MASLOV INDEX AND CLIFFORD ALGEBRAS

Jan Nekovář

The Maslov index of triples of Lagrangean subspaces of a symplectic space over a local field F is considered in the book of G. Lion and M. Vergne [4]. The authors use the Maslov index to construct a central extension of Sp(2n, F) by the Witt group W F of the field F. The cocycle of this extension determines the cocycle of the Weil representation of Sp(2n, F).

In the present work we introduce a generalised Maslov index associated to triples of non-zero vectors in a plane. It takes values in a certain non-commutative group Ñ, which contains K_{2}(F) as a subgroup. The generalised Maslov index can be reduced to this subgroup, giving rise to a central extension of SL(2, F) by K_{2}(F) (the field F is arbitrary). The cocycle of this extension coincides with Matsumoto’s cocycle ([1], [6]). The Witt group (more precisely, its quotient) should be regarded as a “reduction of the group Ñ (mod 2)”. In this work we give an analogous interpretation of the “reduction of Ñ (mod n)”: it is related to a certain class of Z/n-graded algebras, which generalise Clifford algebras.

This work was carried out during my stay at the Steklov Mathematical Institute. I am grateful to Yu.I. Manin for his interest in my work and to A.B. Goncharov for his help with the preparation of the manuscript.

1. Maslov Index (after [4])

The Maslov index arises naturally in the study of invariants of systems of one-dimensional subspaces of a two-dimensional space. Fix the following notation: let F be a field of characteristic char(F) ≠ 2, V = F^{2} a two-dimensional vector space over F, B a symplectic form on V, G = SL(2, F) = Aut(V, B). Denote by

T = \{( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix} )\} ,  B = \{( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )\} ,  U = \{( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} )\} ,

respectively, the standard maximal torus in G, the standard Borel subgroup and its unipotent radical. As {lines in V } = P(V) = G/B, we have

G\{pairs of lines in V } = G\(G/B × G/B) = B\G/B = W,

where W is the Weyl group of G. The non-trivial element of W corresponds to pairs of transversal lines.

Let us consider invariants of triples of lines. We first restrict our attention to the case of general position, when l_{0}, l_{1}, l_{2} are pair-wise transversal lines. After applying suitable g ∈ G, we have

l_{0} = \{[ \begin{matrix}∗ \\ 0 \end{matrix} ]\} ,  l_{1} = \{[ \begin{matrix}0 \\ ∗ \end{matrix} ]\} ,  l_{2} = \{[ \begin{matrix}x \\ ax \end{matrix} ] | x ∈ F \} ,

for some a ∈ F^{∗}. The stabiliser of the pair (l_{0}, l_{1}) is the torus T and the action of an element ( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} ) ∈ T

changes the parameter a to at^{2}. This yields an invariant

m : G\{triples of lines in general position in V } ⟶^{∼} F^{∗}/F^{∗2}
(\{[ \begin{matrix}∗ \\ 0 \end{matrix} ]\} , \{[ \begin{matrix}0 \\ ∗ \end{matrix} ]\} , \{[ \begin{matrix}x \\ ax \end{matrix} ]\}) ↦ a.

Let l_{0}, l_{1}, l_{2}, l_{3} be four lines in general position. We can again assume that l_{0} and l_{1} are the axes of

coordinates and that l_{2} = \{[ \begin{matrix}x \\ ax \end{matrix} ]\}, l_{3} = \{[ \begin{matrix}x \\ bx \end{matrix} ]\}, in which case

m_{012} = m(l_{0}, l_{1}, l_{2}) = a,  m_{013} = b,  m_{023} = \frac{ab}{a − b},  m_{123} = a − b,
m_{012} m_{013}^{−1} m_{023} m_{123}^{−1} = 1 ∈ F^{∗}/F^{∗2},

hence m is a 2-cocycle. In fact, a stronger statement holds: let ⟨a⟩ (where a ∈ F^{∗}/F^{∗2}) be the class of the one-dimensional quadratic form x ↦ ax^{2} in the Witt ring W F of the field F. We have the equality in W F

⟨a − b⟩ − ⟨\frac{ab}{a − b}⟩ = ⟨a⟩ − ⟨b⟩ ,

as

(a − b)X^{2} − \frac{ab}{a − b} Y^{2} = a (X − \frac{b}{a − b} Y )^{2} − b (X − \frac{a}{a − b} Y )^{2} .

This identity implies that m is a 2-cocycle, when considered as a function with values in W F. We extend the domain of definition of m as follows: set m(l_{0}, l_{1}, l_{2}) = 0 ∈ W F for any triple of lines l_{0}, l_{1}, l_{2} not in a general position.

Definition. The Maslov index is the above defined function m : G\{triples of lines in V } ⟶ W F.

Proposition 1. The Maslov index is a skew-symmetric 2-cocycle.

Proof. The skew-symmetry follows from the definition. The cocycle relation has been verified in the case of general position; it is trivial in all other cases.

Corollary. Fix a line l ∈ P(V). The formula (g_{1}, g_{2}) ↦ m(l, g_{1}l, g_{1}g_{2}l) then defines a 2-cocycle on G with values in W F, hence a central extension

1 ⟶ W F ⟶? ⟶ G ⟶ 1.

2. Reduction of the Maslov Index

We shall reduce the Maslov index to a certain subgroup of W F. An oriented line is a pair l̃= (l, v), where l ∈ P(V) and v ∈ (l − {0})/F^{∗2}. The space P̃(V) of oriented lines is naturally identified with G/B̃,

where B̃ = \{( \begin{matrix}a^{2} & ∗ \\ 0 & a^{−2} \end{matrix} )\} ⊂ B. As before, we have

G\{pairs of oriented lines in V } = B̃\G/B̃.

This set projects onto B\G/B = Z/2 with fibres F^{∗}/F^{∗2} and admits a natural structure of an abelian group, which is closely related to the Witt ring; its definition follows.

Let us first recall basic facts about the structure of W F (see [3]). Every quadratic space Q has a well-defined dimension dim(Q) = n ∈ N and discriminant d(Q) ∈ F^{∗}/F^{∗2}, but it is only n (mod 2) and d_{±}(Q) = (−1)^{n(n−1)/2}d(Q) that descend to the Witt ring. Denote by IF the kernel of the homomorphism dim : W F ⟶ Z/2. As an abelian group, IF is generated by the forms ⟨1, −a⟩, that is by the norm forms attached to quadratic extensions. Set

QF = {(e, a) | e ∈ Z/2, a ∈ F^{∗}/F^{∗2}}

with the abelian group law

(e, a) · (e′, a′) = (e + e′, (−1)^{ee′}aa′).

Proposition 2 (see [3]). The map s = (dim, d_{±}) induces an isomorphism of exact sequences

0 ⟶ IF/I^{2}F ⟶ W F/I^{2}F ⟶ W F/IF ⟶ 0
↓^{≀ d_{±}}  ↓^{≀ s}  ↓^{≀ dim}
0 ⟶ F^{∗}/F^{∗2} ⟶  QF  ⟶ Z/2 ⟶ 0.

We define a lifting : QF ⟶ W F of the morphism s by the formulas

{ \atop (0, a)} = ⟨1, −a⟩ , { \atop (1, a)} = ⟨a⟩ .

As a set, QF has the same structure as B̃\G/B̃, for the following reason: if we denote

T̃ = \{( \begin{matrix}a^{2} & 0 \\ 0 & a^{−2} \end{matrix} )\} ⊂ T, N = \{( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix} ) , ( \begin{matrix}0 & ∗ \\ ∗ & 0 \end{matrix} )\} ⊂ G,

then B̃\G/B̃ is naturally identified with N/T̃ and the formulas

( \begin{matrix}0 & A \\ −A^{−1} & 0 \end{matrix} ) T̃ ↦ (1, A), ( \begin{matrix}A & 0 \\ 0 & A^{−1} \end{matrix} ) T̃ ↦ (0, A)

define an isomorphism N/T̃ ⟶^{∼} QF. Putting everything together we obtain an identification

n : {pairs of oriented lines in V } = B̃\G/B̃ = N/T̃ ⟶^{∼} QF

and, composing with the lifting : QF ⟶ W F, a map

ñ : {pairs of oriented lines in V } ⟶ W F.

An explicit formula for n: if l̃_{i} = (l_{i}, v_{i}), then

n_{01} = n(l̃_{0}, l̃_{1}) = \{ \begin{matrix}(0, \frac{v_{1}}{v_{0}} ) , & l_{0} = l_{1} \\ (1, −B(v_{0}, v_{1})) , & l_{0} ≠ l_{1}. \end{matrix}

Proposition 3. For every triple of oriented lines (l̃_{0}, l̃_{1}, l̃_{2}) we have

m_{012} = ñ_{01} − ñ_{02} + ñ_{12} (mod I^{2}F).

Proof. If the three lines are not in general position, then both sides vanish. In the case of general position we can assume that v_{0} = [ \begin{matrix}x \\ 0 \end{matrix} ], v_{1} = [ \begin{matrix}0 \\ y \end{matrix} ], v_{2} = [ \begin{matrix}z \\ az \end{matrix} ]. In this case we have

m_{012} − ñ_{01} + ñ_{02} − ñ_{12} = ⟨a, xy, −axz, −yz⟩ ∈ Ker(dim, d_{±}) = I^{2}F. □ We define the reduced Maslov index of a triple (l̃_{0}, l̃_{1}, l̃_{2}) of oriented lines by the formula

m̃_{012} = m̃(l̃_{0}, l̃_{1}, l̃_{2}) = m_{012} − ñ_{01} + ñ_{02} − ñ_{12}.

Corollary. m̃ : G\{triples of oriented lines in V } ⟶ I^{2}F is a 2-cocycle cohomologous to m (more precisely, to the lift of m via the projection P̃(V) ⟶ P(V)).

3. Relation to K-theory

Let us recall the structure of I^{2}F. As an abelian group, I^{2}F is generated by the classes of forms ⟨1, −a⟩ ⊗ ⟨1, −b⟩ = ⟨1, −a, −b, ab⟩. Such forms are the reduced norms on quaternion algebras. Denote by (\frac{a,b}{F})_{2} the quaternion algebra

X^{2} = a,  Y^{2} = b,  Y X = −XY

over F, and by Quat(F) the subgroup of the Brauer group of F generated by the classes of quaternion algebras (of course, Quat(F) ⊂ Br(F)_{2} is an abelian group of exponent 2). Let BW(F) be the Brauer-Wall group of the field F, i.e. the group of similitude classes of Z/2-graded central simple algebras over F (in the graded sense, see [3]). The abelian group law in BW(F) is given by the graded tensor product of algebras:

(aˆ⊗b)(a′ ˆ⊗b′) = (−1)^{deg(b)deg(a′)}aa′ ⊗ bb′.

Proposition 4 (see [3]). The functor C : Q ↦ C(Q) which associates to each quadratic space its Clifford algebra induces a homomorphism of abelian groups C : W F ⟶ BW(F) with kernel Ker(C) = I^{3}F and a commutative diagram with exact rows

0 ⟶ I^{2}F/I^{3}F ⟶ W F/I^{3}F ⟶ W F/I^{2}F ⟶ 0
↓^{≀}  ↓^{≀ C}  ↓^{≀ s}
0 ⟶ Quat(F) ⟶ Clif(F) ⟶  QF  ⟶ 0
∩ ↓  ∩ ↓  ∥
0 ⟶ Br(F) ⟶ BW(F) ⟶  QF  ⟶ 0,

in which Clif(F) denotes the image of C and the isomorphism I^{2}F/I^{3}F ⟶^{∼} Quat(F) is given by the formula ⟨1, −a⟩ ⊗ ⟨1, −b⟩ ↦ (\frac{a,b}{F})_{2}.

According to Proposition 4, m̃ induces a 2-cocycle

G\{triples of oriented lines in V } ⟶ I^{2}F/I^{3}F ⟶^{∼} Quat(F).

Recall that the Milnor groups K_{n}^{M}(F) are generated by the symbols {a_{1}, …, a_{n}} (a_{i} ∈ F^{∗}) which are multiplicative in each argument and which satify the relation {⋯, a, 1 − a, ⋯} = 1.

Proposition 5 (see [7]). The map

{a_{1}, …, a_{n}} (mod 2) ↦ ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ (mod I^{n+1}F)

induces a surjective homomorphism of abelian groups s_{n} : K_{n}^{M}(F)/2 ⟶ I^{n}F/I^{n+1}F.

Milnor conjectured that the maps s_{n} are all isomorphisms. For n = 0, 1 this follows from Proposition 2. The Merkurjev-Suslin theorem [5] implies that the composite homomorphism

K_{2}(F)/2 = K_{2}^{M}(F)/2⟶^{s_{2}}I^{2}F/I^{3}F ↪ Br(F)_{2}

is an isomorphism, i.e. Quat(F) = Br(F)_{2}.

4. Generalised Maslov Index

We are going to modify the above construction to obtain a 2-cocycle with values in K_{2}(F). We replace B̃ by the unipotent group U and we consider the basic affine space G/U = V − {0} of non-zero vectors in V. There is a canonical identification

n : G\{pairs of non − zero vectors in V } = U\G/U = N,

where N is the normaliser of the torus T. This means that the group QF should be replaced by its extension N. We must construct, however, an analogue of the group BW(F). The latter abelian group has the following structure.

Proposition 6 (see [3]). The extension

0 ⟶ Br(F) ⟶ BW(F) ⟶ QF ⟶ 0

is given by the cocycle

c′((0, a), (0, b)) = c′((1, a), (1, b)) = (\frac{a, b}{F})_{2} , c′((0, a), (1, b)) = c′((1, b), (0, a)) = (\frac{a, −b}{F})_{2}

Definition of the group Ñ: set

(0, A) = ( \begin{matrix}A & 0 \\ 0 & A^{−1} \end{matrix} ) , (1, A) = ( \begin{matrix}0 & A \\ −A^{−1} & 0 \end{matrix} ) ∈ N.

The formulas

\begin{matrix}c((0, A), (0, B)) = −c((1, A), (1, B)) = −\{A, B\} \\ c((0, A), (1, B)) = c((1, B), (0, A)) = −\{A, −B\} \end{matrix}

define a 2-cocycle, which gives rise to a central extension

1 ⟶ K_{2}(F) ⟶ Ñ ⟶ N ⟶ 1

together with a section : N ⟶ Ñ satisfying ñ·n′ = c(n, n′) nn′. According to Proposition 6, the subgroup Clif(F) of BW(F) is equal to “the reduction of Ñ (mod 2)”, i.e. to the image of Ñ via the homomorphisms s_{2} : K_{2}(F) ⟶ Quat(F), N ⟶ N/T̃ = QF. Definition of the generalised Maslov index: for v_{0}, v_{1}, v_{2} ∈ G/U set

m_{012} = m(v_{0}, v_{1}, v_{2}) = \binom{}{n_{12}n_{02}^{−1}n_{01}}^{−1} ∈ Ñ (Maslov index)

m̃_{012} = ñ_{12}(ñ_{02})^{−1}ñ_{01} m_{012} ∈ K_{2}(F) (reduced Maslov index)

Note that, for n, n′ ∈ N, we have

m(U, nU, nn′U) = 1 ∈ Ñ,  m̃(U, nU, nn′U) = c(n, n′),

since n_{01} = n, n_{02} = nn′, n_{12} = n′ and ñ · n′ = c(n, n′) nn′ (where c is the cocycle for the extension Ñ). Formulas for the Maslov index:

(1) u, v linearly independent, A, B ∈ F^{∗}: m̃(u, v, Au + Bv) = {A, B}.

(2) u, v linearly independent, A ∈ F^{∗}, B = B(u, v): m̃(u, Au, v) = −m̃(u, v, Au) = m̃(v, u, Au) = {A, −B}.

(3) A, B ∈ F^{∗}: m̃(u, Au, Bu) = {−B, A}.

These formulas follow directly from the definitions. Recall that the function

n : G\{pairs of non − zero vectors in V } ⟶ N

is given by the formulas

n(u, Au) = ( \begin{matrix}A & 0 \\ 0 & A^{−1} \end{matrix} ) = (0, A) ∈ N, n(u, v) = ( \begin{matrix}0 & −B^{−1} \\ B & 0 \end{matrix} ) = (1, −B^{−1}) ∈ N.

We set, using the previous notation, x(u, Au) = A^{−1}, x(u, v) = B.

Proposition 7. For any triple of non-zero vectors v_{0}, v_{1}, v_{2} in V we have

m̃_{012} = \{\frac{x(v_{0}, v_{1})}{x(v_{0}, v_{2})}, \frac{x(v_{1}, v_{2})}{x(v_{0}, v_{2})}\} = \{\frac{x_{01}}{x_{02}} , \frac{x_{12}}{x_{02}} \} .

Proof. In the case of general position we use the identity

{ \atop (1, A)}{ \atop (1, B)}^{−1}{ \atop (1, C)} = ; (1, \frac{AC}{B} ) \{\frac{C}{B} , \frac{A}{B} \}

for A = −x_{12}^{−1}, B = −x_{02}^{−1}, C = −x_{01}^{−1}. The remaining cases are similar.

Theorem 1. The map m̃ : G\{triples of non − zero vectors in V } ⟶ K_{2}(F) is a 2-cocycle.

Proof. The function m̃ is almost skew-symmetric: it changes sign if we exchange two arguments in general position; in the case of exchanging linearly dependent vectors u, Au one has to add an additional term {A, −1}. It follows that the coboundary

(δm̃)_{0123} = m̃_{012} m_{013}^{−1} m̃_{023} m_{123}^{−1}

is fully skew-symmetric and that it is enough to consider only the case of linearly dependent v_{0}, v_{1} and the case of four vectors in general position.

If v_{0} = Av_{1}, then x_{01} = x_{02}x_{12}^{−1} = x_{03}x_{13}^{−1} = A and

m̃_{012} m_{013}^{−1} = {A, x_{03}x_{02}^{−1}},  m̃_{012} m_{013}^{−1} m̃_{023} = {x_{23}x_{03}^{−1}A, x_{03}x_{02}^{−1}} = {x_{23}x_{13}^{−1}, x_{13}x_{12}^{−1}} = m̃_{123}.

In the case of general position the function m̃ is GL(2)-invariant, so we can assume that v_{0} = [ \begin{matrix}1 \\ 0 \end{matrix} ], v_{1} = [ \begin{matrix}0 \\ 1 \end{matrix} ],

v_{2} = [ \begin{matrix}A \\ B \end{matrix} ], v_{3} = [ \begin{matrix}C \\ D \end{matrix} ]. Consider a new quadruple of vectors v′_{0} = v_{0}, v′_{1} = v_{1}, v′_{2} = [ \begin{matrix}1 \\ 1 \end{matrix} ], v′_{3} = [ \begin{matrix}AC^{−1} \\ BD^{−1} \end{matrix} ].

Multiplicativity of \{ , \} implies that (δm̃)(v_{0}, v_{1}, v_{2}, v_{3}) = (δm̃)(v′_{0}, v′_{1}, v′_{2}, v′_{3}). However, for the vectors [ \begin{matrix}1 \\ 0 \end{matrix} ],

[ \begin{matrix}0 \\ 1 \end{matrix} ], [ \begin{matrix}1 \\ 1 \end{matrix} ], [ \begin{matrix}X \\ Y \end{matrix} ] we have

\begin{matrix}m̃_{012} = 1,  m̃_{013} = \{X, Y \}, & m̃_{023} = \{X − Y, Y \},  m̃_{123} = \{Y − X, X\}, \\ (δm̃)_{0123} & = \{1 − \frac{Y}{X} , \frac{Y}{X} \} = 1.  □ \end{matrix}

Note that we had not used the Steinberg relation {A, 1 − A} = 1 before in its full force, only its consequence {A, −A} = 1. The relation itself can be reconstructed during the construction: one requires that m̃(v_{0}, v_{1}, v_{2}) = 1 for all triples of vectors whose endpoints are collinear.

The above construction then has to be modified as follows.

(1) Suppose that we are given a bimultiplicative function [ , ] : F^{∗} ⊗ F^{∗} ⟶ A with values in an abelian group A, which satisfies the relation [X, −X] = 0.

(2) Using the function [ , ] instead of { , }, define a central extension 1 ⟶ A ⟶ Ñ ⟶ N ⟶ 1 and a lifting : N ⟶ Ñ using the same formulas as before.

(3) For v_{0}, v_{1}, v_{2} ∈ G/U define

m̃_{012} = ñ_{12}(ñ_{02})^{−1}ñ_{01} \binom{}{n_{12}n_{02}^{−1}n_{01}}^{−1} ∈ A.

(4) Let X = {(v_{0}, v_{1}, v_{2}) ∈ (G/U)^{3} | ∃g ∈ G, ∃u_{ij} ∈ gUg^{−1} v_{i} = u_{ij}v_{j}} be the set of the triples of vectors whose endpoints are collinear. We divide A by the Steinberg relation: let \overline{A} = A/(subgroup generated by m̃_{012}(X)).

(5) The induced function \overline{m} : G\(G/U × G/U × G/U) ⟶ \overline{A} then turns out to be a 2-cocycle.

It is natural to ask whether the construction (1)–(5) can be applied to the groups SL(n, F) as an alternative to Matsumoto’s construction ([1], [6]). According to Proposition 7, the cocycle m̃ coincides with Matsumoto’s cocycle for SL(2) in Kubota’s form ([1], [2]). For n > 2, however, the function m̃ defined according to (1)–(4) will coincide with Matsumoto’s cocycle only on those triples of elements of G/U which are “not in a very general position”.

5. Z/n-graded Clifford Algebras

The construction of the generalized Maslov index m̃ with values in K_{2}(F) implies that the corresponding central extension

1 ⟶ K_{2}(F) ⟶? ⟶ SL(2, F) ⟶ 1

is mapped via the morphism K_{2}(F) ⟶ K_{2}(F)/2⟶^{s_{2}}Quat(F) to the extension with kernel Quat(F) ⟶^{∼} I^{2}F/I^{3}F, which is given by the usual (reduced) Maslov index.

We are now going to give an interpretation of the objects related to K_{2}(F)/n for n > 2. Even though the definition of “the Witt ring of forms of degree n” is not known in this case, there exists a natural analogue of the group W F/I^{3}F, namely the group of generalised Clifford algebras defined below.

Let F be a field of characteristic prime to n, which contains the group μ_{n} of the n-th roots of unity. Fix once for all a primitive root of unity ζ ∈ μ_{n}. Any ordered set of elements a_{1}, …, a_{N} ∈ F^{∗} determines an F -algebra A = ⟨a_{1}, …, a_{N}⟩ with generators X_{1}, …, X_{N} and relations

X_{i}^{n} = a_{i},  X_{j}X_{i} = ζX_{i}X_{j}  (i < j).

The algebra A has a natural Z/n-grading, for which deg(X_{i}) = 1 for all i.

We define the graded tensor product Aˆ⊗B of two Z/n-graded algebras A, B as follows: as a vector space it coincides with A ⊗ B and the multiplication is defined by the formula

(a ⊗ b) (a′ ⊗ b′) = ζ^{deg(b)deg(a′)}aa′ ⊗ bb′.

We have ⟨a_{1}, …, a_{N}⟩ = ⟨a_{1}⟩ ˆ⊗ ⋯ ˆ⊗ ⟨a_{N}⟩; in particular, dim ⟨a_{1}, …, a_{N}⟩ = n^{N}.

The algebra ⟨a, b⟩ is the standard cyclic central simple algebra over F. Its class (a, b)_{n,ζ} in the Brauer group Br(F) depends on the choice of ζ, but

(a, b)_{n} = (a, b)_{n,ζ} ⊗ ζ ∈ Br(F) ⊗ μ_{n}

does not depend on ζ. The map {a, b} ↦ (a, b)_{n} defines a homomorphism K_{2}(F) ⟶ Br(F) ⊗ μ_{n}; denote by Cyc_{n}(F) its image.

Before we start developing structure theory of the algebras ⟨a_{1}, …, a_{N}⟩, recall an elementary lemma:

Lemma. Let A be a central simple algebra over F. If A = ⊕A_{i} admits a Z/n-grading, then there exists an invertible element z ∈ A_{0} such that

A_{i} = {a ∈ A | az = ζ^{i}za}.

The element z is determined uniquely up to a scalar multiple and z^{n} = F^{∗}.

Proof. The automorphism f(a) = ζ^{−deg(a)}a of A is inner, by the Skolem-Noether theorem: f(a) = zaz^{−1}. As f^{n} = id, z^{n} is contained in the centre F of A, which also contains the ambiguity of the choice of z. □

Denote by d(A) the image of z^{n} in F^{∗}/F^{∗n}; it is uniquely determined by the Z/n-grading. In other words, the centraliser Z_{A}(A_{0}) of the subalgebra A_{0} in A is isomorphic to ⟨d(A)⟩.

For any F -algebra A let (A) be the Z/n-graded algebra with (A)_{0} = A and (A)_{i} = 0 (i ≠ 0).

Proposition 8. Let A = ⟨a_{1}, …, a_{N}⟩.

(1) If N ≡ 1 (mod 2), then A ⟶^{∼} (A_{0})ˆ⊗ ⟨d⟩ = (A_{0}) ⊗ ⟨d⟩, A_{0} is a central simple algebra over F, [A_{0}] ⊗ ζ ∈ Cyc_{n}(F) and Z_{A}(A_{0}) = ⟨d⟩.

(2) If N ≡ 0 (mod 2), then A is a central simple algebra over F, [A] ⊗ ζ ∈ Cyc_{n}(F) and Z_{A}(A_{0}) = ⟨d(A)⟩.

Remarks. The usual tensor product A ⊗ B denotes the algebra with the usual multiplication

(a ⊗ b) (a′ ⊗ b′) = aa′ ⊗ bb′.

The symbol [A] denotes the class of a central simple algebra A in the Brauer group Br(F). Proof. We use induction on N. The case N = 1 is trivial. We perform the induction step N ↦ N + 1. Case (1): N ≡ 1 (mod 2). In this case

Aˆ⊗ ⟨b⟩ = (A_{0}) ⊗ (⟨d⟩ ˆ⊗ ⟨b⟩) = (A_{0}) ⊗ ⟨d, b⟩

is indeed a central simple algebra over F whose class lies in Cyc_{n}(F). Case (2): N ≡ 0 (mod 2). Let x ∈ ⟨b⟩_{1} be a element satisfying x^{n} = b and let z ∈ A_{0} be an element defining the grading, as in Lemma. Set B = Aˆ⊗ ⟨b⟩. Then y = z ⊗ x generates the centre Z(B) ⟶^{∼} ⟨d⟩ of the algebra B, as y^{n} = z^{n}x^{n} = d(A)b. It follows that B = (B_{0}) ⊗ ⟨d⟩ and Z_{B}(B_{0}) = ⟨d⟩. By the induction hypothesis, we have A = C ⊗ ⟨u, v⟩, where C ⊂ A_{0} and [C] ⊗ ζ ∈ Cyc_{n}(F), so it is enough to consider the case A = ⟨u, v⟩. The algebra A is generated by two elements X, Y satisfying X^{n} = u, Y^{n} = v, Y X = ζXY; then \overline{X} = −X ⊗ x^{−1} and \overline{Y} = −Y ⊗ x^{−1} generate B_{0}, \overline{Y} \overline{X} = ζ\overline{X}\overline{Y}, \overline{X}^{n} = −ub^{−1} and \overline{Y}^{n} = −vb^{−1}, hence B_{0} = ⟨−ub^{−1}, −vb^{−1}⟩ and [B_{0}] ⊗ ζ ∈ Cyc_{n}(F), as claimed. □

As a consequence, we obtain that each algebra A = ⟨a_{1}, …, a_{N}⟩ determines a triple of invariants

C(A) = (e(A), d(A), D) ∈ Z/2 × F^{∗}/F^{∗n} × Cyc_{n}(F)
e(A) = N (mod 2),  ⟨d(A)⟩ = Z_{A}(A_{0}),  D = \{ \begin{matrix}[A] ⊗ ζ, & N ≡ 0 (mod 2) \\ [A_{0}] ⊗ ζ, & N ≡ 1 (mod 2) \end{matrix}

The set S of all algebras ⟨a_{1}, …, a_{N}⟩ forms a semi-group with respect to the graded tensor product ˆ⊗. The set of triples (e, d, D) inherits the group structure from Ñ via the homomorphisms ( , )_{n} : K_{2}(F) ⟶ Cyc_{n}(F) and F^{∗} ⟶ F^{∗}/F^{∗n}:

\begin{aligned}(0, d, D) · (0, d′, D′) & = (0, dd′, DD′(d, d′)_{n}^{−1}) \\ (0, d, D) · (1, d′, D′) & = (1, dd′, DD′(d, −d′)_{n}^{−1}) \\ (1, d, D) · (0, d′, D′) & = (1, dd^{′−1}, DD′(−d, d′)_{n}) \\ (1, d, D) · (1, d′, D′) & = (0, −dd^{′−1}, DD′(d, d′)_{n}) . \end{aligned}

The signs in the definition of the cocycle c have been chosen in order to make the following statement hold.

Proposition 9. For A, B ∈ S we have:

(1) C(⟨a, b⟩) = (0, −ab^{−1}, (a, b)_{n}).

(2) C(Aˆ⊗B) = C(A) · C(B).

(3) C(A) = 1 ⟺ there exists a Z/n-graded vector space V and an isomorphism A ⟶^{∼} End(V) (the grading on End(V) is defined by the formula Hom(V_{i}, V_{j}) ⊂ (End(V))_{j−i}).

(4) For A = ⟨a_{1}, …, a_{N}⟩ put A^{∘} = ⟨−a_{N}, …, −a_{1}⟩. Then C(A) · C(A^{∘}) = C(A^{∘}) · C(A) = 1.

Proof. (1) The grading is defined by the element z = −XY^{−1}, where X^{n} = a, Y^{n} = b, Y X = ζXY, hence z^{n} = −X^{n}Y^{−n} = −ab^{−1}.

(2) Thanks to the associativity of both products, it is enough to consider the case A = ⟨a_{1}, …, a_{N}⟩, B = ⟨b⟩. Case (1): N ≡ 1 (mod 2). In this case C(A) = (1, d(A), [A_{0}] ⊗ ζ) and, by (1) and the proof of Proposition 8,

C(Aˆ⊗ ⟨b⟩) = (0, d(A′), ([A_{0}] ⊗ ζ)(d(A), b)_{n}),

where A′ = ⟨d(A), b⟩. Case (2): N ≡ 0 (mod 2). Using again the proof of Proposition 8, it is enough to consider the case N = 2, A = ⟨u, v⟩, when we have d(Aˆ⊗B) = d(A)d(B) and

[(Aˆ⊗B)_{0}] ⊗ ζ = (−ub^{−1}, −vb^{−1})_{n} = (u, v)_{n}(−uv^{−1}, −b)_{n}^{−1} = ([A] ⊗ ζ) (d(A), −d(B))_{n}^{−1}.

(3) If A ⟶^{∼} End(V), then Z(A) = F ⟹ N ≡ 0 (mod 2) and A is a central simple algebra over F with a trivial class in Br(F). The grading is defined by the element z = ∑ ζ^{i}p_{i}, where p_{i} : V ⟶ V_{i} is the natural projector, hence z^{n} = 1. Conversely, if C(A) = 1, then N ≡ 0 (mod 2), A ⟶^{∼} End(V) in the non-graded sense and the grading is defined by an element z satisfying z^{n} = 1. As μ_{n} ⊂ F, z must be of the form ∑ ζ^{i}p_{i}, where p_{i} are projectors.

(4) It is enough to consider the case N = 1; the statement then follows from (1) and (2). □

We say that two algebras A, B ∈ S are similar if there exist Z/n-graded vector spaces V, W and an isomorphism Aˆ⊗End(V) ⟶^{∼} B ˆ⊗End(W). According to Proposition 9, the similitude classes of the algebras ⟨a_{1}, …, a_{N}⟩ form a group with respect to ˆ⊗, which will be denoted by C_{n}(F) (“the Clifford algebras of degree n”). It is a natural generalisation of the group of Clifford algebras Clif(F) ⊂ BW(F).

Putting everything together, we obtain the following statement.

Theorem 2. There exists an exact sequence

1 ⟶ Cyc_{n}(F) ⟶ C_{n}(F) ⟶ Q_{n}(F) ⟶ 1,

where Q_{n}(F) is the group of pairs (e, a), e ∈ Z/2, a ∈ F^{∗}/F^{∗n} with multiplication

(e, a) · (e′, a′) = (e + e′, (−1)^{ee′}a(a′)^{(−1)^{a}}) .

As a final remark, we note that Cyc_{n}(F) = Br(F)_{n} ⊗ μ_{n}, according to [5].

References

[1] D.A. Kazhdan, S.J. Patterson, Metaplectic forms, Publ. Math. de l’I.H.E.S. 59 (1984), 35–142. Correction: ibid, 62 (1985), 419.

[2] T. Kubota, On automorphic functions and the reciprocity law in a number field, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 2, Kinokuniya, Tokyo 1969. Russian translation: Sbornik Matematika 14:6 (1970), 3–37.

[3] T.Y. Lam, The algebraic theory of quadratic forms, Benjamin, Reading Mass. 1973.

[4] G. Lion, M. Vergne, The Weil representation, Maslov index and theta series, Progress in Mathematics 6, Birkhäuser, Boston 1980.

[5] A.S. Merkurjev, A.A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046.

[6] J. Milnor, Introduction to algebraic K-theory, Annals of Mathematics Studies 72, Princeton University Press, Princeton 1971.

[7] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1969/1970), 318–344.

Mathematical Institute of the Czechoslovak Academy of Sciences, Prague