Representations of Lie Algebras and Partial Differential Equations

2.9 MB·

by  Xiaoping Xu

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Representations of Lie Algebras and

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T Partial Differential Equations
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m Xiaoping Xu
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Institute of Mathematics
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Academy of Mathematics and System Sciences
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6 Chinese Academy of Sciences
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0 55 Zhongguancun Dong Lu
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1 Beijing 100190, P. R. China
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2016
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Preface
Symmetry is an important phenomenon in the natural world. Lie algebra is not purely
an abstract mathematics but a fundamental tool of studying symmetry. In fact, Norwe-
gian mathematician Sorphus Lie introduced Lie group and Lie algebra in 1874 in order
to study the symmetry of differential equations. Lie algebras are the infinitesimal struc-
tures (bones) of Lie groups, which are symmetric manifolds. Lie theory has extensive and
important applications in many other fields of mathematics, such as geometry, topology,
number theory, control theory, integrable systems, operator theory, and stochastic pro-
cess etc. Representations of finite-dimensional semisimple Lie algebras play fundamental
roles in quantum mechanics. The controllability property of the unitary propagator of
an N-level quantum mechanical system subject to a single control field can be described
in terms of the structure theory of semisimple Lie algebras. Moreover, Lie algebras were
used to explain the degeneracies encountered in genetic codes as the result of a sequence
of symmetry breakings that have occurred during its evolution. The structures and repre-
sentations of simple Lie algebras are connected with solvable quantum many-body system
in one-dimension. Our research also showed that the representation theory of Lie algebras
is connected with algebraic coding theory.
The existing classical books on finite-dimensional Lie algebras, such as the ones by
Jacobson and by Humphreys, purely focus on the algebraic structures of semisimple Lie
algebras and their finite-dimensional representations. Explicit irreducible representations
of simple Lie algebras had not been addressed extensively. Moreover, the relations of
Lie algebras with the other subjects had not been narrated that much. It seems to us
that a book on Lie algebras with more extensive view is needed in coupling with modern
development of mathematics, sciences and technology.
Thisbookismainlyanexpositionoftheauthor’sworksandhisjointworkswithhisfor-
mer students on explicit representations of finite-dimensional simple Lie algebras, related
partial differential equations, linear orthogonal algebraic codes, combinatorics and alge-
braic varieties. Various oscillator generalizations of the classical representation theorem
on harmonic polynomials are presented. New functors from the representation category
of a simple Lie algebra to that of another simple Lie algebra are given. Partial differential
equations play key roles in solving certain representation problems. The weight matrices
i
ii PREFACE
of the minimal and adjoint representations over the simple Lie algebras of types E and F
are proved to generate ternary orthogonal linear codes with large minimal distances. New
multi-variable hypergeometric functions related to the root systems of simple Lie algebras
are introduced in connection with quantum many-body system in one-dimension. Certain
equivalent combinatorial properties on representation formulas are found. Irreducibility
of representations are proved directly related to algebraic varieties.
Thisbookisself-containedwiththeminimalprerequisiteofcalculusandlinearalgebra.
It consists of three parts. The first part is mainly the classical structure and finite-
dimensional representation theory of finite-dimensional semisimple Lie algebras, written
with Humphreys’ book “Introduction to Lie Algebras and Representation Theory” as
the main reference, where we give more examples and direct constructions of simple Lie
algebras of exceptional types, revise some arguments and prove some new statements.
This part serves as the preparation for the main context. The second part is our explicit
representation theory of finite-dimensional simple Lie algebras. Many of the irreducible
representations in this part are infinite-dimensional and some of them are even not of
highest-weight type. Certain important natural representation problems are solved by
means of solving partial differential equations. In particular, some of our irreducible
presentations arecompletely characterized by invariantpartialdifferential equations. New
representation functors are constructed from inhomogeneous oscillator representations of
simple Lie algebras, which give fractional representations of the corresponding Lie groups,
such as the projective representations of special linear Lie groups and the conformal
representations of complex orthogonal groups. The third part is an extension of the
second part. First we give supersymmetric generalizations of the classical representation
theorem on harmonic polynomials. They can also be viewed as certain supersymmetric
Howe dualities. Then we present our theory of Lie theoretic codes. Finally, we talk about
root related integrable systems and our new multi-variable hypergeometric functions,
which are natural multi-variable analogues of classical Gauss hypergeometric functions.
The corresponding hypergeometric partial differential equations are found.
PartofthisbookhasbeentaughtformanytimesattheUniversityofChineseAcademy
of Sciences. The book can serve as a research reference book for mathematicians and
scientists. It can also be treated as a text book for students after a proper selection of
materials.
Xiaoping Xu
Beijing, P. R. China
2016
Introduction
Algebraic study of partial differential equations traces back to Norwegian mathematician
Sophus Lie [Lie], who invented the powerful tool of continuous groups (known as Lie
groups) in 1874 in order to study symmetry of differential equations. Lie’s idea has been
carriedonmainlybythemathematiciansintheformerstatesofSovietUnion, EastEurope
and some mathematicians in North America. Now it has become an important mathe-
matical field known as “group analysis of differential equations,” whose main objective is
to find symmetry group of differential equations, related conservation laws and similarity
solutions. One of the classical long-standing problems in the area was to determine all
invariant partial differential equation under the natural representations of classical groups
(It was also mentioned by Olver [Op]). The problem was eventually settled down theo-
retically by the author [X5], where we found the complete set of functional generators for
the differential invariants of classical groups.
Gel’fand, Dikii and Dorfman [GDi1, GDi2, GDo1-GDo3] introduced in 1970s a theory
of Hamiltonian operators in order to study the integrability of nonlinear evolution partial
differential equations. Our first experience with partial differential equation was in the
works [X3, X4, X9, X10, X11] on the structure of Hamiltonian operators and their su-
persymmetric generalizations. In particular, we [X10] proved that Lie conformal algebras
in the sense of Kac [Kv2] are equivalent to linear Hamiltonian operators as mathemat-
ical structures. Since Borcherds’ vertex algebras are determined by their positive parts
(modes in physics terminology) of vertex operators that form Lie conformal algebras (e.g.,
cf. [Kv2, X6]), our result essentially established an equivalence between vertex algebras
and Hamiltonian operators. The result was later generalized by Barakat, De Sole and Kac
[BDK] to vertex poisson algebras. Moreover, we used the techniques and thoughts from
Lie algebras to solve various physical partial differential equations such as the equation of
transonic gas flows, the equation of geopotential forecast, the nonlinear Schro¨dinger equa-
tions, the Navier-Stokes equations and the classical boundary layer equations (cf. [X21]
and the references therein). In this book, we present an explicit representation theory of
finite-dimensional simple Lie algebras and show that partial differential equations can be
used to solve representation problems of Lie algebras.
Abstractly, a Lie algebra G is a vector space with a bilinear map [ , ] : G G G
· · × →
iii
iv INTRODUCTION
such that
[u,v] = [v,u], [u,[v,w]] = [[u,v],w]+[v,[u,w]], u,v,w G. (0.1)
− ∈
A Lie algebra G is called simple if it does not contain any nonzero “invariant” proper
subspace ; that is
I
[u, ] , u G. (0.2)
I ⊂ I ∈
Finite-dimensional complex simple Lie algebras were classified by Killing and Cartan in
later nineteen century. Fora vector space V, we denote by V∗ the space of linear functions
on V. It turns out that such a Lie algebra G must contain a subspace H, called toral
Cartan subalgebra, such that
G = G , G = u G [h,u] = α(h)u for h H , (0.3)
α α
{ ∈ | ∈ }
α∈H∗
M
and G = H, where
0
Φ = α H∗ 0 G = 0 (0.4)
α
{ ∈ \{ } | 6 { }}
is called the root system of G. Moreover,
dimG = 1 for α Φ (0.5)
α
∈
and there exists a subset Φ of Φ containing a basis Π = α ,...,α of H∗ such that
+ 1 n
{ }
Φ = Φ Φ , Φ Φ = (0.6)
+ + + +
− − ∅
[ \
and for any β Φ ,
+
∈ n
β = k α with 0 k Z, (0.7)
i i i
≤ ∈
i=1
X
where Z is the ring of integers. Furthermore, there exists h [G ,G ] such that
i ∈ αi −αi
α (h ) = 2 for i = 1,2,...,n. (0.8)
i i
It turns out that the structure of G is completely determined by the matrix (α (h )) ,
i j n×n
which is called the Cartan matrix of G. Consequently, there are only nine classes of finite-
dimensionalsimpleLiealgebras,whicharecalledoftypesA ,B ,C ,D ,G ,F ,E ,E ,E .
n n n n 2 4 6 7 8
The first four are infinite series, called “classical Lie algebras.” The last five are fixed sim-
ple Lie algebras, called “exceptional types.” Using the root lattices Λ = Zα of type
r α∈Π
A and the Coxeter automorphism of Λ , we constructed in [X1] two families of self-dual
r P
complex lattices that are also real self-dual (unimodular) lattices. These lattices can be
used in studying geometry of numbers and conformal field theory.
Since every finite-dimensional complex simple Lie algebra must contain a toral Cartan
subalgebra, it is natural to ask if there exists a complex simple Lie algebra that does
INTRODUCTION v
not contain any toral Cartan subalgebra. In [X7, X8], we constructed six families of
infinite-dimensional complex simple Lie algebras without any toral Cartan subalgebras.
For a vector space M, we denote by EndM the space of all linear transformations on
M. A representation ν of a Lie algebra G on M is a linear map from G to EndM such
that
ν([u,v]) = ν(u)ν(v) ν(v)ν(u) for u,v G. (0.9)
− ∈
The space M is called a G-module. In this book, we sometimes use the notions:
ν(ξ) = ξ , ν(ξ)(w) = ξ(w) for ξ G, w M. (0.10)
M
| ∈ ∈
A subspace N of a G-module M is called a submodule of M if
ξ(w) N for ξ G, w N . (0.11)
∈ ∈ ∈
If M dose not contain any proper nonzero submodule, we say that M is an irreducible
G-module. Let G be a Lie algebra with a decomposition (0.3) satisfying (0.4), (0.6) and
(0.7). A G-module M is called a weight module if
M = M , M = w M h(w) = µ(h)w . (0.12)
µ µ
{ ∈ | }
µ∈H∗
M
The set
Λ(M) = µ H∗ M = 0 (0.13)
µ
{ ∈ | 6 { }}
is called the weight set of M and the elements in Λ(M) are called the weights of M. A
vector in M with µ Λ(M) is called a weight vector. A nonzero weight vector w is called
µ
∈
a singular vector if
ξ(w) = 0 for ξ G . (0.14)
β
∈
β[∈Φ+
If M is generated by a singular vector v with weight λ, we call M a highest-weight G-
module and λ the highest weight of M.
Finite-dimensional representations of a complex finite-dimensional simple Lie algebra
were essentially determined by Cartan in early twenty century. The approaches were
simplified and further developed by Weyl in 1920s. Let G be a finite-dimensional complex
simple Lie algebra and take the settings in (0.3)-(0.8). Set
Λ+ = λ H∗ 0 λ(h ) Z for i = 1,...,n . (0.15)
i
{ ∈ | ≤ ∈ }
It turns out that any finite-dimensional G-module is a direct sum of its irreducible sub-
modules and any finite-dimensional irreducible G-module is a highest-weight module with
its highest weight in Λ+. Conversely, for any element λ Λ+, there exists a unique finite-
∈
dimensional irreducible G-module with highest weight λ. The above conclusion is now
called Weyl’s theorem on completely reducibility. For any finite-dimensional G-module M,
µ(h ) Z for i 1,...,n , µ Λ(M). (0.16)
i
∈ ∈ { } ∈
vi INTRODUCTION
Write Λ(M) 0 = µ ,µ ,...,µ . We found in [X20] that the set
1 2 m
\{ } { }
(µ (h ),µ (h ),...,µ (h )) i = 1,...,n (0.17)
1 i 2 i m i
{ | }
spans a binary or ternary orthogonal code for certain G and M. In particular, we showed
that when G is of type F , E , E and E , andM is of minimal dimension or M = G with
4 6 7 8
left multiplication as the representation, the set (0.17) spans a ternary orthogonal code
with large minimal distance d, which can be used to correct Jd/2K errors in information
technology.
Amoduleofafinite-dimensional simpleLiealgebraiscalledcuspidalifitisnotinduced
from its proper “parabolic subalgebras.” Infinite-dimensional irreducible weight modules
of finite-dimensional simple Lie algebras with finite-dimensional weight subspaces had
been intensively studied by the authors in [BBL, BFL, BHL, BL1, BL2, Fs, Fv, Mo].
In particular, Fernando [Fs] proved that such modules must be cuspidal or parabolically
induced. Moreover, such cuspidal modules exist only for special linear Lie algebras and
symplectic Lie algebras. A similar result was independently obtained by Futorny [Fv].
Mathieu[Mo] proved thatthese cuspidal modules areirreducible components inthetensor
modules of their multiplicity-free modules with finite-dimensional modules. Thus the
structures of irreducible weight modules of finite-dimensional simple Lie algebra with
finite-dimensional weight subspaces were essentially determined by Fernando’s result in
[Fs] and Methieu’s result in [Mo]. However, explicit presentations of such modules were
almost unknown.
An important feature of this book is the connection between representations of simple
Lie algebras and partial differential equations. Let E be the square matrix with 1 as its
r,s
(r,s)-entry and 0 as the others. Denote by R the field of real numbers. Let n 3 be an
≥
integer. The compact orthogonal Lie algebra o(n,R) = R(E E ), whose
1≤r<s≤n r,s − s,r
representation on the polynomial algebra A = R[x ,...,x ] is given by (E E ) =
1 Pn r,s − s,r |A
x ∂ x ∂ . Denote by A the subspace of homogeneous polynomials in A with degree
r xs− s xr k
k. Recall that the Laplace operator ∆ = ∂2 + +∂2 and its corresponding invariant
x1 ··· xn
η = x2 +x2 + +x2. It is well known that the subspaces
1 2 ··· n
H = f A ∆(f) = 0 (0.18)
k k
{ ∈ | }
of harmonic polynomials form irreducible o(n,R)-submodules and
∞
A = ηiH (0.19)
k
i,k=0
M
is a direct sum of irreducible submodules. In other words, the irreducible submodules
are characterized by the Laplace operator ∆ and its dual invariant η gives the complete
INTRODUCTION vii
reducibility of the polynomial algebra A. The above conclusion is called the classical
theorem on harmonic polynomials.
The Navier equations
ι ∆(~u)+(ι +ι )( t )(~u) = 0 (0.20)
1 1 2
∇ ·∇
are used to describe the deformation of a homogeneous, isotropic and linear elastic
medium intheabsence ofbodyforces, where~uisann-dimensional vector-valued function,
= (∂ ,∂ ,...,∂ ) is the gradient operator, ι and ι are Lam´e constants with ι > 0,
∇ x1 x2 xn 1 2 1
2ι +ι > 0 and ι +ι = 0. In fact, t is the well-known Hessian operator. Math-
1 2 1 2
6 ∇ ·∇
ematically, the above system is a natural vector generalization of the Laplace equation.
In [X16], we found methods of solving linear flag partial differential equations for polyno-
mial solutions. Cao [Cb] used a method of us to prove that the subspaces of homogeneous
polynomial-vector solutions are exactly direct sums of three explicitly given irreducible
o(n,R)-submodules if n = 4 and of four explicitly given irreducible o(n,R)-submodules if
6
n = 4. Thisgaveavector generalizationoftheclassical theoremonharmonicpolynomials.
Denote by B = C[x ,...,x ,y ,...,y ] the polynomial algebra in x ,...,x and y ,...,y
1 n 1 n 1 n 1 n
over the field C of complex numbers. Fix n ,n 1,...,n with n n . We redefine
1 2 1 2
∈ { } ≤
degx = 1, degx = 1, degy = 1, degy = 1 (0.21)
i j r s
− −
for i 1,...,n , j n + 1,..,n , r 1,...,n and s n + 1,...,n . Denote
1 1 2 2
∈ { } ∈ { } ∈ { } ∈ { }
by B the subspace of homogeneous polynomials with degree ℓ in x ,...,x and
hℓ1,ℓ2i 1 { 1 n}
degree ℓ in y ,...,y . We deform the Laplace operator n ∂ ∂ to the operator
2 { 1 n} i=1 xi yi
n1 n2 Pn
∆˜ = x ∂ + ∂ ∂ y ∂ (0.22)
− i yi xr yr − s xs
Xi=1 r=Xn1+1 s=Xn2+1
and its dual n x y to the operator
i=1 i i
P n1 n2 n
η = y ∂ + x y + x ∂ . (0.23)
i xi r r s ys
Xi=1 r=Xn1+1 s=Xn2+1
Define
H = f B ∆˜(f) = 0 . (0.24)
hℓ1,ℓ2i { ∈ hℓ1,ℓ2i | }
Luo and the author [LX1] constructed a new representation on B for the simple Lie
algebra G of type A such that the operators in (0.22) and (0.23) commute with the
n−1
elements in G and used a method in [X16] to prove that H with ℓ ,ℓ Z such
|B hℓ1,ℓ2i 1 2 ∈
that ℓ + ℓ n n + 1 δ are irreducible G-modules. Moreover, B =
1 2 ≤ 1 − 2 − n1,n2 hℓ1,ℓ2i
∞ ηm(H ) is a decomposition of irreducible G-submodules. This establishes
m=0 hℓ1−m,ℓ2−mi
a Z2-graded analogue of the classical theorem on harmonic polynomials for the simple Lie
L
algebra of type A .
n−1
viii INTRODUCTION
Set
B = B , H = H for k Z. (0.25)
hki hℓ,k−ℓi hki hℓ,k−ℓi
∈
ℓ∈Z ℓ∈Z
X X
Luoandtheauthor[LX2]extendedtheaboverepresentationofG onB toarepresentation
ofthesimpleLiealgebraG oftypeD onB andprovedthatH withn n +1 δ
1 n hki 1− 2 − n1,n2 ≥
k Z are irreducible G -modules and B = ∞ ηi(H ) is a decomposition of
∈ 1 hki i=0 hk−2ii
irreducible G -submodules. This gives a Z-graded analogue of the classical theorem on
1 L
harmonic polynomials for the simple Lie algebra of type D .
n
Let B′ = C[x ,x ,...,x ,y ,...,y ] be the polynomial algebra in x ,x ,...,x and
0 1 n 1 n 0 1 n
y ,...,y . We define deg x = 1 and take (0.21). Write the deformed Laplace opera-
1 n 0
tor
n1 n2 n
∆˜′ = ∂2 2 x ∂ +2 ∂ ∂ 2 y ∂ (0.26)
x0 − i yi xr yr − s xs
Xi=1 r=Xn1+1 s=Xn2+1
and its dual operator
n1 n2 n
η′ = x2 +2 y ∂ +2 x y +2 x ∂ . (0.27)
0 i xi r r s ys
Xi=1 r=Xn1+1 s=Xn2+1
Set
∞
B′ = B xi, H ′ = f B′ ∆˜′(f) = 0 . (0.28)
hki hk−ii 0 hki { ∈ hki | }
i=0
X
Luo and the author [LX2] extend the above representation of G on B to a representa-
1
tion of the simple Lie algebra G of type B on B′ and proved that H ′ with k Z
2 n hki ∈
are irreducible G -modules and B′ = ∞ (η′)i(H ′ ) is a decomposition of irre-
2 k∈Z i=0 hki
ducible G -submodules. This is a Z-graded analogue of the classical theorem on harmonic
2 L L
polynomials for the simple Lie algebra of type B .
n
When n < n , the bases of the subspaces H , H and H ′ had been obtained.
1 2 hℓ1,ℓ2i hki hki
Using Fourier transformation, we can identify the subspaces H , H and H ′ with
hℓ1,ℓ2i hki hki
the subspaces of homogeneous solutions for the corresponding usual Laplace equations in
certain spaces of generalized functions. In [LX3], Luo and the author extended the above
representation of G on B to a representation of the simple Lie algebra G of type C on
3 n
B and proved that if n < n or k = 0, the subspace B with k Z are irreducible
1 2 hki
6 ∈
G -modules, and when n = n , the subspace B is a direct sum of two explicitly given
3 1 2 h0i
irreducible G -submodules.
3
The above irreducible modules except n = n = n in the case of type A are
1 2 n−1
explicit infinite-dimensional weight modules of finite-dimensional weight subspaces. They
are not unitary. The irreducible modules for G ,G and G are generically neither of
1 2 3
highest-weight type nor of unitary type. In [LX4], Luo and the author generalized the
above results for G,G and G to those for the corresponding Lie superalgebras. Bai
1 2
INTRODUCTION ix
[B1, B2] proved that some of irreducible representations in [LX1-LX3] have distinguished
Gelfand-Kirillov dimensions.
Dickson [D] (1901) first realized that there exists an E -invariant trilinear form on its
6
27-dimensional basic irreducible module, which corresponds to the unique fundamental
cubic polynomial invariant and constant-coefficient differential operator. We proved in
[X17] that the space of homogeneous polynomial solutions with degree m for the cubic
Dickson invariant differential equation is exactly a direct sum of Jm/2K + 1 explicitly
determined irreducible E -submodules and the whole polynomial algebra is a free module
6
over the polynomial algebra in the Dickson invariant generated by these solutions. Thus
we obtained a cubic E -generalization of the classical theorem on harmonic polynomials.
6
Next we want to describe our new multi-variable hypergeometric functions of type
A and their connections with the representations of type-A simple Lie algebras and the
Calogero-Sutherland model. For a C and 0 < n Z, we denote
∈ ∈
(a) = a(a+1) (a+n 1), (a) = 1. (0.29)
n 0
··· −
The well-known Gauss hypergeometric function is
∞
(α) (β)
F (α,β;γ;z) = n nzn. (0.30)
2 1
n!(γ)
n
n=0
X
Many well known elementary functions and orthogonal polynomials are special cases
of F (α,β;γ;z) (e.g., cf. [WG, AAR, X21]). It satisfies the classical hypergeometric
2 1
equation
z(1 z)y′′ +[γ (α+β +1)z]y′ αβy = 0. (0.31)
− − −
Denoting D = z d , we can rewrite the above equation as
dz
d
(γ +D) (y) = (α+D)(β +D)(y). (0.32)
dz
Moreover, it has the differential property:
d αβ
F (α,β;γ;z) = F (α+1,β +1;γ +1;z) (0.33)
2 1 2 1
dz γ
Furthermore, the Euler integral representation
Γ(γ) 1
F (α,β;γ;z) = tβ−1(1 t)γ−β−1(1 zt)−αdt (0.34)
2 1
Γ(β)Γ(γ β) − −
− Z0
holds in the z plane cut along the real axis from 1 to .
∞
Let N be the addtive semigroup of nonnegative integers and let
Γ = Nǫ (0.35)
A q,p
1≤p<q≤n
X