In this document, we unify the informal mathematical language for a definition to:

\optional{Let \placeholder{\(X\)} be a \placeholder{\vocab{concept \(X\)}}. }

A \newvocab{\placeholder{concept \(Z\)}} is a set/pair/triple/tuple \((Z, \mathtt {op}, ...)\), satisfying:

1. \placeholder{\(Z\)} is a \placeholder{\vocab{concept \(Y\)}} \optional{ over \placeholder{\(X\)} under \placeholder{op} }.
2. \placeholder{formula} for all \placeholder{elements in \(Z\)} \optional{(\vocab{ \placeholder{property} })}.
3. \optional{for each \placeholder{element} in \placeholder{\vocab{concept \(X\)}}, } there exists \placeholder{element} such that \placeholder{formula} for all \placeholder{elements in concept \(Z\)}.
4. \optional{\placeholder{op} is called \placeholder{\vocab{op name}}, }for all \placeholder{elements in \(Z\)}, we have
   1. \placeholder{formula}
   2. \placeholder{formula}
   \optional{(\vocab{ \placeholder{property} })}.

By default, \placeholder{\(X\)} is a set, \placeholder{op} is a binary operation on \placeholder{\(X\)}.

A group is a pair \((G, *)\), satisfying:

1. \((a * b) * c = a * (b * c)\) for all \(a, b, c \in G\) (associativity).
2. there exists \(1 \in G\) such that \[1 * a = a * 1 = a\] for all \(a \in G\).
3. for each \(a \in G\), there exists \(a^{-1} \in G\) such that \[a * a^{-1} = a^{-1} * a = 1\].

It then follows that \(1\), the identity element, is unique, and that for each \(g \in G\) the inverse \(g^{-1}\) is unique. A group G is abelian, or commutative, if \(g * h = h * g\) for all \(g, h \in G\).

In literatures, the binary operation of a group is usually called a product. It's denoted by juxtaposition \(g h\), and is understood to be a mapping \((g, h) \mapsto g * h\) from \(G \times G\) to \(G\).

Here we use an explicit symbol for the operation. It can be denoted multiplicatively as \(*\) in Group or additively as \(+\) in AddGroup, where the identity element will be denoted by \(1\) or \(0\), respectively.

Sometimes we use notations with subscripts (e.g. \(*_G\), \(1_G\)) to indicate where they are.

\(\textsf {Mathlib}\) encodes the mapping \(G \times G \to G\) as \(G \to G \to G\), it is understood to be \(G \to (G \to G)\), that sends \(g \in G\) to a mapping that sends \(h \in G\) to \(g * h \in G\).

Furthermore, a mathematical construct, e.g. a group, is represented by a "type", as Lean has a dependent type theory foundation, see [carneiro2019type] and [ullrich2023extensible, sec. 3.2].

A monoid is a pair \((R, *)\), satisfying:

1. \((a * b) * c = a * (b * c)\) for all \(a, b, c \in R\) (associativity).
2. there exists an element \(1 \in R\) such that \(1 * a = a * 1 = a\) for all \(a \in R\)
i.e. \(1\) is the multiplicative identity (neutral element).

A ring is a triple \((R, +, *)\), satisfying:

1. \(R\) is a commutative group under \(+\).
2. \(R\) is a monoid under \(*\).
3. for all \(a, b, c \in R\), we have
   1. \(a * (b + c) = a * b + a * c\)
   2. \((a + b) * c = a * c + b * c\)
   (left and right distributivity over \(+\)).

In applications to Clifford algebras \(R\) will be always assumed to be commutative.

A division ring is a ring \((R, +, *)\), satisfying:

1. \(R\) contains at least 2 elements.
2. for all \(a \neq 0\) in \(R\), there exists a multiplicative inverse \(a^{-1} \in R\) such that
\[ a * a^{-1} = a^{-1} * a = 1 \]

Let \(R\) be a commutative ring. A module over \(R\), called an \(R\)-module, is a pair \((M, \bullet )\), satisfying:

1. M is a group under \(+\).
2. \(\bullet : R \to M \to M\) is called scalar multiplication, for every \(a, b \in R\), \(x, y \in M\), we have
   1. \(a \bullet (x + y) = a \bullet x + b \bullet y\)
   2. \((a + b) \bullet x = a \bullet x + b \bullet x\)
   3. \(a * (b \bullet x)=(a * b) \bullet x\)
   4. \(1_R \bullet x = x\)

The notation of scalar multiplication is generalized as heterogeneous scalar multiplication HMul in \(\textsf {Mathlib}\): \[ \bullet : \alpha \to \beta \to \gamma \] where \(\alpha \), \(\beta \), \(\gamma \) are different types.

If \(R\) is a division ring, then a module \(M\) over \(R\) is called a vector space.

For generality, \(\textsf {Mathlib}\) uses Module throughout for vector spaces, particularly, for a vector space \(V\), it's usually declared as

For definitions/theorems about it, and most of them can be found under Mathlib.LinearAlgebra e.g. LinearIndependent.

A submodule \(N\) of \(M\) is a module \(N\) such that every element of \(N\) is also an element of \(M\). If \(M\) is a vector space then \(N\) is called a subspace.

The dual module \(M^* : M \to _{l[R]} R\) is the \(R\)-module of all linear maps from \(M\) to \(R\).

Let \((\alpha , +_\alpha , *_\alpha )\) and \((\beta , +_\beta , *_\beta )\) be rings. A ring homomorphism from \(\alpha \) to \(\beta \) is a map \(\mathit {1} : \alpha \to _{+*} \beta \) such that

1. \(\mathit {1}(x +_{\alpha } y) = \mathit {1}(x) +_{\beta } \mathit {1}(y)\) for each \(x,y \in \alpha \).
2. \(\mathit {1}(x *_{\alpha } y) = \mathit {1}(x) *_{\beta } \mathit {1}(y)\) for each \(x,y \in \alpha \).
3. \(\mathit {1}(1_{\alpha }) = 1_{\beta }\).

Isomorphism \(A \cong B\) is a bijective homomorphism \(\phi : A \to B\) (it follows that \(\phi ^{-1} : B \to A\) is also a homomorphism).

Endomorphism is a homomorphism from an object to itself, denoted \(\operatorname {End}(A)\).

Automorphism is an endomorphism which is also an isomorphism, denoted \(\operatorname {Aut}(A)\).

Let \(R\) be a commutative ring. An algebra \(A\) over \(R\) is a pair \((A, \bullet )\), satisfying:

1. \(A\) is a ring under \(*\).
2. there exists a ring homomorphism from \(R\) to \(A\), denoted \(\mathit {1} : R \to _{+*} A\).
3. \(\bullet : R \to M \to M\) is a scalar multiplication
4. for every \(r \in R\), \(x \in A\), we have
   1. \(r * x = x * r\) (commutativity between \(R\) and \(A\))
   2. \(r \bullet x = r * x\) (definition of scalar multiplication)
   where we omitted that the ring homomorphism \(\mathit {1}\) is applied to \(r\) before each multiplication.

Following literatures, for \(r \in R\), usually we write \(\mathit {1}_A(r) : R \to _{+*} A\) as a product \(r \mathit {1}_A\) if not omitted, while they are written as a call to algebraMap _ _ r in \(\textsf {Mathlib}\), which is defined to be Algebra.toRingHom r.

The definition above (adopted in \(\textsf {Mathlib}\)) is more general than the definition in literature (e.g. [jadczyk2019notes, 1.6]):

Let \(R\) be a commutative ring. An algebra \(A\) over \(R\) is a pair \((M, *)\), satisfying:

1. \(A\) is a module \(M\) over \(R\) under \(+\) and \(\bullet \).
2. \(A\) is a ring under \(*\).
3. For \(x, y \in A, a \in R\), we have
\[ a \bullet (x * y) = (a \bullet x) * y = x * (a \bullet y) \]

See Implementation notes in Algebra for details.

What's simply called algebra is actually associative algebra with identity, a.k.a. associative unital algebra. See the red herring principle for more about such phenomenon for naming, particularly the example of (possibly) nonassociative algebra.

Let \(A\) and \(B\) be \(R\)-algebras. \(\mathit {1}_A\) and \(\mathit {1}_B\) are ring homomorphisms from \(R\) to \(A\) and \(B\), respectively. A algebra homomorphism from \(A\) to \(B\) is a map \(f : \alpha \to _{a} \beta \) such that

1. \(f\) is a ring homomorphism
2. \(f(\mathit {1}_{A}(r)) = \mathit {1}_{B}(r)\) for each \(r \in R\)

Let \(R\) be a non-commutative ring, \(r\) an arbitrary equivalence relation on \(R\). The ring quotient of \(R\) by \(r\) is the quotient of \(R\) by the strengthen equivalence relation of \(r\) such that for all \(a, b, c\) in \(R\):

1. \(a + c \sim b + c\) if \(a \sim b\)
2. \(a * c \sim b * c\) if \(a \sim b\)
3. \(a * b \sim a * c\) if \(b \sim c\)

As ideals haven't been formalized for the non-commutative case, \(\textsf {Mathlib}\) uses RingQuot in places where the quotient of non-commutative rings by ideal is needed. The universal properties of the quotient are proven, and should be used instead of the definition that is subject to change.

Let \(X\) be an arbitrary set. An free \(R\)-algebra on \(X\) (or "generated by \(X\)"), named \(A\), is the ring quotient of the following inductively constructed set \(A_{\mathtt {pre}}\)

1. for all \(x\) in \(X\), there exists a map \(X \to A_{\mathtt {pre}}\).
2. for all \(r\) in \(R\), there exists a map \(R \to A_{\mathtt {pre}}\).
3. for all \(a, b\) in \(A_{\mathtt {pre}}\), \(a + b\) is in \(A_{\mathtt {pre}}\).
4. for all \(a, b\) in \(A_{\mathtt {pre}}\), \(a * b\) is in \(A_{\mathtt {pre}}\).

1. there exists a ring homomorphism from \(R\) to \(A_{\mathtt {pre}}\), denoted \(R \to _{+*} A_{\mathtt {pre}}\).
2. \(A\) is a commutative group under \(+\).
3. \(A\) is a monoid under \(*\).
4. left and right distributivity under \(*\) over \(+\).
5. \(0 * a \sim a * 0 \sim 0\).
6. for all \(a, b, c\) in \(A\), if \(a \sim b\), we have
   1. \(a + c \sim b + c\)
   2. \(c + a \sim c + b\)
   3. \(a * c \sim b * c\)
   4. \(c * a \sim c * b\)
   (compatibility with the ring operations \(+\) and \(*\))

Similar to remark 1.2.6, what we defined here is the free (associative, unital) \(R\)-algebra on \(X\), it can be denoted \(R\langle X \rangle \), expressing that it's freely generated by \(R\) and \(X\), where \(X\) is the set of generators.

Let \(R, S\) be rings, \(M\) an \(R\)-module, \(N\) an \(S\)-module. A linear map from \(M\) to \(N\) is a function \(f : M \to _{l} N\) over a ring homomorphism \(\sigma : R \to _{+*} S\), satisfying:

1. \(f(x + y) = f(x) + f(y)\) for all \(x, y \in M\).
2. \(f(r \bullet x) = \sigma (r) \bullet f(x)\) for all \(r \in R\), \(x \in M\).

The set of all linear maps from \(M\) to \(M'\) is denoted \(\operatorname {Lin}(M, M')\), and \(\operatorname {Lin}(M)\) for mapping from \(M\) to itself. \(\operatorname {Lin}(M)\) is an endomorphism.

Let \(A\) be a free \(R\)-algebra generated by module \(M\), let \(\iota : M \to A\) denote the map from \(M\) to \(A\). An tensor algebra over \(M\) (or "of \(M\)") \(T\) is the ring quotient of the free \(R\)-algebra generated by \(M\), by the equivalence relation satisfying:

1. for all \(a, b\) in \(M\), \(\iota (a + b) \sim \iota (a) + \iota (b)\).
2. for all \(r\) in \(R\), \(a\) in \(M\), \(\iota (r \bullet a) \sim r * \iota (a)\).

The definition above is equivalent to the following definition in literature (e.g. [jadczyk2019notes, 1.7]):

Let \(M\) be a module over \(R\). An algebra \(T\) is called a tensor algebra over \(M\) (or "of \(M\)") if it satisfies the following universal properties:

1. \(T\) is an algebra containing \(M\) as a submodule, and it is generated by \(M\),
2. Every linear mapping \(\lambda \) of \(M\) into an algebra \(A\) over \(R\), can be extended to
a homomorphism \(\theta \) of \(T\) into \(A\).

Let \(R\) be a ring, \(M\) an \(R\)-module.

An bilinear form \(B\) over \(M\) is a map \(B : M \to M \to R\), satisfying:

1. \( B(x + y, z) = B(x, z) +B(y, z) \)
2. \( B(x, y + z) = B(x, y) +B(x, z) \)
3. \( B(a \bullet x, y) = a * B(x, y)\)
4. \( B(x, a \bullet y) = a * B(x, y)\)

Let \(R\) be a commutative ring, \(M\) a \(R\)-module.

An quadratic form \(Q\) over \(M\) is a map \(Q : M \to R\), satisfying:

1. \( Q(a \bullet x) = a * a * Q(x)\) for all \(a \in R, x \in M\).
2. there exists a companion bilinear form \(B : M \to M \to R\), such that \(Q(x + y) = Q(x) + Q(y) + B(x, y)\)

In [jadczyk2019notes], the bilinear form is denoted \(\Phi \), and called the polar form associated with the quadratic form \(Q\), or simply the polar form of \(Q\).

This notion generalizes to commutative semirings using the approach in [izhakian2016supertropical].

Let \(M\) be a module over a commutative ring \(R\), equipped with a quadratic form \(Q: M \to R\).

Let \(\iota : M \to _{l[R]} T(M)\) be the canonical \(R\)-linear map for the tensor algebra \(T(M)\).

Let \(\mathit {1} : R \to _{+*} T(M)\) be the canonical map from \(R\) to \(T(M)\), as a ring homomorphism.

A Clifford algebra over \(Q\), denoted \(\mathcal {C}\kern -2pt\ell (Q)\), is the ring quotient of the tensor algebra \(T(M)\) by the equivalence relation satisfying \(\iota (m)^2 \sim \mathit {1}(Q(m))\) for all \(m \in M\).

The natural quotient map is denoted \(\pi : T(M) \to \mathcal {C}\kern -2pt\ell (Q)\) in some literatures, or \(\pi _\Phi \)/\(\pi _Q\) to emphasize the bilinear form \(\Phi \) or the quadratic form \(Q\), respectively.

In literatures, \(M\) is often written \(V\), and the quotient is taken by the two-sided ideal \(I_Q\) generated from the set \(\{ v \otimes v - Q(v) \vert v \in V \}\). See also Clifford algebra [pr-spin].

As of writing, \(\textsf {Mathlib}\) does not have direct support for two-sided ideals, but it does support the equivalent operation of taking the ring quotient by a suitable closure of a relation like \(v \otimes v \sim Q(v)\).

Hence the definition above.

This definition and what follows in \(\textsf {Mathlib}\) is initially presented in [wieser2022formalizing], some further developments are based on [grinberg2016clifford], and in turn based on [bourbaki2007algebra] which is in French and never translated to English.

The most informative English reference on [bourbaki2007algebra] is [jadczyk2019notes], which has an updated exposition in [jadczyk2023bundle].

Let \(V\) be a vector space \(\mathbb R^n\) over \(\mathbb R\), equipped with a quadratic form \(Q\).

Since \(\mathbb R\) is a commutative ring and \(V\) is a module, the definition above applies.

We denote the canonical \(R\)-linear map to the Clifford algebra \(\mathcal {C}\kern -2pt\ell (M)\) by \(\iota : M \to _{l[R]} \mathcal {C}\kern -2pt\ell (M)\).

It's denoted \(i_\Phi \) or just \(i\) in some literatures.

Given a linear map \(f : M \to _{l[R]} A\) into an \(R\)-algebra \(A\), satisfying \(f(m)^2 = Q(m)\) for all \(m \in M\), called is Clifford, the canonical lift of \(f\) is defined to be a algebra homomorphism from \(\mathcal {C}\kern -2pt\ell (Q)\) to \(A\), denoted \(\operatorname {lift} f : \mathcal {C}\kern -2pt\ell (Q) \to _{a} A\).

Given \(f : M \to _{l[R]} A\), which is Clifford, \(F = \operatorname {lift} f\) (denoted \(\bar {f}\) in some literatures), we have:

1. \(F \circ \iota = f\), i.e. the following diagram commutes:

   

2. \(\operatorname {lift}\) is unique, i.e. given \(G : \mathcal {C}\kern -2pt\ell (Q) \to _{a} A\), we have: \[ G \circ \iota = f \iff G = \operatorname {lift} f\]

The universal property of the Clifford algebras is now proven, and should be used instead of the definition that is subject to change.

An Exterior algebra over \(M\) is the Clifford algebra \(\mathcal {C}\kern -2pt\ell (Q)\) where \(Q(m) = 0\) for all \(m \in M\).

Same as the previous section, let \(M\) be a module over a commutative ring \(R\), equipped with a quadratic form \(Q: M \to R\).

We also use \(m\) or \(m_1, m_2, \dots \) to denote elements of \(M\), i.e. vectors, and \(x, y, z\) to denote elements of \(\mathcal {C}\kern -2pt\ell (Q)\).

Grade involution, intuitively, is negating each basis vector.

Formally, it's an algebra homomorphism \(\alpha : \mathcal {C}\kern -2pt\ell (Q) \to _{a} \mathcal {C}\kern -2pt\ell (Q)\), satisfying:

1. \(\alpha \circ \alpha = \operatorname {id}\)
2. \(\alpha (\iota (m)) = - \iota (m)\)

for all \(m \in M\).

That is, the following diagram commutes:

It's called main involution \(\alpha \) or main automorphism in [jadczyk2019notes], the canonical automorphism in [gallier2008clifford].

It's denoted \(\hat {m}\) in [lounesto2001clifford], \(\alpha (m)\) in [jadczyk2019notes], \(m^*\) in [chisolm2012geometric].

Grade reversion, intuitively, is reversing the multiplication order of basis vectors. Formally, it's an algebra homomorphism \(\tau : \mathcal {C}\kern -2pt\ell (Q) \to _{a} \mathcal {C}\kern -2pt\ell (Q)^{\mathtt {op}}\), satisfying:

1. \(\tau (m_1 m_2) = \tau (m_2) \tau (m_1)\)
2. \(\tau \circ \tau = \operatorname {id}\)
3. \(\tau (\iota (m)) = \iota (m)\)

That is, the following diagram commutes:

It's called anti-involution \(\tau \) in [jadczyk2019notes], the canonical anti-automorphism in [gallier2008clifford], also called transpose/transposition in some literature, following tensor algebra or matrix.

It's denoted \(\tilde {m}\) in [lounesto2001clifford], \(m^\tau \) in [jadczyk2019notes] (with variants like \(m^t\) or \(m^\top \) in other literatures), \(m^\dagger \) in [chisolm2012geometric].

Clifford conjugate is an algebra homomorphism \({*} : \mathcal {C}\kern -2pt\ell (Q) \to _{a} \mathcal {C}\kern -2pt\ell (Q)\), denoted \(x^{*}\) (or even \(x^\dagger \), \(x^v\) in some literatures), defined to be: \[ x^{*} = \operatorname {reverse}(\operatorname {involute}(x)) = \tau (\alpha (x)) \] for all \(x \in \mathcal {C}\kern -2pt\ell (Q)\), satisfying (as a \(*\)-ring):

1. \((x + y)^{*} = (x)^{*} + (y)^{*}\)
2. \((x y)^{*} = (y)^{*} (x)^{*}\)
3. \({*} \circ {*} = \operatorname {id}\)
4. \(1^{*} = 1\)

Note: In our current formalization in \(\textsf {Mathlib}\), the application of the involution on \(r\) is ignored, as there appears to be nothing in the literature that advocates doing this.

Clifford conjugate is denoted \(\bar {m}\) in [lounesto2001clifford] and most literatures, \(m^\ddagger \) in [chisolm2012geometric].