression is exactly the same as an m-degression. Thus, everym-degression is a strayingm-degression.

Definition 3.5.15. LetS be a (possibly infinite) set. Recall that a permutation of S means a bijection from S to S.

A finitary permutation of S means a bijection from S to S which fixes all but finitely many elements of S. (Thus, all permutations of S are finitary permutations if S is finite.)

Notice that the finitary permutations of a given set S form a group (under compo-sition).

Definition 3.5.16. Let m ∈ Z. Let (i0, i1, i2, ...) be a straying m-degression. If no two elements of this sequence (i₀, i₁, i₂, ...) are equal, then there exists a unique finitary permutation π of N such that i_π⁻¹₍₀₎, i_π⁻¹₍₁₎, i_π⁻¹₍₂₎, ...

is an m-degression.

This finitary permutation π is called the straightening permutation of (i0, i1, i2, ...).

Definition 3.5.17. Let m ∈ Z. Let (i0, i1, i2, ...) be a straying m-degression. We define the meaning of the term v_i0 ∧v_i1 ∧v_i2 ∧...as follows:

- If some two elements of the sequence (i₀, i₁, i₂, ...) are equal, thenv_i0∧v_i1∧v_i2∧...

is defined to mean the element 0 of ∧

∞ 2 ^,mV.

- If no two elements of the sequence (i0, i1, i2, ...) are equal, then vi0∧vi1∧vi2∧...

is defined to mean the element (−1)^πvi_π−1(0)∧vi_π−1(1)∧vi_π−1(2)∧...of∧

∞

2 ^,mV, where π is the straightening permutation of (i₀, i₁, i₂, ...).

Note that whenever (i₀, i₁, i₂, ...) is an m-degression (not just a straying one), then the value of v_i0 ∧v_i1 ∧v_i2 ∧... defined according to Definition 3.5.17 is exactly the symbolvi0∧vi1∧vi2∧...of Definition 3.5.12 (because no two elements of the sequence (i₀, i₁, i₂, ...) are equal, and the straightening permutation of (i₀, i₁, i₂, ...) is id). Hence, Definition 3.5.17 does not conflict with Definition 3.5.12.

Definition 3.5.18. Let m∈Z. Let b₀, b₁, b₂, ... be vectors in V which satisfy b_i =vm−i for sufficiently largei.

Then, let us define the wedge product b₀∧b₁∧b₂∧...∈ ∧

∞

2 ^,mV as follows:

Find an integerM ∈N such that everyi≥M satisfies b_i =vm−i. (ThisM exists by the condition that b_i =vm−i for sufficiently largei.)

For every i ∈ {0,1, ..., M −1}, write the vector b_i as a C-linear combination P

j∈Z

λ_i,jv_j (with λ_i,j ∈Cfor all j).

Now, defineb₀∧b₁∧b₂∧...to be the element X

(j0,j1,...,jM−1)∈Z^M

λ0,j0λ1,j1...λM−1,jM−1vj0∧vj1∧...∧vjM−1∧vm−M∧vm−M−1∧vm−M−2∧...

of∧

∞

2 ^,mV. Here,vj0∧vj1∧...∧vjM−1∧vm−M∧vm−M−1∧vm−M−2∧...is well-defined, since (j₀, j₁, ..., j_M−1, m−M, m−M −1, m−M −2, ...) is a strayingm-degression.

Note that this element b₀∧b₁ ∧b₂ ∧... is well-defined (according to Proposition 3.5.19 (a) below).

We refer to b₀ ∧b₁∧b₂ ∧... as the (infinite) wedge product of the vectors b₀, b₁, b₂, ....

Note that, for any strayingm-degression (i₀, i₁, i₂, ...), the value of v_i0 ∧v_i1∧v_i2∧...

defined according to Definition 3.5.18 equals the value of v_i0 ∧v_i1 ∧v_i2 ∧... defined

according to Definition 3.5.17. Hence, Definition 3.5.18 does not conflict with Definition 3.5.17.

We have the following easily verified properties of the infinite wedge product:

Proposition 3.5.19. Letm ∈Z. Let b₀, b₁, b₂, ... be vectors inV which satisfy b_i =vm−i for sufficiently largei.

(a)The wedge productb₀∧b₁∧b₂∧...as defined in Definition 3.5.18 is well-defined (i. e., does not depend on the choice of M).

(b) For any straying m-degression (i₀, i₁, i₂, ...), the value of v_i0 ∧v_i1 ∧v_i2 ∧...

defined according to Definition 3.5.18 equals the value of v_i0 ∧v_i1 ∧v_i2 ∧...defined according to Definition 3.5.17.

(c)The infinite wedge product is multilinear. That is, we have b₀ ∧b₁∧...∧bk−1∧(λb+λ⁰b⁰)∧b_k+1∧b_k+2∧...

=λb₀∧b₁∧...∧bk−1∧b∧b_k+1∧b_k+2∧...

+λ⁰b₀∧b₁∧...∧bk−1∧b⁰∧b_k+1∧b_k+2∧... (112) for all k ∈ N, b₀, b₁, b₂, ... ∈ V, b, b⁰ ∈ V and λ, λ⁰ ∈ C which satisfy (b_i =v_m−i for sufficiently largei).

(d) The infinite wedge product is antisymmetric. This means that if b₀, b₁, b₂, ... ∈ V are such that (b_i =vm−i for sufficiently large i) and (two of the vectors b₀, b₁, b₂, ...are equal), then

b₀∧b₁∧b₂∧...= 0. (113)

In other words, when (at least) two of the vectors forming a well-defined infinite wedge product are equal, then this wedge product is 0.

(e) As a consequence, the wedge product b₀∧b₁∧b₂∧... gets multiplied by −1 when we switch b_i with b_j for any two distinct i∈N and j ∈N.

(f ) If π is a finitary permutation of N and b₀, b₁, b₂, ...∈ V are vectors such that (b_i =vm−i for sufficiently largei), then the infinite wedge productb_π(0)∧b_π(1)∧b_π(2)∧ ... is well-defined and satisfies

b_π(0)∧b_π(1)∧b_π(2)∧...= (−1)^π ·b₀∧b₁∧b₂∧.... (114)

Now, we can define the action of gl_∞ on ∧

∞

2 ^,mV just as we wanted to:

Definition 3.5.20. Let m ∈ Z. Define an action of the Lie algebra gl_∞ on the vector space ∧

∞

2 ^,mV by the equation a *(v_i0 ∧v_i1 ∧v_i2 ∧...) =X

k≥0

v_i0 ∧v_i1 ∧...∧v_ik−1 ∧(a * v_ik)∧v_ik+1 ∧v_ik+2∧...

for all a∈gl_∞ and all m-degressions (i₀, i₁, i₂, ...) (and by linear extension). (Recall thata * v =avfor everya∈gl_∞andv ∈V, due to how we defined thegl_∞-module V.)

Of course, this definition is only justified after showing that this indeed is an action.

But this is rather easy. Let us state this as a proposition:

Proposition 3.5.21. Letm ∈Z. Then, Definition 3.5.20 really defines a represen-tation of the Lie algebra gl_∞ on the vector space ∧

∞

2 ^,mV. In other words, there exists one and only one action of the Lie algebra gl_∞ on the vector space ∧

∞ 2 ^,mV such that all a ∈gl_∞ and allm-degressions (i0, i1, i2, ...) satisfy

a *(v_i0 ∧v_i1 ∧v_i2 ∧...) =X

k≥0

v_i0 ∧v_i1 ∧...∧v_ik−1 ∧(a * v_ik)∧v_ik+1∧v_ik+2 ∧....

The proof of this proposition (using the multilinearity and the antisymmetry of our wedge product) is rather straightforward and devoid of surprises. I will show it nevertheless, if only because I assume every other text leaves it to the reader. Due to its length, it is postponed until Subsection 3.5.4.

Proposition 3.5.21 shows that the action of the Lie algebra gl_∞ on the vector space

∧

∞

2 ^,mV in Definition 3.5.20 is well-defined. This makes ∧

∞

2 ^,mV into a gl_∞-module.

Computations in this module can be somewhat simplified by the following “comparably basis-free” formula¹⁰⁸:

Proposition 3.5.22. Letm ∈Z. Let b₀, b₁, b₂, ... be vectors inV which satisfy b_i =vm−i for all sufficiently large i.

Then, every a ∈gl_∞ satisfies a *(b₀∧b₁∧b₂∧...) =X

k≥0

b₀∧b₁∧...∧bk−1∧(a * b_k)∧b_k+1∧b_k+2∧....

We can also explicitly describe this action on elementary matrices and semiinfinite wedges:

Proposition 3.5.23. Let i ∈ Z and j ∈ Z. Let m ∈ Z. Let (i₀, i₁, i₂, ...) be a straying m-degression (so thatv_i0 ∧v_i1 ∧v_i2 ∧...∈ ∧

∞ 2 ^,mV).

(a) If j /∈ {i0, i1, i2, ...}, then Ei,j *(vi0 ∧vi1 ∧vi2 ∧...) = 0.

(b) If there exists aunique `∈N such that j =i<sub>`</sub>, then for this` we have E<sub>i,j</sub> *(v<sub>i0</sub> ∧v<sub>i1</sub> ∧v<sub>i2</sub> ∧...) = v<sub>i0</sub> ∧v<sub>i1</sub> ∧...∧v<sub>i`−1</sub> ∧v_i∧v_{i`+1</sub>∧v<sub>i`+2} ∧...

(In words: If v_j appears exactly once as a factor in the wedge product v_i0 ∧v_i1 ∧ v_i2∧..., thenE_i,j *(v_i0 ∧v_i1 ∧v_i2 ∧...) is the wedge product which is obtained from vi0 ∧vi1 ∧vi2∧... by replacing this factor byvi.)

108I’m saying “comparably” because the condition thatbi=v_m−ifor all sufficiently largeiis not basis-free. But this should not come as a surprise, as the definition of∧

∞

2 ^,mV itself is not basis-free to begin with.

Since we have given ∧

∞

2 ^,mV agl_∞-module structure for everym ∈Z, it is clear that

∧

∞

2 V = L

m∈Z

∧

∞

2 ^,mV also becomes a gl_∞-module.

3.5.4. Proofs

Here are proofs of some of the unproven statements made in Subsection 3.5.3:

Proof of Proposition 3.5.21. The first thing we need to check is the following:

Assertion 3.5.21.0: Let a ∈ gl_∞. Let b0, b1, b2, ... be vectors in V which satisfy

b_i =vm−i for all sufficiently large i.

(a)For everyk∈N, the infinite wedge productb₀∧b₁∧...∧bk−1∧(a * b_k)∧

bk+1∧bk+2∧... is well-defined.

(b)All but finitely manyk∈Nsatisfyb0∧b1∧...∧bk−1∧(a * bk)∧bk+1∧ b_k+2∧...= 0. (In other words, the sum

X

k≥0

b₀∧b₁∧...∧bk−1∧(a * b_k)∧b_k+1∧b_k+2∧...

converges in the discrete topology.)

The proof of Assertion 3.5.21.0 can easily be supplied by the reader. (Part (a) is

clear, since the property of the sequence (b0, b1, b2, ...) to satisfy (bi =vm−i for all sufficiently large i) does not change if we modify one entry of the sequence. Part (b) requires showing

clear, since the property of the sequence (b0, b1, b2, ...) to satisfy (bi =vm−i for all sufficiently large i) does not change if we modify one entry of the sequence. Part (b) requires showing

Im Dokument Pavel Etingof: 18.747: Infinite-dimensional Lie algebras (Spring term 2012 at MIT) (Seite 188-200)