Two-step nilpotent Lie algebras

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66-13,254 SCHEUNEMAN, John Herman, 1937TW O-STEP NIL POTENT LIE ALGEBRAS. Purdue U n iversity, Ph.D ., 1966 M athem atics

U n iversity M icrofilm s, Inc., A n n Arbor, M ich igan

TWO-STEP NILPOTENT LIE ALGEBRAS

A Thesis Submitted to the Faculty of Purdue University by John Herman Scheuneraan In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy June 1966

G rad . School Form N o. 9 R evised

PURDUE

I

UNIVERSITY

G ra d u a te S c h o o l T h is i s to c e r t if y th a t th e t h e s i s p r e p a r e d B y ___________ \ J o t f A/

E n title d

^>C t t E U N F M A f J ---------------

NIL

PoT B r i T ____ I l f

C o m p lie s w ith th e U n iv e r s it y r e g u la tio n s and th at it m e e t s th e a c c e p te d s ta n d a r d s o f th e G ra d u a te S c h o o l w ith r e s p e c t to o r ig in a lit y and q u a lity F o r th e d e g r e e of:

__________ Ph- D .______________________ S ig n ed by th e fin a l e x a m in in g c o m m itte e : , c h a ir m a n

A p p ro v ed by th e h ea d o f s c h o o l o r d e p a r tm e n t: ■ T t s y

T o th e lib r a r ia n : T h is t h e s i s i s not to b e r e g a r d e d a s c o n fid e n tia l

P r o f e s s o r in c h a r g e o f th e t h e s i s

ACKNOWLEDGEMENTS

The author is most grateful to the Purdue Division of Mathematical Sciences, to the Purdue Research Foundation, and especially to his thesis advisor, Louis Auslander, for the moral and financial support they have lent him during his graduate studies.

Ill

TABLE OF CONTENTS Page A B S T R A C T ............................................ INTRODUCTION

..................

iv 1

THE DUALITY . . ......................................

4

The Cohomology Ring of a Lie A l g e b r a ............ Definition and First Properties of N* . . . . . . The Involutive P r o p e r t y ........................ The Main Result of Chapter I I ..................

4 B 12 15

THE INVARIANT........................................

17

The Function I .................................. The Universal Enveloping Algebra . . . . . . . . . The Main Result of Chapter I I I ..................

17 19 21

EXAMPLES.............................................

25

The Case dim N = 6, dim N» = 2 .................. The Hessian. . . . . . . . . . . . . . . . . . . . Certain Algebras N with dim N = B, dim N r = 2 . . The Case Where (dim N - dim N T) is Odd . . . . . . Certain Algebras N With dim N = 10, dim N f = 5 . . Certain Three-Step Nilpotent Lie Algebras . . . . FURTHER RESULTS ...................................... {l(B)} is not a Complete Set of Invariants . . . . Extending the Definition of the Invariant . . . . A Duality for Certain Almost-Algebraic Algebras .

25 29 30 32 33 35 3# 3& 39 42

CONCLUSIONS..........................................

47

LIST OF R E F E R E N C E S .................................

49

A P P E N D I X ............................................

50

VITA

52

ABSTRACT

Scheuneman, John Herman. Ph.D. Purdue University, June, 1966. Two-step Nilpotent Lie Algebras. Major Professor: Louis Auslander. This thesis is a start toward a classification of nilpotent Lie algebras.

There are two main results.

First

it is proved that, in the set of two-step nilpotent Lie algebras, there is a correspondence N following properties:

■» N* with the

1) if dim N = n + r and dim N f = r,

then dim N* = n + in(n - 1) - r and dim N*T = in(n - 1) - r; 2) (N*)* is isomorphic to N; and only if

3)

and N2 are isomorphic if

and N 2* are isomorphic.

Second, an

isomorphism invariant for two-step nilpotent Lie algebras is defined.

Examples are given to demonstrate the power of the

two main results in studying two-step nilpotent Lie algebras. The thesis ends with several miscellaneous results.

Among

these is an extension of the definition of the invariant to include certain three-step nilpotent Lie algebras, and a discussion of the difficulties involved in any further extension. discussed.

The limitations of the invariant are also

1

CHAPTER I

INTRODUCTION

A Lie algebra over the field K is an algebra over K whose bilinear product, denoted by [ ,] , satifies the following conditions: 1) [X,Y] = - [Y,X] 2) [x ,[y ,z]] identity).

(anticommutativity), and

+ [Z,[X,Y]]

+ [Y,[Z,X]]

=0

(the Jacobi

The Lie algebra N is called nilpotent in case

the descending central series (N1 ) of ideals in N defined by N^ = N, N^*^ = [n ,N^] descends to the zero ideal in a finite number of steps.

We say that N is two-step

nilpotent in case [N,N’j = 0 (note:

abelian Lie algebras,

i.e., those N with N T - 0, are considered to be two-step nilpotent, according to our definition). Over certain fields, the classical theory of Lie algebras admirably classifies semisimple Lie algebras, and describes the way in which an arbitrary Lie algebra is built from a semisimple algebra and a solvable Lie algebra. However, regardless of what field is under consideration, little is known about solvable Lie algebras.

Among these

are the nilpotent Lie algebras, about which we are especially ignorant.

In many respects, the semisimple and

nilpotent cases are at opposite extremes (with the solvable

2

case in between), and the classical results and methods yield no information about the nilpotent case. Malcev, in the excellent paper n0n a Class of Homogeneous Spaces* {Reference (V]), was the first to give an indication of the interesting phenomena that occur in the nilpotent case.

He gives the following examples:

1) an

example of a nilpotent Lie algebra over the reals having no basis with rational constants of structure, and 2) a nilpotent Lie algebra over the reals having two bases with rational constants of structure, but such that the rational Lie algebras defined by these bases are not rationally isomorphic (of course, they are isomorphic as real algebras). These examples arose in connection with Malcev's geometrical results in A major part of M

some of which we now describe.

is concerned with compact homogeneous

spaces tyjQ, where $ is a discrete subgroup of the nilpotent Lie group

.

Such homogeneous spaces are called

nilmanifolds.

The following connections (due to Malcev)

between nilmanifolds and nilpotent Lie algebras furnish motivation beyond the purely algebraic for our study of these algebras. subgroup

1) A nilpotent Lie group V\ has a discrete

such that

is compact if and only if the Lie

algebra of H has a basis with rational constants of structure.

2)

Hence, to each nilmanifold there is

associated a rational nilpotent Lie algebra, and,

3 conversely, each basis of a rational Lie algebra N gives rise to a nilmanifold to which N is associated.

3)

If

and N2 are the rational nilpotent Lie algebras associated with

and

YI

2/

& 2 *

r®spectively, then

and N2 are

isomorphic (as rational algebras) if and only if Hl/$L and are both finite coverings of a third nilmanifold. In this thesis, we concern ourselves with nilpotent Lie algebras of finite dimension over subfields of the complex numbers.

For the most part, we restrict attention to the

two-step nilpotent case, and seek to answer the following questions inspired by Malcev's examples.

Given a field K,

how many of these algebras of a given dimension over K are there?

Given an algebra N over the field K and a subfield

K f of K, does N have a basis whose constants of structure are in K *? Our two main results are the duality theorem of Chapter II and the isomorphism invariant of Chapter III.

In Chapter

IV, we give a few examples to demonstrate the power of the main results in answering the questions above.

In Chapter

V, we extend our methods slightly to cover certain three-step nilpotent Lie algebras, and discuss the difficulties involved in further extension of these methods. The limitations of the invariant are also discussed in Chapter V.

CHAPTER II

THE DUALITY

This chapter is devoted to the definition and study of a certain correspondence N — ► N* of two-step nilpotent Lie algebras.

This correspondence is non-trivial and has the

property that (N*)* is isomorphic to N, so it is called "the duality" of two-step nilpotent Lie algebras.

The

duality has the effect of halving the number of cases which must be considered in a survey of all two-step nilpotent Lie algebras, a fact which proves to be very helpful (See Chapter IV D), Our definition of N* is given in terms of the cohomology ring of N,

For the reader*s convenience, we

review this concept in Section A.

In Sections B and C, we

describe most of the useful properties of N*.

The chapter

culminates in Theorem 3 of Section D,

A,

The Cohomology Ring of a Lie Algebra

We first give a staccato review of the terminology and facts necessary to define the cohomology ring H(N) of the Lie algebra N, page 104.

Proofs may be found in [2], beginning on

Following this review, and ending Section A, is

a proof of the most important fact about cohomology rings,

5 namely, that homomorphisms of Lie algebras induce homomorphisms of their cohomology rings. Let N be a Lie algebra over the field K.

A p-cochain,

p = 1,2,...,dim N, is an alternating p-linear map f: Nx...xN —

K.

The set CP(N) of all p-cochains is a

vector space under pointwise addition and scalar multiplication by elements of K.

The linear map

cP(N) — ► CP4*1 (N) is defined by Sf(X1 ,...,Xp+.1 ) = - y The kernel of

(-i)

p = 1,2,...,dim N. b

Elements of ZP (N) are called p-cocycles. of CP“^(N) in CP(N) is denoted by BP(N),

p = 2,3,...,dim N.

B^(N) is defined to be the zero

subspace of C-*-(N).

Elements of BP (N) are called

p-coboundaries.

).

in CP (N) is denoted by ZP (N),

h

The image under

* f ( [Xj ,Xjj|,x^,... ,Xj,... ,3^,...

Since

b ^

-

0, we have Bp ^ Zp .

The

quotient space Zp/bP is called the p^*1 cohomology group of N and is denoted by HP(N), p = 1,2,...,dim N. p-cocycle f under the natural map Zp —

The image of a

Hp is called the

cohomology class of f and is denoted by 7. The cup product of cochains, defined as follows:

_

u:Cp x Cq — *-cP+(*, is

if f t Cp and geC^, then f u g is the

(p + q)-cochain given by f o g(Xi,... >Xp^q) — alt f (X^ j.. .,Xp)g(Xp_^-^ ,. ..,Xp^q). ”

sgnTt f(X TI(1) »•••

Tt (p )

(p+1) »• • •

7t(p+q) ) •

6 Our* first observation is that, if f]_,f2 £ then (f^ + f g g

= f^ ^ g t f2 ^ g.

anc* g e CP,

Next, it is not hard

to verify that if f e CP and g e CP, then ^(f u g ) = 6f

g.

u

This fact, plus a

induces a product, again

, of cohomology class according to the formula

u

7 u g = fug

if 7

t

HP and "g € HP.

Next, the cup product

of cohomology classes is extended by linearity to the whole dimN vector space H(N) = T ~ ® HP(N).

H(N), together with the

pH. cup product on H(N), is a ring called the cohomology ring of N with coefficients in K. Let h: N]_ —

N2 be a homomorphism of Lie algebras.

Define the linear map h*: CP(N2 ) — ^-CPfNj) as follows: if f € Cp (N2 ) and Xx ,...,Xp € Nlt let h*f(X1}...,Xp) = f (hXx Lemma 1 Proof

h* b

=

b

hXp ).

h*

h*( & f )(Xlf... ,Xp+1) = 6f (hXl9... ,hXpfl) _ if ±f(( ^lX LMj j)hXkJ,hX1 ,...,'^ ,hXkJ , , *.., hX j t ■*■,^hXk »• ■•» hxp+1 J

—

±f (h [xj »Xk} »bXj ,•.., hXj ,... ,hXk ,..., hXp^.^_) — ^

±h f ( ^Xj ,Xk] *X^,... ,Xj ,. ••,Xk ,. ..

)

3 < &

=

6(h*f)(x1 ,...,xp+1).

Again, let h:

N2 be a homomorphism.

If

7 f € HP (N2 ), let h*f = Corollary 1 Proof

W

f

€ HP (N1 ).

h* is a linear map of H^Ng) into HP (N^).

We need only show that, if 7 € HP (N2 ), then the

definition of h*T is independent of the representative of f. This is true since, according to Lemma 1, hk (f +

b y )

-

T F f

+ p y y

=

W

f

+

b

H*y =

W

f

So far, we have defined h*: hP(N2 ) — •■■■*p = 1,2,...,dim N2 .

) for

Next, we extend n* to all of H(N2 ) by

linearity. Theorem 1

Let h: N-^ —

N2 be a homomorphism.

Then the

linear map h*: H(N2 ) — *~H(N^) is also a homomorphism. Proof

We first show that h*(f ^ g) = h*"f

^

h^g if

f £ Cp (N2 ) and g £ Cq {N2 ): h*(f u g) (Xx ,.. .,^p+q) = f u gthX-L,.. .,hXp+q) = alt f (hX-L,.. .,hXp )g(hXp+1,... ,hXp+q) = alt h*f(X1 ,...,Xp )h*g(Xp+1,...,Xp+q) = h*f u h*g(X1 ,...,Xp+q). Therefore, h^{f u g) - K*f u h^g.

Also,

T^TFTTgT = h*(7-v~g), by definition of h*, and h*(f u g) - h*(f v_» g) = h«{7 cohomology product. h*f

g), by definition of the

For the same reasons,

httg = h#f ^ l#g = h^T ^ h’S'g.

Therefore,

h*(7 v g) = h*T o h ^ g if f 6 Hp (N2 ) and g € Hq (N2 ). theorem now follows from the linearity of h*.

The

8 Corollary 2

If h is an isomorphism, then h* is an

isomorphism. Proof

According to Theorem 1, both h* and (h "-*-)* are

homomorphisms.

It is easy to verify that if h^: N]_ —

and h 2: N 2 -■-» N ^ are homomorphisms, then (h2 o h-^)* = h^* o h 2^.

Therefore, (h”^-)* - (h*)-1, so h*

is an isomorphism.

B.

Definition and First Properties of N*

We begin with a definition of N* and a preliminary lemma about N*.

Then we introduce several notations which

will be used constantly in the computations in this section and the next.

This section ends with a series of lemmas

whose purpose is to explicitly compute bases for N* and the derived algebra N**. Definition 1

Let N be a two-step nilpotent Lie algebra.

N* is defined to be the algebra whose underlying vector space is the subspace H^N) $ H^(N) u H^CN) of H(N) and whose product, denoted by [,], is defined by 1) [u,v] = u w v if u,v € H 1(N), and 2) £u,v} = 0 if u or v is in H 1(N) Lemma 2 Proof

u

H X(N).

N* is a two-step nilpotent Lie algebra. Since [u,[v,w]] = 0 for any choice of u, v, w £ H*,

the Jacobi identity is satisfied. j-jj follows from the fact that if

The anticommutativity of and ^ are 1-cochains,

9 then

f 1

u f 2(X1,X2) = f1(X1)f2(X2) - f 1(X2)f2(X1 ) =

s - f 2 uf^(X^,X2).

We see that N* is two-step nilpotent

since the derived algebra, H^(N) o Hb(N), is central. Let N be a two-step nilpotent Lie algebra of dimension (n + r) over the field K whose derived algebra N f is of dimension r.

Select a comolement N c to N f in N and a basis

{ X p ... ,Xn} of N c.

From among the vectors {[Xi.Xj]; i < j}

in N', select a basis [X ir>X 3r] = 7 ±rJrJ of N '- Denote by Sj the set of pairs (it,jt ), ip < Jt anci b ” l»2,...,r, involved in the definition of this basis for N ’.

Let

S c = {(&k»bk)5 ak < bk» k = 1,2,...,s = Jn(n - 1) - rj be the rest of the pairs (i,j) with 1 - i < j - n. central in N (i.e.,

The Y ’s are

[ ^ i ^ t * ^ = 0) since N is two-step

nilpotent, and the remaining products of elements of our r

5T*

chosen basis are given by Lx ak »x bk J “

(ak,bk ) £ S c and each

■i ■? xtJt

€ K.

on N by gk (Nc) = 0, and gk (Yitjt) =

I

H h D ak>bkY i tJt> where

t=l Define the 1-cochains gk & kt, k - l,2,...,r.

Define the 1-cochains f^ on K by f^(H') - 0, and Y i^x j^ ~

^ij*

~ 1,2,... ,n.

Lemma 3

Let f^, i = l,2,...,n, be the cochains defined

above.

Then {i*]_,... ff n^ is a basis for Hl(N).

Proof

Since zero is the only 1-coboundary, h !(N) is

identical with Z^N).

A 1-cochain f is a 1-cocycle if and

10

uat ,bt*at

Then

t'bt.

u

Proof: We show that the 2-cochain

&gk agrees with the

2-cochain on the right for all pairs (X^,Xj), i < j, and hence the two functions are equal.

First note that

f± u fjUicXi) - fi (Xk )fj(X1 ) - ^(XiJfjtXk) is zero whenever i - j, k - 1, except when i = k and j = 1, and that f± ^ fjtXi.Xj) = 1. a)

Consider the values of the two functions at

(Xit ,Xjt ), (it.Jt) =

=

e

^kt#

fik u fjk (Xit ,Xjt) agrees with b)

i ^ ( x it>x it) = ^ ( [ X i t ,xjt]} =

S j .

v^ew

tlie remark above,

&k1-» so the function on the right

6 gk at (Xit,Xjt ), {it ,jt ) £ Sj.

Suppose (ak ,bk ) £ Sc.

Then

Sgm (Xak ,Xbk ) = §m( [xak >xbk]) / M i = Sm^ak.bkMj! +

M r + DakfbkYirJr*

imJm - Dak ,bk »

By tbe remark above, the im j_ function on the right also takes the value D&™ bk at (Xak,Xbk), (ak ,bk ) 6 Sc .

We have shown that £ gk and the

function on the right agree at all (Xj^Xj), i < j, so they are equal.

11

Corollary 3

Each

igk is in Z ’(N) ^ Z T(N).

Cqrollarg-it

fik ^ fjk = -

Da*,btfat ^ fbt

t*l Just apply the natural map Z2 — *-H2 to each side of

Proof

the equation in the statement of Lemma 4. Lemma 5

Let gk , k = l,2,...,r, be as defined above.

Then

^£gkj is linearly independent. Pfoof Y ~

Suppose ^

ak

=

oth erwords,

thefunction

ak Sgk vanishes on all (XjjX-j), and inparticular

(Xit ,Xjt ), (it ,jt ) £ Sj.

Therefore,

0 = 21 &k ^ gk (xit>Xl V = X Lemma 6 Proof

on

akSk(YitjV = at*

■{&gk} spans B2 (N )• Let g e B^(N), so g =

if, for some 1-cochain f. r

Let ck = f(Yi j ). k k

We will show that g =

k=I

First, note that g(X,Y) = element of N». £ ® c-fc.

ck igk .

/

ck 6gk (X,Y) = 0 if X or Y is an

Next, suppose (it,jt.) £ Sj. g(Xit ,Xjt) -

= ct, and

ak i gk (Xit ,Xjt)= ^

akgk bm )

ck&k(

k=.

g■

) ^-x*0 T

^"1^1

2_ Ckgk^Dftm»bmYilO*l + ••• + Dam»braYirOr^

=

■*■1^1 •^•rJr = clDam »bm + *** * crDara,bm * Thus the functions g and ^ cSg^ agree everywhere, so they are equal• Lemma 7 ~~

Let

f aj_

X»

and

f h ^ ,

u

be defined as on page 6. ^fa^

t - l,2,...,s = j|n(n - 1) - r Then the set of cohomology classes

^*b-t,’ (at»bt) 6 S q , t — 1,2,. ..,s = in(n - 1) — r^

forms a basis for HY (N) o H^-(N). Proof

We first observe that Z^-u Z^- has dimension

|n(n - 1) and basis ^f^ u f j;i < jj*. dim H^-

u

Therefore,

= £n(n - 1) - dim (B2 A (Z^ ^ Z^-)).According

to Lemmas 5 and 6,

{&Sk\

According to Corollary 3,

is a basis for B2 , B2

so

dim B2 - r.

Z^ u Z^, so

dim(B2 0 (Z^ u Z1 )) = r, and therefore dim HY u H-^ = ^n(n - 1) - r. { * 5 7 "C7~?bt } sPans

Corrollary 4 states that

U H^-, and therefore, since the number

of elements in this set is equal to dim a basis for H^- u

U

H^, this set is

H-*-.

C.

The Involutive Property

The aim of this section is to prove Corollary 5, which states that (N*)* is isomorphic to N. ■?%

4f.

obtained as follows.

This result is

Given N and the special type of basis

13 B of N defined in the preceding section on page 9, we define a two-step nilpotent Lie algebra N^(B) by giving the constants of structure of the "dual basis" B* of N*(B). These constants of structure are defined in terms of those of the basis B of N. isomorphic to N.

It is easy to see that (N*(B))*(B*) is

Using some of the results of the preceding

section, it is also easy to see that N*(B) is isomorphic to N*.

Hence (N*)* is isomorphic to N. As usual, let N be a two-step nilpotent Lie algebra

over K.

Let B = |x^,...

... »Yir jr| be the basis

for N defined in the preceding section on page 9. [Y itJt-N] = °» [x it>x 3t] =

for

C S I>

t = l,2,...,r, ana [xak,Xbk] = Y" D ak»bk ak ^bkT H it^Jt 't t=I (ak,bk ) 6 S c, k - 1,2,...,s = |n(n - 1) - r. Definition 2

Then

Notation will be as above.

f o r

The two-step

nilpotent Lie algebra N*(B) is defined as follows.

The

underlying vector space is the vector space over K with basis B* =

... ,Un ,Waib]L,... ,Wagbg| , s = £n(n - 1) - r.

The Lie product [ , ]on N*(B) is defined by 1) [D ak .°bk] = »akbk ^ r i then [Uit7U jtJ =

V

To clarify Definition 2, we give a pictorial description of how the multiplication table defining N*(B)

14 is obtained from the multiplication table attached to the basis of N.

From this description, it will be clear that

(N*(B) JMB*) has a basis whose constants of structure are identical with those of B, and hence (N*(B))*(B*) is isomorphic to N. Let M be the "matrix of constants of structure" of the basis B of N.

In the appropriate ordering of rows (which

correspond to brackets correspond to

i < j) and columns (which M has the form

n = I , where D is -t-h ^t J"t the matrix whose k t Uil entry is Da^jbk*

With the same

ordering of rows and with columns corresponding to W a ^ b ^ 3 * the matrix M* of constants of structure of the basis B* of N*(B) has the form

IT=

15 Theorem 2 Proof

N* and N*(B) are isomorphic.

According to Lemmas 3 and 7, respectively,

■f^,...,fnj- is a basis for H^fN) and ~ fai

u

^bi> ••• »^as u ^bs) Is a hasis for Ffl(N) u h!(N). fn ,7ai u f'bi»***»?as aj>...)a-^y*..>a^) c) I(a^j...,a i—1»a i t ta jf a

>»• • ? an) — I(a^j*..j3jj) if

t ( K and

i ^ j,

Lemma 9

The function I satisfies the

formulas:

following reduction

n

1) I(®i> • * • ja n ) ■■ ^

(—X ) +1ak I(a^j • • • *& k ,• • • jfijj)

k-1 2) I(a^,...,a^) — “ ^ k... ^a^}.. •ja^) .

1) Write I(a^,...,an) in the form

I(a^,...,an) = aiEi + •*• t an®n» where Ek is an expression in all the a Ts except ak.

Inspection of Ek reveals that

19 Ejc “ (—1)^ 2)

*

The second formula is obtained by iterating the first

one, as follows: n I(a^,•. •tan ) “ ^

(”1)

(3^)•••>®k**'*»an^

n k-l ~ ^ ^ (-1 )^c'^^‘ajc^ ^ (—1) t+^atI(a2,

»aj^»•• •>an)

n + y (-1) atl(ai» •••>ak» ••*»at* ***,an ^ t=k+l ^ (-1) k **• ,an^

kTt (-1) * ^ (a^a^. - a-t-a^Ifa^,...,ajc,...,at,...,an ).

B.

The Universal Enveloping Algebra

This section begins with two definitions:

first we

define the universal enveloping algebra of a Lie algebra; then we define the concept of a compatible basis for a two-step nilpotent Lie algebra.

Using a compatible basis,

we then give a simple realization of the universal enveloping algebra of a two-step nilpotent Lie algebra. This realization is used often in the sequel to facilitate computations• Let N be a Lie algebra over K.

Let T° = K and denote

20

by T*, i ** 1, the i-fold tensor product N$... « N of the underlying vector space of N. T = / © iiT)

The tensor algebra

is an associative algebra with respect to the

tensor product. Definition 4

Let J be the ideal in T generated by

elements of the form X 0 Y - Y & X are in T* = N.

all

[x,Yj, where X and Y

The associative algebra T/J is called the

universal enveloping algebra of N and is denoted by U(N). Definition 5 over K.

Let N be a two-step nilpotent Lie algebra

A basis for N is said to be compatible if it is the

union of a basis for N' and a basis for a complement to derived algebra N'.

the

In the sequel, we will denote

compatiable bases by |x^,...,Xn ,Y^,...,Yr^, where the Y's will be understood to be a basis for the derived algebra. We now give a realization of the universal enveloping algebra of a two-step nilpotent Lie algebra using a compat­ ible basis.

Let N be a two-step nilpotent Lie algebra.

Let

■^Yi,...,Yr } be a basis for N 1, and let {X-^t...,XnJ be a basis for a complement to N'. is a compatible basis for N. linear combination of Y's.

Then B = {x^,...,Xn ,Y^,...,Yrj Note that eachjx^jX^] is a

It is easy to see that the

universal enveloping algebra of N is isomorphic to the algebra of polynomials in the non-commuting variables X ^ ,...,Xn ,Y^,...,Yr , where these variables satisfy the

_

21

following commutation relations: ~ Y^X^, and Y^Y^ - Y jY^. compatible bases relations.

- XJti = [x±x J ,

The virtue

of our use of

lies in the simplicity of the commutation

Note that we may consider the underlying vector

space of N to be

C.

contained in U(N).

The Main Result of Chapter III

Let N be a two-step nilpotent Lie algebra and let B = {^1,...,Xn,Y^,...,Yr} be a compatible basis for N.

This

section is a study of the element I(Xj,...,Xn ) of the universal enveloping algebra of N.

Our first result is that

K X i ^ . ^ X n ) can be expressed as a commutative polynomial, called the symmetric form of I(X^,.•.,Xn )• form is denoted by 1(B). about compatible bases.

This symmetric

Next follows a pair of lemmas After that is a definition of an

equivalence relation, "K-equivalence", in the set of polynomials of several variables over K.

Finally, we come

to the main result of Chapter III, which states that if B^ and B2 are compatible bases of isomorphic two-step nilpotent Lie algebras, then ItB^) and K B 2 ) are K-equivalent. Lemma 10

Let N be a two-step nilpotent Lie algebra, and

let B s ^X]_,... ,Xn ,Y^,... ,Yr} be a compatible basis for N. Let I be the function of Section A applied to the associative algebra U(N). 1)

If n is even, then I(X^,...,Xn ) can be expressed as a

22

polynomial in the Y ’s, 2)

If n is odd, then I(X-^,... ,Xn ) can be expressed in the

form I(X2 »...,Xn ) = xipi ♦ ••• + xnpn* where each P is a polynomial in the Y ’s. Proof

a)

I(XX ) = X x and I(X1 ,X2) =

which is a polynomial in Y ’s. n = 1,2.

X - y X 2

- X ^

=

Hence the lemma is true for

According to Lemma 9,

so the lemma follows by induction, since each [x^,Xj] is a linear form in the Y ’s. The expression for I(X-p ...,Xn ) in Lemma 10 is called the symmetric form of I(Xp...,Xn), because all the variables in each term commute with one another.

While

polynomials in non-commuting variables do not have unique expressions, their symmetric forms (if they have them) are unique up to ordering of variables, and these symmetric forms are indistinguishable from commutative polynomials. Definition 6

Let B = -jx^,... ,Xn,Y^,... >Yr| be a compatible

basis for a two-step nilpotent Lie algebra N.

We denote by

1(B) the symmetric form of I ( X p ...,Xn). 1(B) has the following properties:

It is a homogeneous

polynomial of degree £n if n is even and £(n + 1) if n is odd; it depends on r variables if n is even and on (r + n) variables if n is odd, where r = dim N 1; the coefficients of

23 1(B) are polynomials in the constants of structure of B. These facts ijiay all be easily proved by induction using the reduction formulas of Lemma 9. The following lemma about compatible bases is needed in the proofs to come* Lemma 11

Let B = {x-^,... ,Xn,Y-^,... ,Yrj- be a compatible

basis for the two-step nilpotent Lie algebra N.

Then any

Lie algebra isomorphic to N has a basis whose constants of structure are identical with those of some compatible basis B1

.,Xn ’,Y^’,... ,Yr’} of N, where the X 1’s are

linear combinations of the X*s (and,of course, the Y Tts are linear combinations of the Y ’s, since B T is a compatible basis). Proof

Let f: N.— *- M be an isomorphism.

Let

C = {^i,...,Un ,V^,...,Vrj be a compatible basis for M. Y± » = f-1 ^ )

and X ±* = f“1 (Ui ).

Let

Then

is a compatible basis for N whose constants of structure are identical with those of C. X^* = X^» + W^, where Xi 1 is in the span of the X Ts, and WjL 6 N f.

Therefore, [X^jZ] = [^i1*2]* for anX Z 6 N, so

the compatible basis B 1 = ^Xj, *,... , 1 ^ TjY-j*,... ,Yr 'j- has the same constants of structure as C. Definition 7 polynomials in

Let f (Z-^ ,... jZ^ ) and

g{Z-^

, , . ,

JZ m

)

be

f and g are said to be

K-equivalent in case f(Z^,...,Z^ ) - ag(Z^f,...,Zffl!), where

24 a 6 K and a / 0, and

Z ^ 1

= ^

where

€

K and

detft^) ^ 0 . Theorem 4

The K-equivalence class of 1(B), where

B = ^ 2.* • • * >Xn ,Yl> •* * >Yr }

a compatible basis for the

two-step nilpotent Lie algebra N ovexvK j is a-K-isomorphism invariant of N. Proof

Let C be a compatible basis for an algebra M which

is isomorphic to N.

By Lemma 11, the constants of structure

of C are identical with those of some compatible basis B ’ = ^X-^1,.. .,Xn ’,Y^f,... ,Yr 'J of N, where the X T,s span the same complement to N 1 as the X Ts.

Therefore, except for

renaming the variables, I(BT) = 1(G).

To prove the theorem,

we observe that 1(B) and I(B’) are K-equivalent. V

aiix j and Yj-

non-singular.

where

First,

and (h-^*) are

According to Lemma 8, we have

I(B') * alt X1 »X2 »...Xn » = det(a±j) alt X1X2...Xn , so I(B') = det(a^^)1(B).

The left side of the last equation~~is

a polynomial in primed variables and the right side is a polynomial in the unprimed variables; a non-singular linear transformation relates the primed and unprimed variables, so 1(B) and I(Bf) are K-equivalent.

25

I

CHAPTER IV

EXAMPLES

This chapter is devoted to showing how the main results of Chapters II and III may be used to study two-step nilpotent Lie algebras.

In Sections A, C, and E, we give a

few

concrete applications of our results in

order toshow

the

power of our method and to exhibit some

of the

interesting phenomena that occur. introduce the Hessian of a form.

In Section B, we This turns out to be a

useful tool in our computations because it gives a simple way to measure K-equivalence of I(B)’s.

Certain limitations

on the use of the Hessian are overcome in Section D through the use of the duality theorem.

In addition to the above-

mentioned work on the two-step nilpotent case, we study certain three-step nilpotent Lie algebras in Section F.

A.

The Case dim N = 6 , dim N* = 2

In this section, we give a complete survey oftwo-step nilpotent Lie algebras over the complexes and reals which are six-dimensional and have a two-dimensional derived algebra.

We find that over the complexes, there are three

such algebras, and over the reals there are four ( up to isomorphism).

We also find that there are infinitely many

26 non-isomorphic two-step nilpotent Lie algebras over the rationals satisfying these dimension conditions. Lemma 12

A six-dimensional two-step nilpotent Lie algebra

N with dim N T = 2 has a compatible basis {X-^,... ,X^,Y-l ,Y2} with multiplication table [*ix2]

*xi

M d

~ 0

[X l X 3]

=T 2

[j 2X 4] = a Y 2

[X1X4]

=0

^ 3*4] =

+M 2.

We denote the algebra defined by this multiplication table by N(a,b,c). Proof

Start with an arbitrary compatible basis.

By

reordering the X ’s, we may assume either that [x^XgJ and ^XjX^J are linearly independent or that all brackets are zero except [x^Xg] and [x^X^} and these are linearly independent.

If the latter case occurs, use X-^ + X^ instead

of X^ in the basis, so we can always assume the former case. Then let Y^ = (^1^ 2]* ^2 =

* so

mu]-'t^Pl^ca‘ tion

table is tx ix 2] a x i

[ ^ d

= b iT i + b 2x 2

tX lX 3] = Y 2

lX ^ J

“ C 1Y 1 + = 2*2

tX lX J = a lT l + a 2X 2 [X £ J “ d lT l + d 2X 2Now let X ^ ’ = X^ - a^X2, et cetera, to make all the constants of structure zero except those corresponding to a, b, c above. Lemma 13

1) Over the complex numbers, N(0,0,0), N(1,1,0)

27 and N(-l,l,2) are mutually non-isoraorphic. 2) N(a,b>c), where 4ab + c2

?

0, is isomorphic over the

complexes to N(1,1,0). 3) N(a,b,c), where 4ab 4> e2 = 0, but not all of a, b, c are zero, is isomorphic over the complexes to N(l,-1,2). Lemma 14

1) Over the reals, N(0,0,0), N(1,1,0), N(l,-1,0)

and N(-l,l,2) are mutually non-isomorphic. 2) N(a,b,c), where 4ab + c2 >0, is isomorphic to N(1,1,0) over the reals. 3) N(a,b,c), where 4ab + c2 < 0, is isomorphic to N(l,-1,0) over the reals. 4)

N(a,b,c), where 4ab + c^ = o but not all of a, b, c are

zero, is isomorphic over the reals to N(l,-1,2). All of the statements in these two lemmas may be proved by means of simple though lengthy computations.

Rather than

present them, we show how the invariant of Chapter III can be used in the problems of non-existence of isomorphism, parts 13-1 ) and 14-1 ). In the present case, 1(B) = tx lx z][x 3x 4] - [x lx 3] [ V J

+ [¥t][¥3]

2Y]Y2 + Y 2

The discriminants are -4 and 0, so

the forms are not complex-equivalent by Lemma 15.

Therefore,

by the main result of Chapter III, the algebras are not coraplex-isoihorphic • Corollary 7 the reals. -Yx2

«(1,1,0) and N(l,-1,0) are not isomorphic over For, their invariants are Y ^2 +• Y 22 and

Y 22, respectively, and the discriminants are -4 and

4, respectively.

There is no real a such that -4a2 - 4, so

by Lemma 15, the invariants are not real-equivalent.

By the

main result of Chapter III, N(1,1,0) and N(l,-1,0) are not real-isomorphic.

29 We can also get information on rational isomorphism classes among the algebras N(a,b,c).

We find that there are

infinitely many non-isomorphic rational algebras. Corollary

Let p and q be positive primes.

8

Then N(p,l,0)

and N(q,l,0) are not rationally isomorphic unless p = q. (But by Lemma 13~2), they are isomorphic over the reals.) Proof

The invariants are Y ^ _ pYg2 and Y ^ _ q ^ 2,

respectively.

The discriminants are 4p and 4q.

There is no

rational a such that 4a2P - 4q so by the main result of Chapter III, there is no rational isomorphism between N(p,l,0) and N(q,l,0).

B.

The Hessian

Taking our hint from the usefulness of the discriminant in the computations of the preceding section, we now introduce the Hessian of a form. Definition

8

Let f(Z^,...,Zm ) be a form (i.e., a

homogeneous polynomial).

The Hessian of f, denoted by Hf,

is defined by

Note that H is a generalization of the discriminant to forms in arbitrary numbers of variables of arbitrary degree. In fact, it behaves the same as the discriminant under K-equivalence.

30 Theorem 5

(Hesse)

and detft^j)

?

Suppose Zit - ^

t^Z^, where

f K

0, and that f(Z1 ,...,Zm ) = agfZ^',... ,Zm ’),

where a £ K, a ^ 0.

Then

H f ( Z l f ...,Zm ) = a m ( d e t ( t i j ))2Hg(Z1 S . . . , Z m M .

This old result is an easy exercise in the chain rule and linear algebra. Corollary 9

If B and B ’ define K-isomorphic two-step

nilpotent Lie algebras, then the Hessians of their invariants satisfy HI(B) = a ^ ^ H K B 1), where a,b

e

K and

a,b ^ 0, and m is the number of variables in 1(B). We note that HI(B) has coefficients which are polynomials in the constants of structure of B, since this is also true of 1(B), as we noted on page 22.

C.

Certain Algebras N with dim N = B, dim N* = 2

In this section, we study an interesting family of eight-dimensional two-step nilpotent Lie algebras whose derived algebras are two-dimensional.

We show that this

family contains real Lie algebras having no basis with rational constants of structure and complex Lie algebras having no basis with real constants of structure.

We also

show that this family contains uncountably many non-isomorphic real and complex Lie algebras. In order to obtain these results, we make use of the following fact.

31 Lemma 16

Let N be a two-step nilpotent Lie algebra over

the field K and let K' be a subfield of K.

If N has any

basis with constants of structure in the field K», then N has a compatible basis with constants of structure in K(. Proof

Let {Z-p ... ,Zm } be a basis with constants of

structure in K'.

From

and complete this to a basis for N by adjoining elements of ...,Zra| .

We thus obtain a compatible basis for N and

elementary linear algebra arguments show that the constants of structure are in K*. Define the algebra N(d) as follows: B =

is a compatible basis, and [, J

is defined by 1 ) [ X ^ ] = [XjXj = Tl [x3x 5] = [X 6x4] = y2

[X5X6] = dY4 2) all other [XjXj] = 0, i < j, and the Y ’s are central. Then N(d) is a two-step nilpotent Lie algebra whose derived algebra N f is the span of Y^ and Y£* It turns out that in this case 1(B) = dY-^3 +. Y^X^2* (See the appendix for the general expression for I(Xj_f... ,Xg) •)

Therefore,

HI(B) = det Note that, in view of Corollary 9, any algebra which is K-isomorphic to N(d) must have the property that the Hessian

7

of its invariant is equal to a (12dYj

7

-

t * # ?*

k t y

a ^ 0. Suppose algebra N(d)

now that d is irrational.

If tn# r*«

has any basis with rational constant*

structure, then it has a compatible basis B’

m

m

constants of structure, according to Lemma 1 .

>« ,

according to the remark at the end oi oeetion a polynomial

**

-

with rational coefficients.

«

r,

-

impossible for a polynomial to ne simultaneously * *-•''■ polynomial and of the form a2(12dY^2 - AY-,2 I.

■

arr#,

*

*

has no basis with rational constants of structu r*. Likewise, we see that if d is complex but not r*«:, then the complex Lie algebra N(d) has no basis wit

r* »

constants of structure. To conclude this section, we note that u

* beiow#

u

to the field K, then the Lie algebras B(d) snc Bid'* are K-isomorphic if and only if d = d 1.

For if Bid! and it#*

are isomorphic, then according to Corollary ladY-j^2- 4Y22 = a2(12d*Y12 - 4Y22).

D.

we m*#t %#ee

Hence a2 * 1

an* a *

The Case Where (dim N - dim B*j it Qjj

We have seen in Sections A and C how ueet ul the HI(B) is in studying two-step nilootent Lie algnferna. However, there are times when the Hessian gives no information at all.

In particular, if (din B - 44*11* » .*#

odd and (dim N - dim N f) > dim N*, then the Meant©# ©f «f^

32 of its invariant is equal to a2 (12dY^2 - 4Y22 ), a e K, a ^ 0. Suppose now that d is irrational.

If the real Lie

algebra N(d) has any basis with rational constants of structure, then it has a compatible basis B f with rational constants of structure, according to Lemma 16.

Then,

according to the remark at the end of section B, HI(B’) is a polynomial with rational coefficients.

However, it is

impossible for a polynomial to be simultaneously a rational polynomial and of the form a2(12dY^2 - 4Y22).

Hence, N(d)

has no basis with rational constants of structure. Likewise, we see that if d is complex but not real, then the complex Lie algebra N(d) has no basis with real constants of structure. To conclude this section, we note that if d,d* belong to the field K, then the Lie algebras N(d) and N(df) are K-isomorphic if and only if d = d ’.

For if N(d) and N(dT)

are isomorphic, then according to Corollary 9, we must have 12dYx2- 4Y22 » a2(12d»Y12 - 4Y22).

D.

Hence a2 = 1

and d = d».

The Case Where (dim N - dim N f) is Odd

We have seen in Sections A and C how useful the Hessian HI(B) is in studying two-step nilpotent Lie algebras. However, there are times when the Hessian gives no information at all.

In particular, if (dim N - dimNT) is

odd and {dim N - dim N 1) > dim N', then the Hessian of 1(B)

33 is always zero. 1(B) =

The reasons are as follows.

+ ••• + ^rpr» where the P Ts are polynomials in

the Y fs.

The X ^ enter only linearly, so

- o for ^Xi d X j

all X^,Xj.

Therefore, if there are too many X ’s compared to

the number of Y ’s, then the determinant defining the Hessian is zero. The results of Chapter II enable us to use the Hessian to get a more complete survey because they show that studying the isomorphism class of N* is the same as studying that of N, and eitner N or N* has a sufficiently large derived algebra so that the Hessian is not always zero.

E.

Certain Algebras N with dim N = 10. dim N T = 5

One would like to know more about the fact that some two-step nilpotent Lie algebra over the reals do not have bases with rational constants of structure. inquiry runs as follows.

One line of

Suppose N has no basis with

rational constants of structure, but that N has an ideal C of codimension 1 which does have a basis with rational constants of structure.

Is there anything special about C?

In particular, do all bases of C with rational constants of structure have the property that they are rational linear combinations of one another?

In the present section, we

give an example showing that the answer to the last question is nh6".

34 — |x1 ,...,x5,y1,.. .,Y5j. be a i compatible basis for a N(d) defined by

[XiX2] «

[X^J

y3

=

Y

l ¥ 5] “ 0 [x3xj = 0 [X3X5] = Y [x4x5] = r

(*lx 3] s T4

Dvd = y5 [xl*5] - 0 [x2x3] » dY1 central, as usual.

In this case (see the appendix), 1(B) turns out to be 1(B) = X1 (dY12 _ y 22) - X2(Y1Y4 ~ Y 2Y 5> + x 3 so the elements of

T* are semisimple. Definition

Denote by S(M,N;h) the semidirect product of M

and N given by ht M — ►•Der(N).

SCM-^jN^jh^) and S(M2 ,N2;h2)

are called simply isomorphic in case there are isomorphisms

• il(X1 ,X2 ,x3 jX^,X5 ) s =

xl< [X2X3] [x4xj] - M

[X3X5] + tx2x s][x3x4]

- X2 < [xlx3] [X4X 5] * [xlxj [X3X5] ♦ [xlxs] tx3x4]

+X3(xlx2]X4X5 \\ X2x3+[x^j]X2X4' XXX3'X2X5*+[xlX^■x2x3 -X4(Xix2X3Xj v 4X5(X1X2X3X4 3]X2X4.+tXlX4]■x2x3‘ *

-

5)

ii(x1 ,x2 ,x3 ,xit,x5 ,x6 ) = ( [x3x4] X5X6* * X3X5] ([ x ^ J x 5x6; - fob. 1[X2X5] [X4X6 y 3; X 5X6 - x2x5; X3X6 ♦ [X1X4 - x2x4; X3X6 [xix5; X2X3 X4X6 + [xlx6j< x 2x3; x4x 5; [x2x4; X3X5. [XgXj |Xxx 5n X4X6 h h . >5*6 ([X1X3- x 5X6^ - [X2X4 V e + [x2x5; "X1X4 X3X6 < [ v 3; X4X6' [*A]

-

+

( ( (

‘

+ [x3x6ltx4x 5l + f t * ] [Vs] + [X2X6][X3X5] + [x2x6][x3x4] + [x2x 5][x3x j

-

v 5:

+ [XlX6]tX4X5] + txlz6l[X3X5]

-

- [X2X6]' >1 X3. V 5 1 ♦ [x3x4 X1X2 . X5X6^

-

-

;xix4; X3X5 [xix 5; X2X6

♦ p ! X 5][X3X4] + [X1X6][X2X51 + [x4x6 ] [x2x4]

[X3X5^ ( X1X2 X4X6J - K - 4] x2xd - [X1X4’ x2x5; + [x1x5][x2x4 + [X3X6^ < X1X2 '

-

'

+ -

:v5; ! X1X3' [ X1X3" [X1X3J

[X4X5]< X1X2 [X3X6.

X2X6,

[x4xd < X1X2 '[X3X5~ + [x5x6]( X1X2 .i[X3X4

X2X5^

+ +

[XiXjjfXjXj]

lx2x4. + [x4x4][x2x3]

VITA

VITA

John Herman Scheuneman, a citizen of the United States of America, was born in Chicago, Illinois, on August 30, 1937.

He received his early education in Chicago, and in

September, 1955, entered Purdue University to study engineering.

Upon receiving the B. S. degree in Engineering

Sciences from Purdue in June, 1959, he went to work as an engineer for the Continental Can Company in Chicago.

In

February, 1961, he re-entered Purdue to study mathematics, and received the M. S. degree in June, 1962,

He was

married to Janice Dowd, of Longview, Washington, on December 30, 1963.

For the past year, he has been pursuing research

in absentia under the direction of Professor Louis Auslander (formerly of Purdue University) at the City University of New York.