FACTORIZATIONS OF ENVELOPING ALGEBRAS OF THREE DIMENSIONAL LIE ALGEBRAS AND THEIR APPLICATIONS

JUN MORITA

1. Gauss Eliminations

To solve a linear equation generically, for example,

\{ \begin{matrix}a & x & + & b & y & = & r \\ c & x & + & d & y & = & s \end{matrix}

we usually use the so-called Gauss elimination, which can be expressed as follows:

( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}1 & 0 \\ \frac{c}{a} & 1 \end{matrix} ) ( \begin{matrix}a & 0 \\ 0 & \frac{ad−bc}{a} \end{matrix} ) ( \begin{matrix}1 & \frac{b}{a} \\ 0 & 1 \end{matrix} ) .

That is, we see

\begin{matrix}( \begin{matrix}1 & 0 \\ −\frac{c}{a} & 1 \end{matrix} ) ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}a & b \\ 0 & \frac{ad−bc}{a} \end{matrix} ) , \\ ( \begin{matrix}\frac{1}{a} & 0 \\ 0 & \frac{a}{ad−bc} \end{matrix} ) ( \begin{matrix}a & b \\ 0 & \frac{ad−bc}{a} \end{matrix} ) = ( \begin{matrix}1 & \frac{b}{a} \\ 0 & 1 \end{matrix} ) . \end{matrix}

This idea leads to certain decompositions of groups of Lie type.

Let G be a split semisimple algebraic group or a Kac-Moody group (over a field). We take subgroups U, T, V of G as a standard maximal uppertriangular unipotent subgroup, a standard maximal diagonal subgroup, a standard maximal lowertriangular unipotent subgroup, respectively. Roughly speaking, Ω = V T U is an open dense (or very large) subset of G. Then, more precisely we obtain the Gauss decomposition:

G = UV T U.

There is also a strong version. Namely,

G = Z(G) ∪ ⋃_{g∈G} g(V hU)g^{−1}

for all h ∈ T, where Z(G) is the center of G. This is called a strong Gauss decomposition of G. Using such a decomposition, we can immediately find that every noncentral element can be expressed as a product of two unipotent elements.

Philosophically we can think of an additive version of these decompositions. Here we introduce some idea of additive Gauss decompositions.

2. Additive Gauss Decompositions

Let F be a field of characteristic 0, and L be a Lie algebra over F. The universal enveloping algebra of L is denoted by U(L). Let V be an L-module.

Theorem (cf. [1]). Ler L be a three dimensional Lie algebra over F. If L is generated by two elements x and y, Then, U(L) = ∑_{i,j,k} x^{i}y^{j}x^{k}. Furthermore, if both x and y act on V as locally nilpotent operators, then V is locally finite as an L-module.

The key formulas to establish the above theorem are as follows:

(A_{k}) yxy^{k} ≡ \frac{k}{k+1}xy^{k+1} + \frac{1}{k+1}y^{k+1}x mod U_{k},
(B_{k}) y^{k}xy ≡ \frac{1}{k+1}xy^{k+1} + \frac{k}{k+1}y^{k+1}x mod U_{k},
(C_{k}) yU_{k} ⊂ U_{k+1}, U_{k} ⊂ U_{k+1},

where U_{k} = ∑_{0≤m≤k}(F xy^{m} + F y^{m}x + F y^{m}).

Proof of Theorem. Suppose that L is generated by two elements x, y. Let z = [x, y]. Now we want to show U(L) = ∑_{i,j,k≥0} F x^{i}y^{j}x^{k}. Put 𝔛 = ∑_{i,j,k≥0} F x^{i}y^{j}x^{k} ⊆ U(L) and let U_{k} be defined as above. Clearly x𝔛 ⊆ 𝔛, 𝔛x ⊆ 𝔛 and U_{k} ⊆ 𝔛 for all k ≥ 0. We claim

\begin{matrix}y(x^{ℓ}y^{m}x^{n}) ∈ 𝔛, \\ z(x^{ℓ}y^{m}x^{n}) ∈ 𝔛. \end{matrix}

and show this by induction on ℓ. If ℓ = 0, then we see y(y^{m}x^{n}) ∈ 𝔛 and using (A_{m}) we get

z(y^{m}x^{n}) = (xy − yx)(y^{m}x^{n})
= xy^{m+1}x^{n} − yxy^{m}x^{n}
∈ F xy^{m+1}x^{n} + (F xy^{m+1} + F y^{m+1}x + U_{m})x^{n} ⊆ 𝔛.

Let ℓ > 0. Then, we obtain, using our inductive assumption, that

\begin{aligned}y(x^{ℓ}y^{m}x^{n}) & = & (xy − z)(x^{ℓ−1}y^{m}x^{n}) \\ & ∈ & x𝔛 + 𝔛 ⊆ 𝔛 \end{aligned}

and, letting [z, x] = ax+by +cz for a, b, c ∈ F, we also get using our inductive assumption that

\begin{aligned}z(x^{ℓ}y^{m}x^{n}) & = & (xz + ax + by + cz)(x^{ℓ−1}y^{m}x^{n}) \\ & ∈ & x𝔛 + 𝔛 + 𝔛 + 𝔛 ⊆ 𝔛. \end{aligned}

Hence, our induction method shows y𝔛 ⊆ 𝔛. Since 𝔛 is a left ideal of U(L) containing 1, we obtain 𝔛 = U(L). Therefore,

U(L) = 𝔛 = ∑_{i,j,k≥0} F x^{i}y^{j}x^{k} = U(F x) U(F y) U(F x).

Q.E.D.

The decomposition described in the above theorem may be rewritten as

U(L) = U(F x) U(F y) U(F x) = U(F y) U(F x) U(F y),

which we call an additive Gauss decomposition. If we take

e = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , f = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )

as generators of sℓ_{2} = sℓ_{2}(F), then we have

U(sℓ_{2}) = U(F e) U(F f) U(F e) = U(F f) U(F e) U(F f).

If ℌ = F x ⊕ F y ⊕ F z is a Heisenberg Lie algebra with [x, y] = z and [x, z] = [y, z] = 0, then we also have

U(ℌ) = U(F x) U(F y) U(F x) = U(F y) U(F x) U(F y).

We note that a three dimensional Lie algebra L is not two generated if and only if L is abelian or isomorphic to the Lie algebra M = F x ⊕ F y ⊕ F z satisfying [x, y] = 0, [x, z] = x, [y, z] = y. Hence, almost all three dimensional Lie algebras have additive Gauss decompositions.

3. Applications

Let 𝔤 be a Kac-Moody Lie algebra or a Borcherds Lie algebra over F with the so-called standard Cartan subalgebra 𝔥, and 𝔤_{+} (resp. 𝔤_{−}) the standard maximal positive (resp. negative) nilpotent subalgebra of 𝔤 corresponding to positive (resp. negative) roots with respect to 𝔥. Using additive Gauss decompositions of sℓ_{2} and ℌ, we obtain

U(𝔤) = U(𝔤_{±}) U(𝔤_{∓}) U(𝔤_{±}).

Hence, all Kac-Moody Lie algebras and all Borcherds Lie algebras have additive Gauss decompositions.

If e and f are locally nilpotent on an sℓ_{2}-module V, then h is diagonalizable on V, since V is a locally finite sℓ_{2}-module. Furthermore, we take standard generators

e_{1}, …, e_{n}, h_{1}, …, h_{n}, f_{1}, …, f_{n}

for the derived subalgebra 𝔤′ = [𝔤, 𝔤] of a Kac-Moody Lie algebra 𝔤. If the e_{i} and the f_{i} are all locally nilpotent on a 𝔤-module V, then the h_{i} are simultaneously diagonalizable on V. Hence, this gives a sufficient condition for a 𝔤-module V to be integrable.

Let L = L_{−1}⊕L_{0}⊕L_{1} be a three graded Lie algebra of dimension three with dim L_{ε} = 1 for ε = 0, ±1. If U(L) = U(L_{1}) U(L_{−1}) U(L_{1}), then L ≃ sℓ_{2} or L ≃ ℌ. This gives a new caracterization of sℓ_{2} and ℌ.

We can also apply our method to quantum groups. Let C be the field of complex numbers, and we fix a nonzero element q ∈ C with the property that q^{m} ≠ 1 for all m = 1, 2, 3 . . .. We denote by U_{q}(sℓ_{2}) the associative C-algebra generated by t^{±1}, e, f with the following defining relations as usual:

t^{±1}t^{∓1} = 1, tet^{−1} = q^{2}e, tft^{−1} = q^{−2}f, [e, f] = \frac{t − t^{−1}}{q − q^{−1}} .

The algebra U_{q}(sℓ_{2}) is called the quantum group associated with sℓ_{2}. For each natural number k we define a q-integer, [k] = [k]_{q} = (q^{k} − q^{−k})/(q − q^{−1}), as a nonzero complex number, and here we put [0] = 0 for convenience. Then, for n ≥ 0, we obtain:

\begin{aligned}(A_{n}^{+}) & : & fef^{n} & = & \frac{[n]}{q[n+1]}ef^{n+1} & + & \frac{q^{n}}{[n+1]}f^{n+1}e & + & \frac{[n]}{q^{n}}f^{n}t, \\ (B_{n}^{+}) & : & f^{n}ef & = & \frac{1}{q^{n}[n+1]}ef^{n+1} & + & \frac{q[n]}{[n+1]}f^{n+1}e & + & \frac{[n]}{q^{n}}f^{n}t, \\ (A_{n}^{−}) & : & fef^{n} & = & \frac{q[n]}{[n+1]}ef^{n+1} & + & \frac{1}{q^{n}[n+1]}f^{n+1}e & + & q^{n}[n]f^{n}t^{−1}, \\ (B_{n}^{−}) & : & f^{n}ef & = & \frac{q^{n}}{[n+1]}ef^{n+1} & + & \frac{[n]}{q[n+1]}f^{n+1}e & + & q^{n}[n]f^{n}t^{−1}. \end{aligned}

Using these relations, we can establish the following result.

\begin{aligned}U_{q}(sℓ_{2}) & = & (∑_{i,j,k≥0} Ce^{i}f^{j}e^{k}) + (∑_{i,j,k≥0} Ce^{i}f^{j}e^{k})t \\ & = & C[e] C[f] C[e] (C ⊕ Ct). \end{aligned}

As a direct consequence of this, we obtain the following. Let V be an infinite dimensional U_{q}(sℓ_{2})-module. Suppose that both operators e and f are locally nilpotent on V. Then:

(1) For each v ∈ V, the submodule, U_{q}(sℓ_{2})v, generated by v is finite dimensional.

(2) The operator t is diagonalizable on V.

(3) V is a direct sum of finite dimensional irreducible U_{q}(sℓ_{2})-modules. In particular, V is completely reducible.

Using the results (1),(2),(3), we can produce a sufficient condition for a representation of a quantum group to be integrable.

Remark. We note that this article is a survay of the successive papers [1], [5], [6].

References

[1] S. Berman, J. Morita and Y. Yoshii. Some factorizations in universal enveloping algebras of three dimensional Lie algebras and generalizations, Canad. Math. Bull. 45 (2002), 525 – 536.

[2] V. Chernousov, E. W. Ellers and N. Gordeev. Gauss decomposition with prescribed semisimple part: short proof, J. algebra 229 (2000), 314 – 332.

[3] V. G. Kac. Infinite dimensional Lie algebras, 3rd edition, Cambridge University Press, Cambridge, 1990.

[4] R. V. Moody and A. Pianzola. Lie algebras with triangular decompositions, Canadian Math. Soc., Ser. Mono. Adv. Texts, A Wiley-Interscience Publ., New York, 1995.

[5] J. Morita and E. Plotkin. Prescribed Gauss decompositions for Kac-Moody groups over fields, Rend. Sem. Mat. Padova 106 (2001), 153 – 163.

[6] J. Morita and H. Sakaguchi. Some formulae in U_{q}(sℓ_{2}) and diagonalizability, Kyûshû J. Math. 57 (2003), 165 – 173.

Institute of mathematics

University of Tsukuba

Tsukuba, Ibaraki, 305-8571 JAPAN

  E-mail address: morita@math.tsukuba.ac.jp