<< стр. 2(всего 2)СОДЕРЖАНИЕ

A table of the Cliп¬Ђord groups Clp,q for 0 в‰¤ p, q в‰¤ 7 can be found in Kirillov , and for
0 в‰¤ p, q в‰¤ 8, in Lawson and Michelsohn  (but beware that their Clp,q is our Clq,p ). It can
also be shown that
Clp+1,q в‰€ Clq+1,p
and that
Clp,q в‰€ Cl0 ,
p,q+1

frow which it follows that
Spin(p, q) в‰€ Spin(q, p).
34 CHAPTER 1. CLIFFORD ALGEBRAS, CLIFFORD GROUPS, PIN AND SPIN

1.7 The Complex Cliп¬Ђord Algebras Cl(n, C)
One can also consider Cliп¬Ђord algebras over the complex п¬Ѓeld C. In this case, it is well-known
that every nondegenerate quadratic form can be expressed by

О¦C (x1 , . . . , xn ) = x2 + В· В· В· + x2
n 1 n

in some orthonormal basis. Also, it is easily shown that the complexiп¬Ѓcation C вЉ—R Clp,q of
the real Cliп¬Ђord algebra Clp,q is isomorphic to Cl(О¦C ). Thus, all these complex algebras are
n
isomorphic for p+q = n, and we denote them by Cl(n, C). Theorem 1.15 yields the following
periodicity theorem:

Theorem 1.18 The following isomorphisms hold:

Cl(n + 2, C) в‰€ Cl(n, C) вЉ—C Cl(2, C),

with Cl(2, C) = C(2).

Proof . Since Cl(n, C) = C вЉ—R Cl0,n = C вЉ—R Cln,0 , by Lemma 1.15, we have

Cl(n + 2, C) = C вЉ—R Cl0,n+2 в‰€ C вЉ—R (Cln,0 вЉ—R Cl0,2 ) в‰€ (C вЉ—R Cln,0 ) вЉ—C (C вЉ—R Cl0,2 ).

However, Cl0,2 = H, Cl(n, C) = C вЉ—R Cln,0 , and C вЉ—R H в‰€ C(2), so we get Cl(2, C) = C(2)
and
Cl(n + 2, C) в‰€ Cl(n, C) вЉ—C C(2),
and the theorem is proved.
As a corollary of Theorem 1.18, we obtain the fact that

Cl(2k, C) в‰€ C(2k ) and Cl(2k + 1, C) в‰€ C(2k ) вЉ• C(2k ).

The table of the previous section can also be completed as follows:
n 0 1 2 3 4 5 6 7 8
H вЉ• H H(2) R(8) R(8) вЉ• R(8) R(16)
Cl0,n R C H C(4)
R R вЉ• R R(2) C(2) H(2) H(2) вЉ• H(2) H(4)
Cln,0 C(8) R(16)
Cl(n, C) C 2C C(2) 2C(2) C(4) 2C(4) 2C(8)
C(8) C(16).
where 2C(k) is an abbrevation for C(k) вЉ• C(k).

1.8 The Groups Pin(p, q) and Spin(p, q) as double covers
of O(p, q) and SO(p, q)
It turns out that the groups Pin(p, q) and Spin(p, q) have nice topological properties w.r.t.
the groups O(p, q) and SO(p, q). To explain this, we review the deп¬Ѓnition of covering maps
1.8. THE GROUPS PIN(P, Q) AND SPIN(P, Q) AS DOUBLE COVERS 35

and covering spaces (for details, see Fulton , Chapter 11). Another interesting source is
Chevalley , where is is proved that Spin(n) is a universal double cover of SO(n) for all
n в‰Ґ 3.
Since Cp,q is an algebra of dimension 2p+q , it is a topological space as a vector space
p+q в€—
isomorphic to V = R2 . Now, the group Cp,q of units of Cp,q is open in Cp,q , because
x в€€ Cp,q is a unit if the linear map y в†’ xy is an isomorphism, and GL(V ) is open in
в€—
End(V ), the space of endomorphisms of V . Thus, CP,q is a Lie group, and since Pin(p, q)
в€—
and Spin(p, q) are clearly closed subgroups of Cp,q , they are also Lie groups.

Deп¬Ѓnition 1.7 Given two topological spaces X and Y , a covering map is a continuous
surjective map, p: Y в†’ X, with the property that for every x в€€ X, there is some open
subset, U вЉ† X, with x в€€ U , so that pв€’1 (U ) is the disjoint union of open subsets, VО± вЉ† Y ,
and the restriction of p to each VО± is a homeomorphism onto U . We say that U is evenly
covered by p. We also say that Y is a covering space of X. A covering map p: Y в†’ X is
called trivial if X itself is evenly covered by p (i.e., Y is the disjoint union of open subsets
subsets YО± each homeomorphic to X), and nontrivial , otherwise. When each п¬Ѓber, pв€’1 (x),
has the same п¬Ѓnite cardinaly n for all x в€€ X, we say that p is an n-covering (or n-sheeted
covering).

Note that a covering map, p: Y в†’ X, is not always trivial, but always locally trivial (i.e.,
for every x в€€ X, it is trivial in some open neighborhood of x). A covering is trivial iп¬Ђ Y
is isomorphic to a product space of X Г— T , where T is any set with the discrete topology.
Also, if Y is connected, then the covering map is nontrivial.

Deп¬Ѓnition 1.8 An isomorphism П• between covering maps p: Y в†’ X and p : Y в†’ X is a
homeomorphism, П•: Y в†’ Y , so that p = p в—¦ П•.

Typically, the space X is connected, in which case it can be shown that all the п¬Ѓbers
pв€’1 (x) has the same cardinality.
One of the most important properties of covering spaces is the pathвЂ“lifting property, a
property that we will use to show that Spin(n) is path-connected.

Proposition 1.19 (Path lifting) Let p: Y в†’ X be a covering map, and let Оі: [a, b] в†’ X
be any continuous curve from xa = Оі(a) to xb = Оі(b) in X. If y в€€ Y is any point so that
p(y) = xa , then there is a unique curve, Оі: [a, b] в†’ Y , so that y = Оі(a) and
p в—¦ Оі(t) = Оі(t) for all t в€€ [a, b].

Proof . See Fulton , Chapter 11, Lemma 11.6.
Many important covering maps arise from the action of a group G on a space Y . If Y
is a topological space, an action (on the left) of a group G on Y is a map О±: G Г— Y в†’ Y
satisfying the following conditions, where, for simplicity of notation, we denote О±(g, y) by
g В· y:
36 CHAPTER 1. CLIFFORD ALGEBRAS, CLIFFORD GROUPS, PIN AND SPIN

(1) g В· (h В· y) = (gh) В· y, for all g, h в€€ G and y в€€ Y ;

(2) 1 В· y = y, for all в€€ Y , where 1 is the identity of the group G;

(3) The map y в†’ g В· y is a homeomorphism of Y for every g в€€ G.

We deп¬Ѓne an equivalence relation on Y as follows: x в‰Ў y iп¬Ђ y = g В· x for some g в€€ G
(check that this is an equivalence relation). The equivalence class G В· x = {g В· x | g в€€ G} of
any x в€€ Y is called the orbit of x. We obtain the quotient space Y /G and the projection
map p: Y в†’ Y /G sending every y в€€ Y to its orbit. The space Y /G is given the quotient
topology (a subset U of Y /G is open iп¬Ђ pв€’1 (U ) is open in Y ).
Given a subset V of Y and any g в€€ G, we let

g В· V = {g В· y | y в€€ V }.

We say that G acts evenly on Y if for every y в€€ Y there is an open subset V containing y
so that g В· V and h В· V are disjoint for any two distinct elements g, h в€€ G.
The importance of the notion a group acting evenly is that such actions induce a covering
map.

Proposition 1.20 If G is a group acting evenly on a space Y , then the projection map,
p: Y в†’ Y /G, is a covering map.

Proof . See Fulton , Chapter 11, Lemma 11.17.
The following proposition shows that Pin(p, q) and Spin(p, q) are nontrivial covering
spaces unless p = q = 1.

Proposition 1.21 For all p, q в‰Ґ 0, the groups Pin(p, q) and Spin(p, q) are double covers of
O(p, q) and SO(p, q), respectively. Furthermore, they are nontrivial covers unless p = q = 1.

Proof . We know that kernel of the homomorphism ПЃ: Pin(p, q) в†’ O(p, q) is Z2 = {в€’1, 1}.
If we let Z2 act on Pin(p, q) in the natural way, then O(p, q) в‰€ Pin(p, q)/Z2 , and the reader
can easily check that Z2 acts evenly. By Proposition 1.20, we get a double cover. The
argument for ПЃ: Spin(p, q) в†’ SO(p, q) is similar.
Let us now assume that p = 1 and q = 1. In order to prove that we have nontrivial
covers, it is enough to show that в€’1 and 1 are connected by a path in Pin(p, q) (If we had
Pin(p, q) = U1 в€Є U2 with U1 and U2 open, disjoint, and homeomorphic to O(p, q), then в€’1
and 1 would not be in the same Ui , and so, they would be in disjoint connected components.
Thus, в€’1 and 1 canвЂ™t be pathвЂ“connected, and similarly with Spin(p, q) and SO(p, q).) Since
(p, q) = (1, 1), we can п¬Ѓnd two orthogonal vectors e1 and e2 so that О¦p,q (e1 ) = О¦p,q (e2 ) = В±1.
Then,
Оі(t) = В± cos(2t) 1 + sin(2t) e1 e2 = (cos t e1 + sin t e2 )(sin t e2 в€’ cos t e1 ),
1.8. THE GROUPS PIN(P, Q) AND SPIN(P, Q) AS DOUBLE COVERS 37

for 0 в‰¤ t в‰¤ ПЂ, deп¬Ѓnes a path in Spin(p, q), since

(В± cos(2t) 1 + sin(2t) e1 e2 )в€’1 = В± cos(2t) 1 в€’ sin(2t) e1 e2 ,

as desired.
In particular, if n в‰Ґ 2, since the group SO(n) is path-connected, the group Spin(n) is
also path-connected. Indeed, given any two points xa and xb in Spin(n), there is a path
Оі from ПЃ(xa ) to ПЃ(xb ) in SO(n) (where ПЃ: Spin(n) в†’ SO(n) is the covering map). By
Proposition 1.19, there is a path Оі in Spin(n) with origin xa and some origin xb so that
ПЃ(xb ) = ПЃ(xb ). However, ПЃв€’1 (ПЃ(xb )) = {в€’xb , xb }, and so, xb = В±xb . The argument used in
the proof of Proposition 1.21 shows that xb and в€’xb are path-connected, and so, there is a
path from xa to xb , and Spin(n) is path-connected. In fact, for n в‰Ґ 3, it turns out that
Spin(n) is simply connected. Such a covering space is called a universal cover (for instance,
see Chevalley ).
This last fact requires more algebraic topology than we are willing to explain in detail,
and we only sketch the proof. The notions of п¬Ѓbre bundle, п¬Ѓbration, and homotopy sequence
associated with a п¬Ѓbration are needed in the proof. We refer the perseverant readers to Bott
and Tu  (Chapter 1 and Chapter 3, Sections 16вЂ“17) or Rotman  (Chapter 11) for a
detailed explanation of these concepts.
Recall that a topological space is simply connected if it is path connected and ПЂ1 (X) = (0),
which means that every closed path in X is homotopic to a point. Since we just proved that
Spin(n) is path connected for n в‰Ґ 2, we just need to prove that ПЂ1 (Spin(n)) = (0) for all
n в‰Ґ 3. The following facts are needed to prove the above assertion:
(1) The sphere S nв€’1 is simply connected for all n в‰Ґ 3.
SU(2) is homeomorphic to S 3 , and thus, Spin(3) is simply
(2) The group Spin(3)
connected.
(3) The group Spin(n) acts on S nв€’1 in such a way that we have a п¬Ѓbre bundle with п¬Ѓbre
Spin(n в€’ 1):
Spin(n в€’ 1) в€’в†’ Spin(n) в€’в†’ S nв€’1 .

Fact (1) is a standard proposition of algebraic topology and a proof can found in many
books. A particularly elegant and yet simple argument consists in showing that any closed
curve on S nв€’1 is homotopic to a curve that omits some point. First, it is easy to see that
in Rn , any closed curve is homotopic to a piecewise linear curve (a polygonal curve), and
the radial projection of such a curve on S nв€’1 provides the desired curve. Then, we use the
stereographic projection of S nв€’1 from any point omitted by that curve to get another closed
curve in Rnв€’1 . Since Rnв€’1 is simply connected, that curve is homotopic to a point, and so is
its preimage curve on S nв€’1 . Another simple proof uses a special version of the SeifertвЂ”van
KampenвЂ™s theorem (see Gramain ).
Fact (2) is easy to establish directly, using (1).
38 CHAPTER 1. CLIFFORD ALGEBRAS, CLIFFORD GROUPS, PIN AND SPIN

To prove (3), we let Spin(n) act on S nв€’1 via the standard action: x В· v = xvxв€’1 . Because
SO(n) acts transitively on S nв€’1 and there is a surjection Spin(n) в€’в†’ SO(n), the group
Spin(n) also acts transitively on S nв€’1 . Now, we have to show that the stabilizer of any
element of S nв€’1 is Spin(n в€’ 1). For example, we can do this for e1 . This amounts to some
simple calculations taking into account the identities among basis elements. Details of this
proof can be found in MneimnВґ and Testard , Chapter 4. It is still necessary to prove that
e
Spin(n) is a п¬Ѓbre bundle over S nв€’1 with п¬Ѓbre Spin(n в€’ 1). For this, we use the following
results whose proof can be found in MneimnВґ and Testard , Chapter 4:
e

Lemma 1.22 Given any topological group G, if H is a closed subgroup of G and the pro-
jection ПЂ: G в†’ G/H has a local section at every point of G/H, then

H в€’в†’ G в€’в†’ G/H

is a п¬Ѓbre bundle with п¬Ѓbre H.

Lemma 1.22 implies the following key proposition:

Proposition 1.23 Given any linear Lie group G, if H is a closed subgroup of G, then

H в€’в†’ G в€’в†’ G/H

is a п¬Ѓbre bundle with п¬Ѓbre H.

Now, a п¬Ѓbre bundle is a п¬Ѓbration (as deп¬Ѓned in Bott and Tu , Chapter 3, Section 16,
or in Rotman , Chapter 11). For a proof of this fact, see Rotman , Chapter 11, or
MneimnВґ and Testard , Chapter 4. So, there is a homotopy sequence associated with
e
the п¬Ѓbration (Bott and Tu , Chapter 3, Section 17, or Rotman , Chapter 11, Theorem
11.48), and in particular, we have the exact sequence

ПЂ1 (Spin(n в€’ 1)) в€’в†’ ПЂ1 (Spin(n)) в€’в†’ ПЂ1 (S nв€’1 ).

Since ПЂ1 (S nв€’1 ) = (0) for n в‰Ґ 3, we get a surjection

ПЂ1 (Spin(n в€’ 1)) в€’в†’ ПЂ1 (Spin(n)),

and so, by induction and (2), we get

ПЂ1 (Spin(n)) в‰€ ПЂ1 (Spin(3)) = (0),

proving that Spin(n) is simply connected for n в‰Ґ 3.
We can also show that ПЂ1 (SO(n)) = Z/2Z for all n в‰Ґ 3. For this, we use Theorem 1.11
and Proposition 1.21, which imply that Spin(n) is a п¬Ѓbre bundle over SO(n) with п¬Ѓbre
{в€’1, 1}, for n в‰Ґ 2:
{в€’1, 1} в€’в†’ Spin(n) в€’в†’ SO(n).
1.9. MORE ON THE TOPOLOGY OF O(P, Q) AND SO(P, Q) 39

Again, the homotopy sequence of the п¬Ѓbration exists, and in particular, we get the exact
sequence
ПЂ1 (Spin(n)) в€’в†’ ПЂ1 (SO(n)) в€’в†’ ПЂ0 ({в€’1, +1}) в€’в†’ ПЂ0 (SO(n)).
Since ПЂ0 ({в€’1, +1}) = Z/2Z, ПЂ0 (SO(n)) = (0), and ПЂ1 (Spin(n)) = (0), when n в‰Ґ 3, we get
the exact sequence
(0) в€’в†’ ПЂ1 (SO(n)) в€’в†’ Z/2Z в€’в†’ (0),
and so, ПЂ1 (SO(n)) = Z/2Z. Therefore, SO(n) is not simply connected for n в‰Ґ 3.

Remark: Of course, we have been rather cavalier in our presentation. Given a topological
space, X, the group ПЂ1 (X) is the fundamental group of X, i.e., the group of homotopy
classes of closed paths in X (under composition of loops). But ПЂ0 (X) is generally not a
group! Instead, ПЂ0 (X) is the set of path-connected components of X. However, when X is
a Lie group, ПЂ0 (X) is indeed a group. Also, we have to make sense of what it means for the
sequence to be exact. All this can be made rigorous (see Bott and Tu , Chapter 3, Section
17, or Rotman , Chapter 11).

1.9 More on the Topology of O(p, q) and SO(p, q)
It turns out that the topology of the group, O(p, q), is completely determined by the topology
of O(p) and O(q). This result can be obtained as a simple consequence of some standard
Lie group theory. The key notion is that of a pseudo-algebraic group.
Consider the group, GL(n, C), of invertible n Г— n matrices with complex coeп¬ѓcients. If
A = (akl ) is such a matrix, denote by xkl the real part (resp. ykl , the imaginary part) of akl
(so, akl = xkl + iykl ).

Deп¬Ѓnition 1.9 A subgroup, G, of GL(n, C) is pseudo-algebraic iп¬Ђ there is a п¬Ѓnite set of
polynomials in 2n2 variables with real coeп¬ѓcients, {Pi (X1 , . . . , Xn2 , Y1 , . . . , Yn2 )}t , so that
i=1

A = (xkl + iykl ) в€€ G iп¬Ђ Pi (x11 , . . . , xnn , y11 , . . . , ynn ) = 0, for i = 1, . . . , t.

Recall that if A is a complex n Г— n-matrix, its adjoint, Aв€— , is deп¬Ѓned by Aв€— = (A) .
Also, U(n) denotes the group of unitary matrices, i.e., those matrices A в€€ GL(n, C) so
that AAв€— = Aв€— A = I, and H(n) denotes the vector space of Hermitian matrices, i.e., those
matrices A в€€ GL(n, C) so that Aв€— = A. Then, we have the following theorem which is
essentially a reп¬Ѓned version of the polar decomposition of matrices:

Theorem 1.24 Let G be a pseudo-algebraic subgroup of GL(n, C) stable under adjunction
(i.e., we have Aв€— в€€ G whenever A в€€ G). Then, there is some integer, d в€€ N, so that G is
homeomorphic to (G в€© U(n)) Г— Rd . Moreover, if g is the Lie algebra of G, the map
(U, H) в†’ U eH ,
(U(n) в€© G) Г— (H(n) в€© g) в€’в†’ G, given by
is a homeomorphism onto G.
40 CHAPTER 1. CLIFFORD ALGEBRAS, CLIFFORD GROUPS, PIN AND SPIN

Proof . A proof can be found in Knapp , Chapter 1, or MneimnВґ and Testard , Chapter
e
3.
We now apply Theorem 1.24 to determine the structure of the space O(p, q). Let Jp,q be
the matrix
Ip 0
Jp,q = .
0 в€’Iq
We know that O(p, q) consists of the matrices, A, in GL(p + q, R) such that

A Jp,q A = Jp,q ,

and so, O(p, q) is clearly pseudo-algebraic. Using the above equation, it is easy to determine
the Lie algebra, o(p, q), of O(p, q). We п¬Ѓnd that o(p, q) is given by

X1 X2
X1 = в€’X1 , X3 = в€’X3 , X2 arbitrary
o(p, q) =
X2 X3

where X1 is a p Г— p matrix, X3 is a q Г— q matrix and X2 is a p Г— q matrix. Consequently, it
immediately follows that

0 X2
o(p, q) в€© H(p + q) = X2 arbitrary ,
X2 0

a vector space of dimension pq.
Some simple calculations also show that

X1 0 в€ј O(p) Г— O(q).
O(p, q) в€© U(p + q) = X1 в€€ O(p), X2 в€€ O(q) =
0 X2

Therefore, we obtain the structure of O(p, q):

Proposition 1.25 The topological space O(p, q) is homeomorphic to O(p) Г— O(q) Г— Rpq .

Since O(p) has two connected components when p в‰Ґ 1, we see that O(p, q) has four
connected components when p, q в‰Ґ 1. It is also obvious that

X1 0
SO(p, q) в€© U(p + q) = X1 в€€ O(p), X2 в€€ O(q), det(X1 ) det(X2 ) = 1 .
0 X2

This is a subgroup of O(p) Г— O(q) that we denote S(O(p) Г— O(q)). Furthermore, it is easy
to show that so(p, q) = o(p, q). Thus, we also have

Proposition 1.26 The topological space SO(p, q) is homeomorphic to S(O(p)Г—O(q))Г—Rpq .
1.9. MORE ON THE TOPOLOGY OF O(P, Q) AND SO(P, Q) 41

Note that SO(p, q) has two connected components when p, q в‰Ґ 1. The connected
component of Ip+q is a group denoted SO0 (p, q). This latter space is homeomorphic to
SO(q) Г— SO(q) Г— Rpq .
As a closing remark observe that the dimension of all these space depends only on p + q:
It is (p + q)(p + q в€’ 1)/2.

Acknowledgments. I thank Eric King whose incisive questions and relentless quest for the
вЂњessenceвЂќ of rotations eventually caused a level of discomfort high enough to force me to
improve the clarity of these notes. Rotations are elusive!
42 CHAPTER 1. CLIFFORD ALGEBRAS, CLIFFORD GROUPS, PIN AND SPIN
Bibliography

 Michael Artin. Algebra. Prentice Hall, п¬Ѓrst edition, 1991.

 M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra. Addison
Wesley, third edition, 1969.

 Michael F Atiyah, Raoul Bott, and Arnold Shapiro. Cliп¬Ђord modules. Topology, 3,
Suppl. 1:3вЂ“38, 1964.

 Andrew Baker. Matrix Groups. An Introduction to Lie Group Theory. SUMS. Springer,
2002.

 Raoul Bott and Tu Loring W. Diп¬Ђerential Forms in Algebraic Topology. GTM No. 82.
Springer Verlag, п¬Ѓrst edition, 1986.

 Nicolas Bourbaki. Alg`bre, Chapitre 9. ElВґments de MathВґmatiques. Hermann, 1968.
e e e

 T. BrВЁcker and T. tom Dieck. Representation of Compact Lie Groups. GTM, Vol. 98.
o
Springer Verlag, п¬Ѓrst edition, 1985.
Вґ
 Elie Cartan. Theory of Spinors. Dover, п¬Ѓrst edition, 1966.

 Claude Chevalley. Theory of Lie Groups I. Princeton Mathematical Series, No. 8.
Princeton University Press, п¬Ѓrst edition, 1946. Eighth printing.

 Claude Chevalley. The Algebraic Theory of Spinors and Cliп¬Ђord Algebras. Collected
Works, Vol. 2. Springer, п¬Ѓrst edition, 1997.

 Yvonne Choquet-Bruhat and CВґcile DeWitt-Morette. Analysis, Manifolds, and Physics,
e
Part II: 92 Applications. North-Holland, п¬Ѓrst edition, 1989.

 Morton L. Curtis. Matrix Groups. Universitext. Springer Verlag, second edition, 1984.

 Jean DieudonnВґ. Sur les Groupes Classiques. Hermann, third edition, 1967.
e

 William Fulton. Algebraic Topology, A п¬Ѓrst course. GTM No. 153. Springer Verlag, п¬Ѓrst
edition, 1995.

43
44 BIBLIOGRAPHY

 William Fulton and Joe Harris. Representation Theory, A п¬Ѓrst course. GTM No. 129.
Springer Verlag, п¬Ѓrst edition, 1991.

 Jean H. Gallier. Geometric Methods and Applications, For Computer Science and En-
gineering. TAM, Vol. 38. Springer, п¬Ѓrst edition, 2000.

 AndrВґ Gramain. Topologie des Surfaces. Collection Sup. Puf, п¬Ѓrst edition, 1971.
e

 A.A. Kirillov. Spinor representations of orthogonal groups. Technical report, University
of Pennsylvania, Math. Department, Philadelphia, PA 19104, 2001. Course Notes for
Math 654.

 Anthony W. Knapp. Lie Groups Beyond an Introduction. Progress in Mathematics,
Vol. 140. BirkhВЁuser, п¬Ѓrst edition, 1996.
a

 Blaine H. Lawson and Marie-Louise Michelsohn. Spin Geometry. Princeton Math.
Series, No. 38. Princeton University Press, 1989.

 Pertti Lounesto. Cliп¬Ђord Algebras and Spinors. LMS No. 286. Cambridge University
Press, second edition, 2001.

 R. MneimnВґ and F. Testard. Introduction ` la ThВґorie des Groupes de Lie Classiques.
e a e
Hermann, п¬Ѓrst edition, 1997.

 Ian R. Porteous. Topological Geometry. Cambridge University Press, second edition,
1981.

 Joseph J. Rotman. Introduction to Algebraic Topology. GTM No. 119. Springer Verlag,
п¬Ѓrst edition, 1988.

 << стр. 2(всего 2)СОДЕРЖАНИЕ