Classification of Clifford algebras

In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl_1,1(R) and Cl_2,0(R), which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers.

The significance of this result is that the additional structure on a Clifford algebra relative to the "underlying" associative algebra — namely, the structure given by the grade involution automorphism and reversal anti-automorphism (and their composition, the Clifford conjugation) — is in general an essential part of its definition, not a procedural artifact of its construction as the quotient of a tensor algebra by an ideal. The category of Clifford algebras is not just a selection from the category of matrix rings, picking out those in which the ring product can be constructed as the Clifford product for some vector space and quadratic form. With few exceptions, "forgetting" the additional structure (in the category theory sense of a forgetful functor) is not reversible.

Continuing the example above: Cl_1,1(R) and Cl_2,0(R) share the same associative algebra structure, isomorphic to (and commonly denoted as) the matrix algebra M₂(R). But they are distinguished by different choices of grade involution — of which two-dimensional subring, closed under the ring product, to designate as the even subring — and therefore of which of the various anti-automorphisms of M₂(R) can accurately represent the reversal anti-automorphism of the Clifford algebra. These distinguished (anti-)automorphisms are structures on the tensor algebra which are preserved by the "quotient by ideal" construction of the Clifford algebra. The matrix algebra representation admits them, but only the Clifford algebra distinguishes them from other elements of the matrix algebra's (anti-)automorphism group.

Of the four real degrees of freedom in M₂(R)), only one, generated by ${\displaystyle -i\sigma _{2}=\left[{\begin{array}{rr}0&-1\\1&0\end{array}}\right]}$, is reversed by the obvious anti-automorphism of M₂(R), the matrix transpose. (The designation ${\displaystyle -i\sigma _{2}}$ refers to the Pauli matrices, which are here just a conventional way of naming the degrees of freedom in a two by two matrix.) In the case of Cl_2,0(R), where both degrees of freedom in the odd part have positive norm, the obvious assignment of the even subalgebra to the matrices spanned by 1 and ${\displaystyle -i\sigma _{2}}$ (and the odd part to the other two non-identity symmetric degrees of freedom) is consistent with selecting the matrix transpose as the reversal anti-automorphism. In this representation, the grade involution can be expressed as the inner automorphism ${\displaystyle A\rightarrow (-i\sigma _{2})A{(-i\sigma _{2})}^{-1}}$.

But in the case of Cl_1,1(R), the degree of freedom with negative norm lies in the odd part, and the reversal anti-automorphism must instead reverse one of the non-identity degrees of freedom that has positive norm. The general anti-automorphism of a full matrix algebra (over a field other than ${\displaystyle \mathbb {Z} _{2}}$) is easily proven to be an anti-similarity transformation ${\displaystyle A\rightarrow TA^{t}T^{-1}}$ for some invertible matrix T. (These anti-automorphisms form the odd part of a Z₂-graded matrix transformation group of which the inner automorphisms are the even subgroup; together, they form the group of Jordan automorphisms. It's not quite this simple in the case of a matrix group over a general commutative ring or a C^*-algebra.) If we choose to represent the degree of freedom with positive norm in the odd part of the Clifford algebra using one of the two Pauli matrices with real entries, ${\displaystyle \sigma _{1}=\left[{\begin{array}{rr}0&1\\1&0\end{array}}\right]}$, then the non-identity degree of freedom in the even subalgebra corresponds to the remaining Pauli matrix, ${\displaystyle \sigma _{3}=\left[{\begin{array}{rr}1&0\\0&-1\end{array}}\right]}$. The anti-similarity transformation that reverses only this last degree of freedom, which is ${\displaystyle A\rightarrow \sigma _{1}A^{t}\sigma _{1}^{-1}}$, is then an appropriate representation of the reversal anti-automorphism on Cl_1,1(R). In this representation, the grade involution can be expressed as the inner automorphism ${\displaystyle A\rightarrow \sigma _{3}A\sigma _{3}^{-1}}$.

Looking back at the case of Cl_2,0(R), we may observe that "the degree of freedom with negative norm" need not be generated by ${\displaystyle \pm i\sigma _{2}}$; one could just as easily choose ${\displaystyle \pm \left[{\begin{array}{rr}{(1-k^{2})}^{\frac {1}{2}}&-k\\k&-{(1-k^{2})}^{\frac {1}{2}}\end{array}}\right]}$ for any ${\displaystyle \left|k\right|>=1}$, which is a similarity transformation away from ${\displaystyle \pm i\sigma _{2}}$. What matters is not the relationship of matrix degrees of freedom to the matrix transpose but their relationship to the particular anti-similarity transformation that represents the reversal anti-automorphism of the Clifford algebra. (And, of course, to the automorphism chosen as the grade involution, which must be selected from those which have the right signature relative to the chosen reversal anti-automorphism. This is a somewhat subtle point; these two operations on the representation need not commute on the entire matrix algebra in which the representation is embedded! But for the representation to be faithful, they do need to commute on the representation itself, and that's a property that has to be verified separately from their individual properties.)

Looking at the other two Clifford algebras with four real degrees of freedom, Cl_0,2(R) and Cl₁(C), we find that they are distinguished from the above two and from one another by their structure as associative algebras. Both may be represented as subalgebras of the matrix algebra M₄(R), and in a sense as subalgebras of M₂(C). Setting Cl₁(C) aside, one may observe that the same assignment of the even subalgebra to the matrices spanned by 1 and ${\displaystyle -i\sigma _{2}}$ works for Cl_0,2(R) as for Cl_2,0(R), with the odd part spanned by ${\displaystyle -i\sigma _{1}}$ and ${\displaystyle -i\sigma _{3}}$ — complex, not real, 2 by 2 matrices. This is consistent with selecting the complex matrix transpose — not Hermitian conjugation — as the reversal anti-automorphism. In this representation, the grade involution can again be expressed as the inner automorphism ${\displaystyle A\rightarrow (-i\sigma _{2})A{(-i\sigma _{2})}^{-1}}$; it can also be expressed as complex conjugation of the matrix entries (still not Hermitian conjugation). Calling this associative algebra H (Hamilton's quaternions) risks emphasizing the composition of these two operations (the Clifford conjugation, which does indeed coincide here with Hermitian conjugation and thus with the quaternion conjugation) over the others (which, as they invert only one or two of the non-identity degrees of freedom, necessarily involve "breaking the symmetry" of the imaginary quaternions). Perhaps it is arguable whether the reversal anti-automorphism or the Clifford conjugation is "more fundamental", but there can be more to the grade involution than just the composition of the two; a general associative algebra can have outer automorphisms, too.

When extending this analysis to other Clifford algebras, it is important to remember that the identification of general anti-automorphisms with anti-similarity transformations only holds for a full matrix algebra over a (non-${\displaystyle \mathbb {Z} _{2}}$) field. The quaternions H are a division algebra but not a field, and a matrix with quaternion entries is equivalent to a sparse matrix over a complex (or real) field; the general anti-automorphism of such a matrix algebra is still an anti-similarity transformation parameterized by some matrix T in the full matrix algebra, but that T may lie outside the subring representing the matrices with quaternion entries. Similarly, a direct sum of two copies of a matrix algebra — of which the simplest case is the associative algebra underlying Cl_1,0(R), D = R² — may also be represented as a sparse matrix algebra (as when identifying (a, x) with ${\displaystyle a\cdot 1+x\cdot \sigma _{3}}$); but the matrix parameter of a general anti-similarity transformation may not lie in this same algebra. (In this example, the reversal anti-automorphism is simply the identity on Cl_1,0(R), but the Clifford conjugation corresponds to ${\displaystyle A\rightarrow TA^{t}T^{-1}}$ with a T lying outside D, such as ${\displaystyle \sigma _{1}}$ or ${\displaystyle -i\sigma _{2}}$.)

Discussion of (anti-)automorphism groups of an associative algebra over C — and therefore of a Clifford algebra over field C — has to account for two different ways to be "anti-": order reversal of the algebra product and complex antilinearity relative to the scalar product. Complex conjugation of the entries in a particular complex matrix representation of a Clifford algebra, much like the matrix transpose, is not as special as it first seems; what matters is not the relationship of matrix degrees of freedom to the complex conjugation but their relationship to the complex antilinear automorphism that represents "charge conjugation" of all components in the Clifford algebra. (The set of all such complex antilinear automorphisms presumably forms the odd part of a different Z₂-graded matrix transformation group of which the even subgroup is, or perhaps includes, the inner automorphisms; adjoining the anti-similarity transformations as well as the complex antilinear anti-automorphisms, one might obtain a Z₂×Z₂-graded matrix transformation group

The relevant antilinear automorphism is, in any case, restricted by the complex Clifford algebra construction as a quotient of the (covariant) tensor algebra over the field C by the complex bilinear extension of a symmetric real-valued quadratic form on the underlying real (co)vector space to a complexified version of that space. (See section 4.1 of Vaz and da Rocha; the point is that the two-operand quadratic form is by construction not only complex bilinear — not linear in one operand and antilinear in the other — but consistent with a real quadratic form on the R^n subspace associated with the original real axes.) So a statement like "there is essentially only one complex Clifford algebra for each value of the dimension n" is true in the qualified sense that, given a complex Clifford algebra constructed in this way, one can extract "real" subalgebras for any signature p+q=n by choosing different antilinear automorphisms to represent complex conjugation on the complexified (co)vector space. But it is not automatically true that a particular antilinear automorphism has a "natural" representation relative to the otherwise simplest representation as a complex matrix algebra. The complex Clifford algebras for different quadratic form signatures share an associative algebra structure, but may have inequivalent antilinear "charge conjugation" automorphisms as well as reversal anti-automorphisms and grade involution automorphisms.

Returning to Cl₁(C), we find that it is consistent as an associative algebra with C⊕C (the complexification of the representation of Cl_1,0(R) by R⊕R), but that we can represent "charge conjugation" simply as complex conjugation in at most one of the two choices of signature for the real quadratic form. In the representation identifying (a, x) with ${\displaystyle a\cdot 1+x\cdot \sigma _{3}}$, the other signature is obtained by ${\displaystyle A\rightarrow TA^{*}T^{-1}}$ with ${\displaystyle T=\sigma _{1}}$ (or ${\displaystyle -i\sigma _{2}}$, etc.). Since there is only one odd (complex) degree of freedom, these two antilinear automorphisms are related by the grade involution automorphism itself, and the reversal anti-automorphism is the identity; the general case is more complicated, of course.

A representation of a real Clifford algebra using a matrix with complex entries — such as Cl_3,0(R), which has the same structure as an associative algebra over the reals as M₂(C) — has anti-automorphisms as a real algebra that are not represented by anti-similarity transforms on the complex matrix. (In the simplest such case, Cl_0,1(R) represented by C, the reversal anti-automorphism is again simply the identity but the Clifford conjugation corresponds to complex conjugation — in a way that has nothing to do with "charge conjugation".) And although similarity transformations (inner automorphisms) form the entirety of the even subalgebra of the automorphisms of M_n(R) or M_n(C), a general associative algebra may also have outer automorphisms that cannot be represented as similarity transformations, and it is possible a priori for the grade involution of a Clifford algebra representable as M_n(K) to be representable only as an outer, rather than an inner, automorphism of M_n(K) (for non-field K). The reversal anti-automorphism and Clifford conjugation anti-automorphism are related by the grade involution automorphism, but one cannot necessarily recover the correct grade involution automorphism given representations of these two anti-automorphisms on an otherwise adequate matrix algebra.

In short (too late!), one must be cautious with statements like "Cl₂(C) and Cl_3,0(R) are both determined to be M₂(C)," or even "Cl_4,2(R) and Cl_6,0(R) are both represented by M₈(R)." That's really only true at the level of the associative algebra structure, and even then it's an abuse of notation to denote a subalgebra of M₄(R) as M₂(C). Really classifying the Clifford algebras would require assigning each a representative embedding as a subalgebra of a matrix ring over the correct field — e. g., not "M₄(H)" but some appropriately constrained 64-real-dimensional subalgebra of (probably) M₁₆(R) — with a specific choice of grade involution automorphism (especially if it's an outer automorphism), reversal anti-automorphism, and (if the field is C) antilinear charge conjugation automorphism. These are of course only unique up to some group of automorphisms of the full matrix ring.