Algebras over a ring
ClassAssociative algebra over a commutative ring#
An associative algebra over a commutative ring \(R\), or \(R\)-algebra, is a ring \(A\) whose elements can also be multiplied by scalars from \(R\), in a way compatible with the multiplication of \(A\). Polynomial rings, matrix rings, rings of endomorphisms of a module and field extensions are all algebras over a suitable base, and the notion lets one study such rings together with their linear structure over \(R\). Most of commutative algebra and algebraic geometry is phrased in terms of \(R\)-algebras.
Definition
Let \(R\) be a commutative ring. An \(R\)-algebra is a ring \(A\) (associative, with identity, not necessarily commutative) together with a ring homomorphism \(\eta \colon R \to A\), the structure map, whose image lies in the centre of \(A\): that is, \(\eta(r)\,a = a\,\eta(r)\) for all \(r \in R\) and \(a \in A\). Scalar multiplication is then defined by \(r \cdot a = \eta(r)\,a\), and the structure map is recovered from it as \(\eta(r) = r \cdot 1\).
Equivalently, an \(R\)-algebra is a ring \(A\) that is also an \(R\)-module, such that for all \(r \in R\) and \(a, b \in A\)
An algebra satisfies the first identity and the second; conversely, a module structure on a ring satisfying both identities comes from an algebra structure, with structure map \(r \mapsto r \cdot 1\). In other words, multiplication \(A \times A \to A\) is \(R\)-bilinear.
A homomorphism of \(R\)-algebras \(f \colon A \to B\) is a ring homomorphism that commutes with the structure maps, \(f(\eta_A(r)) = \eta_B(r)\); such a map is \(R\)-linear, and conversely an \(R\)-linear map that preserves \(1\) and products is an algebra homomorphism. A subalgebra is a subring that contains the image of the structure map. The \(R\)-algebras and their homomorphisms form the category of \(R\)-algebras.
Examples
- Rings over the integers. Every ring \(A\) is a \(\mathbb{Z}\)-algebra through the unique ring homomorphism \(\mathbb{Z} \to A\), whose image is central, and no other \(\mathbb{Z}\)-algebra structure on \(A\) exists. Thus \(\mathbb{Z}\)-algebras are just rings.
- Commutative algebras. If \(R\) and \(S\) are commutative, then any ring homomorphism \(R \to S\) makes \(S\) an \(R\)-algebra, with that homomorphism as its structure map. So a commutative \(R\)-algebra is the same thing as a commutative ring \(S\) with a ring homomorphism \(R \to S\). Examples are a quotient ring \(R/I\), a localization of \(R\), and an extension field \(L\) of a field \(K\), which is a \(K\)-algebra.
- Polynomials. If \(A\) is an \(R\)-algebra, the polynomial ring \(A[X]\) is an \(R\)-algebra, with \(r\) acting on coefficients; in particular \(R[X]\) is an \(R\)-algebra. The same holds for polynomials in several indeterminates.
- Matrices. For a finite index set with \(n\) elements and an \(R\)-algebra \(A\), the \(n \times n\) matrices with entries in \(A\) form an \(R\)-algebra, with \(\eta(r)\) the scalar matrix \(r I\). For \(n \ge 2\) and \(A = R\) nonzero this algebra is not commutative, though \(\eta(R)\) is still central.
- Endomorphisms. For an \(R\)-module \(M\), the \(R\)-linear maps \(M \to M\) form an \(R\)-algebra under composition, with \(\eta(r)\) the map \(m \mapsto rm\) (see algebra of endomorphisms).
- Products. A product of \(R\)-algebras is an \(R\)-algebra, operations taken componentwise.
- Number systems. The complex numbers form a commutative \(\mathbb{R}\)-algebra of dimension \(2\), and the quaternions a noncommutative \(\mathbb{R}\)-algebra of dimension \(4\) in which every nonzero element is invertible.
Properties
- The structure map need not be injective. For \(n \ge 1\) the ring \(\mathbb{Z}/n\mathbb{Z}\) is a \(\mathbb{Z}\)-algebra whose structure map has kernel \(n\mathbb{Z}\), and the zero ring is an algebra over every \(R\). The map is injective exactly when distinct scalars act differently on \(A\).
- If \(R\) is the zero ring then every \(R\)-algebra is the zero ring, since \(1 = \eta(1) = \eta(0) = 0\) in \(A\).
- Restriction of scalars. If \(A\) is an \(R\)-algebra and \(g \colon S \to R\) is a ring homomorphism from a commutative ring \(S\), then \(A\) becomes an \(S\)-algebra with structure map \(\eta \circ g\).
- New algebras arise from old ones as quotients by two-sided ideals and as tensor products. Others are built from other data: free algebras from a set of generators and tensor algebras from an \(R\)-module; quotients of the tensor algebra give the exterior algebra and Clifford algebras.
Generalizations
Dropping associativity gives non-associative algebras, of which Lie algebras are the main example; dropping the identity gives non-unital algebras, such as the continuous functions with compact support on the real line. In both cases the definition through a module structure and the two compatibility identities still makes sense, while the definition through a central structure map does not: without an identity there is no element \(1\) for \(r\) to act on, and without associativity the identity \((r \cdot a)\,b = r \cdot (ab)\) no longer follows from \(r \cdot a = \eta(r)\,a\). Over a noncommutative base ring \(k\) the usual substitute is a ring \(A\) with a ring homomorphism \(k \to A\) whose image need not be central.
In Mathlib
Mathlib defines algebras with more generality than the textbook version: the base \(R\) is a commutative semiring and \(A\) a semiring, so that, for instance, every semiring is an algebra over the natural numbers. An algebra structure is extra data attached to a pair of existing types rather than a new type, so one ring can carry algebra structures over several bases at once. The data are a scalar multiplication and a ring homomorphism from \(R\) to \(A\), required to land in the centre and to induce the scalar multiplication. Keeping the scalar multiplication as data lets an algebra structure reuse a scalar action that already exists, so that the two actions agree by definition rather than only up to a proof, and two algebra structures with the same structure map are equal. Every algebra is automatically a module, every commutative ring is automatically an algebra over itself, and every ring is automatically an algebra over the integers. Non-unital and non-associative algebras are not instances of this definition; Mathlib describes them by a module structure together with the two compatibility identities above.
Written for Trough, under CC BY-SA 4.0.
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StructureThe category of algebras over a ring#
For a commutative ring \(R\), the category of algebras over \(R\) has as its objects the associative unital algebras over \(R\) and as its morphisms the homomorphisms of \(R\)-algebras. It is the natural home for the constructions of commutative and noncommutative algebra that are characterized by universal properties, such as polynomial rings, free algebras, tensor algebras and tensor products, and it sits between the category of rings and the category of \(R\)-modules, to both of which it maps by forgetting structure.
Definition
An \(R\)-algebra is a ring \(A\), not necessarily commutative, together with a ring homomorphism \(\iota_A \colon R \to A\) whose image lies in the centre of \(A\). Equivalently, \(A\) is a ring and an \(R\)-module at once, with
A homomorphism of \(R\)-algebras \(f \colon A \to B\) is a ring homomorphism, sending \(1\) to \(1\), with \(f(\iota_A(r)) = \iota_B(r)\) for every \(r \in R\); equivalently, a ring homomorphism that is \(R\)-linear. With composition of functions and the identity maps, these data form the category of \(R\)-algebras, often written \(\mathrm{Alg}_R\). It is a category whose objects are sets with extra structure and whose morphisms are functions preserving it.
Examples
- The ring \(R\) itself, with the identity as structure map.
- The polynomial ring \(R[X]\). For every \(R\)-algebra \(A\), sending a homomorphism \(R[X] \to A\) to the image of \(X\) is a bijection onto \(A\), so \(R[X]\) is the free \(R\)-algebra on one generator.
- For an \(R\)-algebra \(A\) and a finite index set of size \(n\), the square matrices of size \(n\) over \(A\) form an \(R\)-algebra, with \(r\) acting through the scalar matrix \(\iota_A(r) I\). For \(n \ge 2\) and \(A\) nonzero this algebra is not commutative.
- A field extension \(L\) of a field \(K\) is a \(K\)-algebra.
- The zero ring, in which \(0 = 1\), is an \(R\)-algebra for every \(R\).
Properties
Initial and terminal objects. For every \(R\)-algebra \(A\), the structure map \(\iota_A\) is a homomorphism of \(R\)-algebras, and it is the only one from \(R\) to \(A\), so \(R\) is an initial object. Dually, every \(R\)-algebra has exactly one homomorphism to the zero algebra, which is therefore terminal.
Limits. The category has all small limits, and the forgetful functor to sets preserves them, so limits are computed on underlying sets: a product is the Cartesian product with componentwise operations, and the equalizer of \(f, g \colon A \to B\) is the subalgebra of elements \(a\) with \(f(a) = g(a)\). Limits are likewise preserved by the forgetful functors to rings and to \(R\)-modules.
Free objects. Forgetting part of the structure gives functors to the category of rings, to the category of \(R\)-modules and to sets, and each of them has a left adjoint. For rings it sends a ring \(A\) to \(R \otimes_{\mathbb{Z}} A\). For the other two the adjoints are free constructions, and these are the two Mathlib provides: the free algebra functor is left adjoint to the forgetful functor to sets, sending a set \(S\) to the free algebra of noncommutative polynomials in the elements of \(S\), and the tensor algebra functor is left adjoint to the forgetful functor to modules, sending a module \(M\) to its tensor algebra.
Isomorphisms. Isomorphisms in the category correspond exactly to isomorphisms of \(R\)-algebras, and a homomorphism whose underlying function is bijective is an isomorphism, since the inverse function is automatically a homomorphism.
Tensor products. The tensor product \(A \otimes_R B\) is again an \(R\)-algebra, and with it, and with \(R\) as unit, the category is monoidal (monoidal category); the swap \(a \otimes b \mapsto b \otimes a\) makes it a symmetric monoidal category. The tensor product is not in general the coproduct: homomorphisms \(A \otimes_R B \to C\) correspond to pairs of homomorphisms \(A \to C\) and \(B \to C\) whose images commute elementwise. Coproducts exist, but they are free products, in which elements of \(A\) and \(B\) need not commute.
Change of base. A ring homomorphism \(\varphi \colon R \to S\) makes every \(S\)-algebra into an \(R\)-algebra, with \(r\) acting as \(\varphi(r)\); this is the restriction of scalars functor from \(S\)-algebras to \(R\)-algebras. Consequently isomorphic base rings have equivalent categories of algebras. Since every ring has a unique ring homomorphism from \(\mathbb{Z}\), and it is central, the category of \(\mathbb{Z}\)-algebras is equivalent to the category of rings.
Related concepts
Restricting to commutative algebras gives the category of commutative \(R\)-algebras, which is equivalent to the category of commutative rings equipped with a ring homomorphism from \(R\). There the commutation condition above is automatic, and the tensor product is the coproduct. Adding a compatible comultiplication and counit gives the category of bialgebras over \(R\), and adding an antipode the category of Hopf algebras over \(R\).
In Mathlib
Mathlib bundles an object as a type together with a ring structure and an \(R\)-algebra structure on it, and wraps each morphism as an algebra homomorphism; composition and identities are those of functions. The base ring is required to be a commutative ring and the algebras to be rings, so algebras over semirings, and nonunital or nonassociative algebras, are not objects of this category. The algebras live in a universe of types fixed as a parameter. Some results are stated only when that universe is the one containing \(R\): the free algebra and tensor algebra adjunctions, the monoidal and symmetric monoidal structures, and the description of commutative algebras and their coproducts.
Written for Trough, under CC BY-SA 4.0.
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Uses 3
DefinitionFree algebra of a commutative ring#
The free algebra on a set \(X\) over a commutative ring \(R\), written \(R\langle X\rangle\), is the associative algebra with identity generated by the elements of \(X\) subject to no relations other than those every associative \(R\)-algebra must satisfy. Its elements are the noncommutative polynomials in variables from \(X\): finite \(R\)-linear combinations of words in those variables, in which \(xy\) and \(yx\) are different monomials. It plays the role for associative algebras that the free group plays for groups: every algebra generated by a family of elements is a quotient of a free algebra, and many algebras of interest are defined by naming generators and relations and dividing a free algebra by the relations.
Definition
Let \(R\) be a commutative ring and \(X\) a set. The free algebra \(R\langle X\rangle\) is an \(R\)-algebra together with a map \(\iota : X \to R\langle X\rangle\) that has the following universal property: for every associative \(R\)-algebra \(A\) with identity, commutative or not, and every function \(f : X \to A\), there is exactly one homomorphism of \(R\)-algebras \(\bar f : R\langle X\rangle \to A\) with \(\bar f \circ \iota = f\). In other words, sending \(\bar f\) to \(\bar f \circ \iota\) gives a bijection between algebra homomorphisms \(R\langle X\rangle \to A\) and functions \(X \to A\). The two halves of this statement are that the extension \(\bar f\) restricts to \(f\) on the generators and that a homomorphism \(g\) with \(g \circ \iota = f\) is equal to \(\bar f\). In particular, two algebra homomorphisms out of \(R\langle X\rangle\) that agree on the generators are equal. As with any universal property, it determines \(R\langle X\rangle\) up to a unique isomorphism compatible with \(\iota\).
Concretely, \(R\langle X\rangle\) can be built as the free module on the set of words \(x_1 x_2 \cdots x_n\) in the alphabet \(X\), the empty word included, with multiplication given on words by concatenation and extended bilinearly. The empty word is the identity, and \(\iota\) sends \(x\) to the word of length one. This construction is the monoid algebra of the free monoid on \(X\) (the free monoid being the set of words under concatenation, and its monoid algebra the free module on it with multiplication extended from the monoid). The free algebra is isomorphic to this monoid algebra, and the words form a basis of \(R\langle X\rangle\) as an \(R\)-module.
Examples
- If \(X\) is empty, the only word is the empty one and \(R\langle X\rangle\) is \(R\) itself.
- If \(X\) has a single element \(t\), the words are the powers \(t^n\), and \(R\langle X\rangle\) is the polynomial ring \(R[t]\). Here the free algebra is commutative, because an algebra generated by one element over a commutative ring is commutative.
- If \(X = \{x, y\}\) and \(R\) is not the zero ring, then \(xy \neq yx\), since they are distinct words and hence distinct basis elements. So, over a nonzero ring, the free algebra on two or more generators is not commutative. Its quotient by the two-sided ideal generated by \(xy - yx\) is the polynomial ring in two variables \(R[x, y]\).
Properties
- Generation. The free algebra is generated as an \(R\)-algebra by the image of \(\iota\). More generally, for any function \(f : X \to A\) into an \(R\)-algebra, the subalgebra generated by the values of \(f\) is the image of \(\bar f\). Consequently an \(R\)-algebra is finitely generated exactly when it receives a surjective algebra homomorphism from the free algebra on some finite set of its elements, so every finitely generated algebra is a quotient of a free algebra by a two-sided ideal.
- Scalars embed. The structure map \(R \to R\langle X\rangle\) is injective: \(r\) and \(s\) have the same image exactly when \(r = s\). When \(R\) is not the zero ring, the map \(\iota\) is injective.
- No zero divisors. Suppose \(R\) has no zero divisors, and let \(a\) and \(b\) be nonzero. If \(u\) and \(v\) are words of maximal length occurring in \(a\) and \(b\), the word \(uv\) arises in the product \(ab\) only as \(u\) times \(v\), so its coefficient is the nonzero product of their coefficients. Hence \(R\langle X\rangle\) has no zero divisors, and it is a domain, a nonzero ring without zero divisors, when \(R\) is an integral domain.
- Rank. For a nonzero commutative ring \(R\), the rank of \(R\langle X\rangle\) as an \(R\)-module is the cardinality of the set of finite words in \(X\); this is countably infinite when \(X\) is finite and nonempty, and equals \(|X|\) when \(X\) is infinite.
- Characteristic. Because \(R\) embeds in \(R\langle X\rangle\), if \(R\) has characteristic \(p\), so does the free algebra (see characteristic \(p\)).
Related concepts
The free algebra is the noncommutative analogue of the polynomial ring in several variables, which is the free commutative algebra on \(X\) and the quotient of \(R\langle X\rangle\) by the ideal generated by all commutators \(xy - yx\). The tensor algebra of a module generalizes it: the tensor algebra of a free module with a basis indexed by \(X\) is isomorphic to the free algebra on \(X\). Algebras presented by generators and relations, such as the exterior algebra, the Clifford algebra and the universal enveloping algebra of a Lie algebra, arise as quotients of a free or tensor algebra by a two-sided ideal.
In Mathlib
Mathlib defines the free algebra over any commutative semiring, not only a commutative ring, and on any type of generators. It is constructed as a quotient of formal expressions built from scalars and generators by sums and products, modulo the relations that make the result an algebra; after the basic operations are set up, the library works through the universal property and an induction principle rather than this construction. The universal property is packaged as a bijection between functions on the generators and algebra homomorphisms, with targets any associative algebra, commutative or not, over the same semiring. The description by words is a theorem, given by the isomorphism with the monoid algebra of the free monoid. When the scalars are a commutative ring, the free algebra is automatically a ring. The rank statement requires a nonzero commutative ring, and the injectivity of the generator map requires nonzero scalars, matching the hypotheses above.
Written for Trough, under CC BY-SA 4.0.
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Uses 4
InstanceTensor product of algebras#
The tensor product of algebras \(A \otimes_R B\) of two associative algebras \(A\) and \(B\) over a commutative ring \(R\) is the tensor product of their underlying modules, made into an \(R\)-algebra by multiplying factor by factor. It is the standard way to combine two algebras into one in which each acts on its own side: it changes the base ring of an algebra (base change), it builds polynomial and matrix algebras over an \(R\)-algebra \(A\) from those over \(R\), and for commutative algebras it is the coproduct, the algebraic counterpart of the fibre product of affine schemes.
Definition
Let \(R\) be a commutative ring and let \(A\) and \(B\) be \(R\)-algebras, not necessarily commutative. The module \(A \otimes_R B\) consists of finite sums of pure tensors \(a \otimes b\). It carries a unique \(R\)-bilinear multiplication with
for all \(a_1, a_2 \in A\) and \(b_1, b_2 \in B\) (the product of pure tensors). Its existence follows from the universal property of the module tensor product, applied first, for fixed \(a_1\) and \(b_1\), to the bilinear map \((a_2, b_2) \mapsto a_1 a_2 \otimes b_1 b_2\), which gives a linear map of left multiplication on \(A \otimes_R B\), and then to the map sending \((a_1, b_1)\) to that left multiplication, which is itself bilinear. The multiplication is associative and has identity \(1 \otimes 1\), because both properties can be checked on pure tensors, where they reduce to the same properties in \(A\) and \(B\). With the structure map \(r \mapsto (r \cdot 1) \otimes 1\), which equals \(r \mapsto 1 \otimes (r \cdot 1)\), this makes \(A \otimes_R B\) an \(R\)-algebra.
The maps \(a \mapsto a \otimes 1\) and \(b \mapsto 1 \otimes b\) are \(R\)-algebra homomorphisms from \(A\) and from \(B\) into \(A \otimes_R B\), and their images commute with each other elementwise.
Universal property
For any \(R\)-algebra \(C\), the \(R\)-algebra homomorphisms \(A \otimes_R B \to C\) correspond bijectively to the pairs of \(R\)-algebra homomorphisms \(f \colon A \to C\) and \(g \colon B \to C\) whose images commute; the homomorphism attached to such a pair sends \(a \otimes b\) to \(f(a)\,g(b)\). When \(C\) is commutative the commuting condition is automatic, and every pair \(f\), \(g\) gives a homomorphism sending \(a \otimes b\) to \(f(a)\,g(b)\). Consequently, for commutative \(R\), \(A\) and \(B\), the algebra \(A \otimes_R B\) together with its two inclusions is a coproduct in the category of commutative \(R\)-algebras.
Examples
- Base ring. \(R \otimes_R A \cong A\) and \(A \otimes_R R \cong A\) as \(R\)-algebras.
- Polynomials. For every \(R\)-algebra \(A\), \(A[X] \cong A \otimes_R R[X]\) as \(R\)-algebras. For a commutative \(R\)-algebra \(A\) and any set of variables, \(A \otimes_R R[X_i] \cong A[X_i]\) (see polynomial in several indeterminates).
- Matrices. For finite index sets of sizes \(m\) and \(n\), \(M_n(A) \cong A \otimes_R M_n(R)\) as \(R\)-algebras, and the Kronecker product gives \(M_m(A) \otimes_R M_n(B) \cong M_{mn}(A \otimes_R B)\).
- Quotients. For a commutative ring \(R\), a commutative \(R\)-algebra \(B\) and an ideal \(I\) of \(R\), \(B \otimes_R (R/I) \cong B/IB\). For positive integers \(m\) and \(n\) this gives \(\mathbb{Z}/m \otimes_{\mathbb{Z}} \mathbb{Z}/n \cong \mathbb{Z}/\gcd(m,n)\); in particular the tensor product of two nonzero rings can be zero.
- Field extensions. If \(K\) is an extension of a field \(k\) and \(f\) is a polynomial over \(k\), then \(K \otimes_k k[X]/(f) \cong K[X]/(f)\). Taking \(k = \mathbb{R}\), \(K = \mathbb{C}\) and \(f = X^2 + 1\), so that \(\mathbb{C} \cong \mathbb{R}[X]/(X^2 + 1)\), gives \(\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C}[X]/(X^2 + 1)\). Since \(X^2 + 1 = (X - i)(X + i)\) over the complex numbers with coprime factors, the Chinese remainder theorem gives \(\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C} \times \mathbb{C}\), which has zero divisors although both factors are fields.
Properties
- Symmetry and associativity. \(A \otimes_R B \cong B \otimes_R A\) by \(a \otimes b \mapsto b \otimes a\), and \((A \otimes_R B) \otimes_R C \cong A \otimes_R (B \otimes_R C)\), both as \(R\)-algebras.
- Generation. The pure tensors generate \(A \otimes_R B\) as an \(R\)-algebra, so two homomorphisms out of it that agree on pure tensors are equal.
- Commutativity. If \(A\) and \(B\) are commutative, so is \(A \otimes_R B\).
- Dimension. If \(A\) and \(B\) are finite-dimensional over a field \(k\), then \(\dim_k (A \otimes_k B) = \dim_k A \cdot \dim_k B\).
- Nontriviality. The Quotients example shows that \(A \otimes_R B\) may be the zero ring. This cannot happen when \(R\), \(A\) and \(B\) are commutative, \(A\) and \(B\) are integral domains, and the structure maps \(R \to A\) and \(R \to B\) are injective: then \(A \otimes_R B\) is nonzero.
Related concepts
Base change along a commutative \(R\)-algebra \(S\) is the special case \(S \otimes_R A\), which is an \(S\)-algebra. The tensor algebra of a module is a different construction: it is the free algebra generated by the module, not a tensor product of two algebras. In the category of all \(R\)-algebras the tensor product is not the coproduct, because the universal property above only applies to pairs of homomorphisms with commuting images.
In Mathlib
Mathlib works in more generality than this article: \(R\) is a commutative semiring and \(A\) and \(B\) are semirings; in that setting, \(A \otimes_R B\) is a ring as soon as \(A\) is. The multiplication itself is defined, with its basic identities, for algebras that need be neither unital nor associative. The \(R\)-algebra structure is a special case of a more general one: if \(A\) is also an algebra over a second commutative semiring \(S\) whose action commutes with that of \(R\), then \(A \otimes_R B\) is an \(S\)-algebra through the left factor, and the instance over \(R\) is this with \(S = R\). The analogous structure through the right factor is not registered automatically, since for \(S \otimes_R S\) the two choices would conflict. The coproduct statement is made in the category of commutative \(R\)-algebras lying in the same universe as \(R\), a size restriction with no mathematical effect. Mathlib's dimension formula is stated for free modules over semirings with the strong rank condition, a class containing all fields, and uses the convention that an infinite-dimensional space has dimension \(0\), so it holds without a finiteness assumption.
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DefinitionTensor algebra of a commutative ring#
The tensor algebra of a module \(M\) over a commutative ring \(R\), written \(T(M)\), is the associative algebra with identity obtained by putting together all tensor powers \(M \otimes \cdots \otimes M\) and multiplying them by juxtaposition. It is the most general associative algebra into which \(M\) maps linearly: every linear map from \(M\) to an associative algebra extends uniquely to an algebra homomorphism out of \(T(M)\). For this reason it is the starting point for many algebras defined by generators taken from a module, such as the exterior algebra, the symmetric algebra, the Clifford algebra and the universal enveloping algebra of a Lie algebra, each of which is a quotient of a tensor algebra.
Definition
Let \(R\) be a commutative ring and \(M\) an \(R\)-module. For \(n \ge 0\) let \(M^{\otimes n}\) be the \(n\)-th tensor power of \(M\), the tensor product of \(n\) copies of \(M\) over \(R\), with \(M^{\otimes 0} \cong R\) and \(M^{\otimes 1} \cong M\) canonically; below these identifications are written as equalities. The tensor algebra of \(M\) is the direct sum
with the multiplication determined on pure tensors by concatenation,
and extended bilinearly. The identity is \(1 \in R = M^{\otimes 0}\), and the inclusion of the summand \(M^{\otimes 1}\) gives an \(R\)-linear map \(\iota : M \to T(M)\), the canonical map. Under \(\iota\), the pure tensor \(x_1 \otimes \cdots \otimes x_n\) is the product \(\iota(x_1) \cdots \iota(x_n)\).
Equivalently, \(T(M)\) is characterized by a universal property: for every associative \(R\)-algebra \(A\) with identity, commutative or not, and every \(R\)-linear map \(f : M \to A\), there is exactly one homomorphism of \(R\)-algebras \(\bar f : T(M) \to A\) with \(\bar f \circ \iota = f\). In other words, composing with \(\iota\) gives a bijection between algebra homomorphisms \(T(M) \to A\) and linear maps \(M \to A\). Here the extension \(\bar f\) agrees with \(f\) on \(M\), and an algebra homomorphism \(g\) satisfies \(g \circ \iota = f\) exactly when \(g = \bar f\); in particular two algebra homomorphisms out of \(T(M)\) that agree on \(\iota(M)\) are equal. The universal property determines \(T(M)\) up to a unique isomorphism compatible with \(\iota\), and the direct sum description is one realization of it: the tensor algebra is isomorphic as an \(R\)-algebra to the direct sum of the tensor powers with the concatenation product.
Examples
- If \(M = 0\), every tensor power of positive degree vanishes and \(T(M) = R\).
- If \(M = R\), each \(R^{\otimes n}\) is free of rank one on \(1 \otimes \cdots \otimes 1 = t^n\), where \(t = \iota(1)\), and \(T(R)\) is the polynomial ring \(R[t]\).
- If \(M\) is free with a basis \((b_i)_{i \in I}\), then \(T(M)\) is the free algebra on \(I\), the algebra of noncommutative polynomials in variables indexed by \(I\): a basis of \(M\) gives an algebra isomorphism from \(T(M)\) to the free algebra on the index set, sending \(\iota(b_i)\) to the \(i\)-th variable. Correspondingly, \(T(M)\) has a basis indexed by the finite words in \(I\), the word \(i_1 i_2 \cdots i_n\) standing for the product \(\iota(b_{i_1}) \cdots \iota(b_{i_n})\). For a vector space \(V\) of dimension \(d\), the degree \(n\) part has dimension \(d^n\); when \(d \ge 2\), the algebra \(T(V)\) is not commutative, since for distinct indices \(i \ne j\) the products \(\iota(b_i)\iota(b_j)\) and \(\iota(b_j)\iota(b_i)\) are distinct basis elements (the words \(ij\) and \(ji\)).
- Over \(R = \mathbb{Z}\), take \(M = \mathbb{Z}/2\). Every tensor power of positive degree is again \(\mathbb{Z}/2\), generated by \(t^n\) where \(t = \iota(\bar 1)\), and \(T(M) \cong \mathbb{Z}[t]/(2t)\). Here \(2\) and \(t\) are both nonzero but \(2t = \iota(2 \cdot \bar 1) = \iota(0) = 0\), so the tensor algebra of a module that is not free can have zero divisors.
Properties
- Generation. \(T(M)\) is generated as an \(R\)-algebra by \(\iota(M)\), and for a linear map \(f : M \to A\), the image of \(\bar f\) is the subalgebra generated by \(f(M)\).
- Grading. \(T(M)\) is a graded algebra indexed by the natural numbers: it is the internal direct sum of the submodules spanned by products of \(n\) elements of \(\iota(M)\), with degrees adding under multiplication. The degree \(n\) part is the image of \(M^{\otimes n}\).
- Embeddings. The structure map \(R \to T(M)\) is injective: \(r\) and \(s\) have the same image exactly when \(r = s\). Indeed, the algebra homomorphism \(T(M) \to R\) that sends \(M\) to \(0\) is a left inverse of it. Likewise \(\iota(x) = \iota(y)\) exactly when \(x = y\), and the images of \(M\) and of \(R\) meet only in \(0\).
- Free modules. If \(M\) is a free \(R\)-module, then so is \(T(M)\). If moreover \(R\) has no zero divisors, neither does \(T(M)\), and \(T(M)\) is a domain when \(R\) is an integral domain; the last example above shows that freeness of \(M\) cannot be dropped from these two statements. For \(R\) nonzero and \(M\) free, the rank of \(T(M)\) is the cardinal sum of the powers \((\operatorname{rank} M)^n\) over \(n \ge 0\).
Related concepts
Many algebras are quotients of \(T(M)\) by a two-sided ideal generated by elements of low degree. The symmetric algebra is the quotient by the relations \(xy = yx\) for \(x, y \in M\); it is the free commutative algebra on \(M\), and for free \(M\) of finite rank it is a polynomial ring in several variables. The Clifford algebra of a quadratic form \(Q\) on \(M\) is the quotient by the relations \(x^2 = Q(x)\), and the exterior algebra is the case \(Q = 0\). For a Lie algebra \(L\) over a commutative ring, the universal enveloping algebra is the quotient of \(T(L)\) by the relations \(xy - yx = [x, y]\). The free algebra on a set \(X\) is the tensor algebra of the free module on \(X\).
In Mathlib
Mathlib defines the tensor algebra for a module over any commutative semiring, not only a commutative ring, and when the scalars form a ring the tensor algebra is automatically a ring. It is not built as a direct sum of tensor powers. Instead it is a quotient of the free algebra on the underlying set of \(M\) by the relations that make the generator map additive and compatible with scalars, which yields the universal property directly; the direct sum description and the grading are theorems proved afterwards. Accordingly the grading is expressed by powers of the image of \(M\) inside the algebra, and the comparison with the direct sum of tensor powers is a separate algebra isomorphism. The universal property is packaged as a bijection between linear maps and algebra homomorphisms, with targets any associative algebra, commutative or not, over the same semiring. The statements about bases, rank and zero divisors carry the freeness and nontriviality hypotheses given above, and the rank is stated for a commutative ring.
Written for Trough, under CC BY-SA 4.0.
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ClassLie algebra#
A Lie algebra is a module over a commutative ring, typically a vector space over a field, equipped with a bilinear product, the bracket, which vanishes on equal arguments and satisfies the Jacobi identity. The bracket measures how far two operations are from commuting. Lie algebras arise as the infinitesimal versions of Lie groups, as algebras of derivations and vector fields, and from any associative algebra through the commutator, and they are used to study symmetry in representation theory, differential geometry and physics.
Definition
Let \(R\) be a commutative ring. A Lie algebra over \(R\) is an \(R\)-module \(L\) together with an \(R\)-bilinear map (a map linear in each argument) \(L \times L \to L\), written \((x, y) \mapsto [x, y]\), such that for all \(x, y, z \in L\)
The second condition is the Jacobi identity. Applying the first condition to \(x + y\) shows that the bracket is antisymmetric: \([x, y] = -[y, x]\) in every Lie algebra. When \(2\) is invertible in \(R\), antisymmetry conversely implies \([x, x] = 0\), but over a ring of characteristic \(2\) the alternating condition is the stronger one. In a Lie algebra Mathlib records that every element has bracket zero with itself and the Jacobi identity in the cyclic form above, and that scalars pull out of the first argument and the second.
A Lie algebra homomorphism is a linear map \(\varphi\) with \(\varphi([x, y]) = [\varphi(x), \varphi(y)]\). A Lie subalgebra is a submodule closed under the bracket, and an ideal is a submodule \(I\) with \([x, y] \in I\) whenever \(x \in L\) and \(y \in I\).
Examples
- Commutators. Every associative algebra \(A\) over \(R\) becomes a Lie algebra with the bracket \([x, y] = xy - yx\). Taking \(A\) to be the algebra of endomorphisms of a module \(V\) gives the general linear Lie algebra \(\mathfrak{gl}(V)\); taking the \(n \times n\) matrices gives \(\mathfrak{gl}_n(R)\).
- Abelian Lie algebras. A Lie algebra is abelian when every bracket is zero. Any module with the zero bracket is one, and since two elements of a ring commute exactly when their commutator vanishes, a commutative algebra gives an abelian Lie algebra.
- Classical matrix Lie algebras. Because the trace of a commutator of square matrices is zero, the matrices of trace zero form a Lie subalgebra \(\mathfrak{sl}_n(R)\) of \(\mathfrak{gl}_n(R)\). Likewise the orthogonal Lie algebra \(\mathfrak{so}_n(R)\) consists of the matrices \(A\) with \(A^{\mathsf T} = -A\).
- The cross product. On \(R^3\) the cross product is bilinear, satisfies \(v \times v = 0\) (hence is antisymmetric) and satisfies the Jacobi identity, so it makes \(R^3\) a Lie algebra; over the real numbers it is isomorphic to \(\mathfrak{so}_3(\mathbb{R})\).
- Derivations. For a commutative \(R\)-algebra \(A\), a derivation is an \(R\)-linear map \(D \colon A \to A\) with \(D(ab) = a\,D(b) + D(a)\,b\). The commutator \(D_1 D_2 - D_2 D_1\) of two derivations is again a derivation, and the derivations of \(A\) form a Lie algebra over \(R\). In geometry, the smooth vector fields on a manifold form a Lie algebra under the Lie bracket of vector fields.
- Lie groups. The tangent space at the identity of a Lie group carries a natural bracket, giving the Lie algebra of the Lie group.
Properties
- The adjoint representation. Each \(x \in L\) defines a linear map \(\operatorname{ad} x \colon L \to L\) by \(\operatorname{ad} x\,(y) = [x, y]\). Together with antisymmetry, the Jacobi identity says that \(\operatorname{ad}[x, y] = \operatorname{ad} x \circ \operatorname{ad} y - \operatorname{ad} y \circ \operatorname{ad} x\), that is, \(\operatorname{ad}\) is a Lie algebra homomorphism from \(L\) to \(\mathfrak{gl}(L)\).
- Solvable and nilpotent Lie algebras. The derived series \(L \supseteq [L, L] \supseteq [[L, L], [L, L]] \supseteq \cdots\) defines solvable Lie algebras, those whose derived series reaches \(0\), and the lower central series \(L \supseteq [L, L] \supseteq [L, [L, L]] \supseteq \cdots\) defines nilpotent Lie algebras in the same way. Every nilpotent Lie algebra is solvable.
- Engel's theorem. A finite-dimensional Lie algebra \(L\) over a field is nilpotent if and only if \(\operatorname{ad} x\) is a nilpotent linear map for every \(x \in L\).
- Representations. A representation of \(L\) on an \(R\)-module \(V\) is a Lie algebra homomorphism \(L \to \mathfrak{gl}(V)\); the adjoint representation is the case \(V = L\).
- Lie's theorem. Over an algebraically closed field of characteristic zero, every nonzero finite-dimensional representation of a finite-dimensional solvable Lie algebra has a common eigenvector.
- Simple Lie algebras. A Lie algebra is simple when it is not abelian and its only ideals are \(0\) and \(L\). For \(n \ge 2\), the algebra \(\mathfrak{sl}_n(\mathbb{C})\) is simple.
- The universal enveloping algebra. Every Lie algebra \(L\) has a universal enveloping algebra \(U(L)\), an associative algebra with a Lie algebra homomorphism \(\iota \colon L \to U(L)\) such that every Lie algebra homomorphism from \(L\) into an associative algebra \(A\) factors as \(F \circ \iota\) for a unique algebra homomorphism \(F \colon U(L) \to A\); that is, composition with \(\iota\) is a bijection between these two sets of homomorphisms. Representations of \(L\) are therefore the same as modules over \(U(L)\).
In Mathlib
As in the definition above, Mathlib works over an arbitrary commutative ring, and it splits the definition in two. A Lie ring is an additive commutative group with a biadditive bracket that vanishes on equal arguments and satisfies the Jacobi identity in the form \([x, [y, z]] = [[x, y], z] + [y, [x, z]]\), which says that each \(\operatorname{ad} x\) acts as a derivation of the bracket. A Lie algebra is then a Lie ring with a module structure over \(R\), with the single extra axiom \([x, t y] = t[x, y]\); compatibility with scalars in the first argument follows from antisymmetry. Ideals are treated as submodules of \(L\) regarded as a representation of itself through the adjoint action. Neither the commutator Lie ring of an associative ring nor the cross-product Lie ring on \(R^3\) is attached automatically; each is a construction switched on where it is needed, since on \(R^3\), which is itself a ring of functions, the two brackets would conflict. Engel's theorem is proved more generally over any commutative ring, for Lie algebras that are Noetherian as modules, which includes the finite-dimensional case over a field.
Written for Trough, under CC BY-SA 4.0.
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AbbreviationExterior algebra#
The exterior algebra of a module \(M\) over a commutative ring \(R\), written \(\Lambda(M)\), is the associative algebra generated by \(M\) subject only to the rule that every element of \(M\) squares to zero. Its product, the wedge product \(\wedge\), is the algebraic form of oriented area and volume: in an \(n\)-dimensional space with basis \(e_1, \dots, e_n\), the wedge of \(n\) vectors is their determinant in that basis times \(e_1 \wedge \cdots \wedge e_n\). The exterior algebra underlies the theory of determinants, alternating multilinear forms and differential forms, and it is the simplest case of a Clifford algebra.
Definition
Let \(R\) be a commutative ring and \(M\) an \(R\)-module. The exterior algebra of \(M\) is the quotient of the tensor algebra \(T(M)\) by the two-sided ideal generated by the elements \(x \otimes x\) for \(x \in M\). It comes with an \(R\)-linear map \(\iota : M \to \Lambda(M)\), and \(\iota\) is injective, so \(M\) is usually identified with its image. The product in \(\Lambda(M)\) is written \(\wedge\), and by construction \(x \wedge x = 0\) for every \(x \in M\).
Equivalently, \(\Lambda(M)\) is characterized by a universal property. Let \(A\) be an associative \(R\)-algebra with identity and \(f : M \to A\) a linear map with \(f(x)^2 = 0\) for all \(x \in M\). Then there is exactly one homomorphism of \(R\)-algebras \(\bar f : \Lambda(M) \to A\) with \(\bar f \circ \iota = f\): composing with \(\iota\) gives a bijection between algebra homomorphisms \(\Lambda(M) \to A\) and linear maps \(M \to A\) whose values square to zero, and an algebra homomorphism \(g\) satisfies \(g \circ \iota = f\) exactly when \(g = \bar f\). The canonical map from \(T(M)\) is the algebra homomorphism that is the identity on \(M\).
For \(n \ge 0\) the \(n\)-th exterior power \(\Lambda^n(M)\) is the submodule of \(\Lambda(M)\) consisting of the \(R\)-linear combinations of products of \(n\) elements of \(M\) (for \(n = 0\), the multiples of \(1\)); it is spanned by the products \(m_1 \wedge \cdots \wedge m_n\). Here \(\Lambda^0(M)\) is the image of \(R\) (a copy of \(R\), since \(R \to \Lambda(M)\) is injective) and \(\Lambda^1(M)\) is the image of \(M\).
Examples
- If \(M = R\), then \(\Lambda(R) = R \oplus R\,e\) with \(e = \iota(1)\) and \(e \wedge e = 0\): the ring of dual numbers \(R[\varepsilon]/(\varepsilon^2)\).
- If \(M = R^2\) with standard basis \(e_1, e_2\), then \(\Lambda(M)\) is free on \(1, e_1, e_2, e_1 \wedge e_2\), and for vectors \(u = a e_1 + c e_2\) and \(v = b e_1 + d e_2\),
$\(u \wedge v = ad\, e_1 \wedge e_2 + cb\, e_2 \wedge e_1 = (ad - bc)\, e_1 \wedge e_2,\)$
the determinant of the matrix with columns \(u\) and \(v\) times \(e_1 \wedge e_2\). - If \(M = R^3\), the coefficients of \(u \wedge v\) on \(e_2 \wedge e_3\), \(e_3 \wedge e_1\) and \(e_1 \wedge e_2\) are the components of the cross product \(u \times v\), and \(u \wedge v \wedge w\) is the determinant of \(u, v, w\) times \(e_1 \wedge e_2 \wedge e_3\). For \(R\) nonzero the exterior powers are free of ranks \(1, 3, 3, 1\).
Properties
- Anticommutativity. Expanding \((x + y) \wedge (x + y) = 0\) gives \(x \wedge y + y \wedge x = 0\) for all \(x, y \in M\). Imposing only the relations \(x \wedge y + y \wedge x = 0\) does not force \(x \wedge x = 0\) in general when \(2\) is not invertible in \(R\): over a field of characteristic \(2\) those relations give the symmetric algebra of a nonzero \(M\), a polynomial ring, in which \(x^2 \neq 0\) for \(x \neq 0\).
- Alternating maps. The map \((m_1, \dots, m_n) \mapsto m_1 \wedge \cdots \wedge m_n\) is multilinear and alternating: it vanishes whenever two of the \(m_i\) are equal. It is universal for this property: for every \(R\)-module \(N\), linear maps \(\Lambda(M) \to N\) correspond to families, one for each \(n\), of alternating \(n\)-linear maps from \(M\) to \(N\). (See alternating map.)
- Grading. \(\Lambda(M)\) is the internal direct sum of the exterior powers \(\Lambda^n(M)\), with degrees adding under multiplication. It is graded commutative: if \(\alpha\) is a product of \(i\) elements of \(M\) and \(\beta\) a product of \(j\) elements, then \(\alpha \wedge \beta = (-1)^{ij}\, \beta \wedge \alpha\).
- Bases and ranks. Let \(M\) be free with a basis \((e_i)_{i \in I}\), and order \(I\) totally. For a finite subset \(S = \{i_1 < \cdots < i_k\}\) put \(e_S = e_{i_1} \wedge \cdots \wedge e_{i_k}\), with \(e_\varnothing = 1\). Then the \(e_S\) form a basis of \(\Lambda(M)\), and the \(e_S\) with \(|S| = n\) form a basis of \(\Lambda^n(M)\). If \(S\) and \(T\) share an element, \(e_S \wedge e_T = 0\). Consequently, if \(R\) is nonzero and \(M\) is free of finite rank \(r\), then \(\Lambda^n(M)\) has rank \(\binom{r}{n}\), so \(\Lambda^n(M) = 0\) for \(n > r\) and \(\Lambda(M)\) has rank \(2^r\).
- Functoriality. A linear map \(f : M \to N\) induces an algebra homomorphism \(\Lambda(f) : \Lambda(M) \to \Lambda(N)\) extending \(f\), with \(\Lambda(f)(m_1 \wedge \cdots \wedge m_n) = f(m_1) \wedge \cdots \wedge f(m_n)\) and \(\Lambda(g) \circ \Lambda(f) = \Lambda(g \circ f)\). \(\Lambda(f)\) is surjective exactly when \(f\) is, and for vector spaces over a field, \(\Lambda(f)\) is injective when \(f\) is.
- Determinants. If \(V\) is a vector space of dimension \(n\), then \(\Lambda^n(V)\) is one-dimensional, so an endomorphism \(T\) of \(V\) acts on it as multiplication by a scalar; that scalar is the determinant of \(T\). Similarly, \(n\) vectors form a basis of \(V\) exactly when their wedge is nonzero; more generally, vectors \(v_1, \dots, v_k\) in a vector space are linearly independent exactly when \(v_1 \wedge \cdots \wedge v_k \neq 0\).
Related concepts
The Clifford algebra of a quadratic form \(Q\) on \(M\) is the quotient of \(T(M)\) by the relations \(x \otimes x = Q(x)\); the exterior algebra is the case \(Q = 0\). Replacing \(x \otimes x\) by the relations \(x \otimes y = y \otimes x\) gives the symmetric algebra instead, a commutative analogue whose degree \(n\) part is spanned by the products \(m_1 \cdots m_n\), now symmetric rather than alternating in their factors. In geometry, the differential forms on a smooth manifold are built pointwise from the exterior algebra of the dual of each tangent space.
In Mathlib
Mathlib defines the exterior algebra of a module over any commutative ring, with no freeness or finiteness assumption on the module, as the Clifford algebra of the zero quadratic form, so that the universal property, the induction principle and functoriality are inherited from the Clifford algebra. The \(n\)-th exterior power is not a separate module but a submodule of the exterior algebra: the \(n\)-th power, in the sense of products of submodules, of the image of \(M\). The universal property is packaged as a bijection between linear maps whose values square to zero and algebra homomorphisms into any associative algebra, commutative or not. The bases require a total order on the index set, used to list each finite subset in increasing order, and the rank formula assumes a nonzero ring and a free module of finite rank.
Written for Trough, under CC BY-SA 4.0.
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Uses 4
DefinitionClifford algebra#
The Clifford algebra of a module \(M\) over a commutative ring \(R\), equipped with a quadratic form \(Q\), is the associative \(R\)-algebra generated by \(M\) subject only to the rule that every vector \(v\) squares to the scalar \(Q(v)\): it is the universal associative algebra in which each vector squares to the value of the quadratic form. The complex numbers, the quaternions, the dual numbers and the exterior algebra all arise as Clifford algebras, and the construction underlies spin groups and spinors in geometry, topology and physics.
Definition
Let \(R\) be a commutative ring, \(M\) an \(R\)-module and \(Q \colon M \to R\) a quadratic form. The Clifford algebra \(\mathrm{Cl}(M, Q)\) is the quotient of the tensor algebra \(T(M)\) by the two-sided ideal generated by the elements \(v \otimes v - Q(v)\cdot 1\) for \(v \in M\) (the Clifford algebra of \(Q\)). Composing the inclusion of \(M\) into \(T(M)\) with the quotient map gives a linear map \(\iota \colon M \to \mathrm{Cl}(M, Q)\), and by construction \(\iota(v)^2 = Q(v)\cdot 1\) for every \(v \in M\).
The algebra is characterized by a universal property. Let \(A\) be an associative \(R\)-algebra and \(f \colon M \to A\) a linear map with \(f(v)^2 = Q(v)\cdot 1\) for all \(v\). Then there is exactly one algebra homomorphism \(\varphi \colon \mathrm{Cl}(M, Q) \to A\) with \(\varphi \circ \iota = f\); in other words, restriction along \(\iota\) is a bijection between algebra homomorphisms out of \(\mathrm{Cl}(M, Q)\) and such linear maps. This determines \(\mathrm{Cl}(M, Q)\) up to unique isomorphism, and the algebra is generated by the image of \(\iota\).
Replacing \(v\) by \(u + v\) in the defining relation gives
so the symmetrized product of two vectors is the scalar given by the polar form of \(Q\) (see polar form). In particular, if \(Q(u+v) = Q(u) + Q(v)\), that is, if \(u\) and \(v\) are orthogonal, then \(\iota(u)\,\iota(v) = -\iota(v)\,\iota(u)\).
When \(M\) is free with basis \(e_1, \dots, e_n\), the ordered products \(e_{i_1} e_{i_2} \cdots e_{i_k}\) with \(i_1 < i_2 < \dots < i_k\), the empty product \(1\) included, form a basis of \(\mathrm{Cl}(M, Q)\) over \(R\), so it is free of rank \(2^n\). If moreover \(Q(x_1 e_1 + \dots + x_n e_n) = a_1 x_1^2 + \dots + a_n x_n^2\), the multiplication is determined by \(e_i^2 = a_i\) and \(e_i e_j = -e_j e_i\) for \(i \neq j\).
Examples
- Zero form. For \(Q = 0\) the relations say that every vector squares to zero, and the result is the exterior algebra; Mathlib defines the exterior algebra of \(M\) as the Clifford algebra of the zero form.
- Zero module. When \(M = 0\) only the scalars remain: the Clifford algebra of the zero module is \(R\) itself.
- Complex numbers. Take \(M = \mathbb{R}\) and \(Q(x) = -x^2\). The element \(i = \iota(1)\) satisfies \(i^2 = -1\), and the Clifford algebra is isomorphic to \(\mathbb{C}\) as a real algebra.
- Quaternions. Take \(M = R^2\) and \(Q(x, y) = c_1 x^2 + c_2 y^2\). Then \(i = \iota(1, 0)\) and \(j = \iota(0, 1)\) satisfy \(i^2 = c_1\), \(j^2 = c_2\) and \(ij = -ji\), and the Clifford algebra is isomorphic to the quaternion algebra with these parameters. Over the real numbers with \(c_1 = c_2 = -1\) this gives Hamilton's quaternions.
- Dual numbers. Take \(M = R\) and \(Q = 0\), the zero form on a module of rank one, so this is the exterior algebra of \(R\). Then the Clifford algebra is the algebra of dual numbers \(R[\varepsilon]\) with \(\varepsilon^2 = 0\).
Properties
- Functoriality. A linear map \(g \colon M_1 \to M_2\) with \(Q_2 \circ g = Q_1\) is an isometry, and every isometry induces an algebra homomorphism of Clifford algebras sending \(\iota(v)\) to \(\iota(g(v))\); an isometric isomorphism induces an algebra isomorphism.
- Even and odd parts. Products of an even number of vectors span the even part, and products of an odd number span the odd part. The algebra is the direct sum of these two submodules, and multiplication adds parities, making \(\mathrm{Cl}(M, Q)\) a \(\mathbb{Z}/2\)-graded algebra. The even part is therefore a subalgebra.
- One dimension up. Let \(Q'\) be the quadratic form \(Q'(v, r) = Q(v) - r^2\) on \(M \oplus R\). Then \(\mathrm{Cl}(M, Q)\) is isomorphic to the even subalgebra of \(\mathrm{Cl}(M \oplus R, Q')\). Also, the even subalgebras for \(Q\) and for \(-Q\) are isomorphic.
- Involutions. The grade involution is the algebra homomorphism from \(\mathrm{Cl}(M, Q)\) to itself that sends each vector \(v\) to \(-v\); it fixes the even part and negates the odd part. Reversion is the linear map that fixes vectors and reverses the order of products, sending \(ab\) to the reverse of \(b\) times the reverse of \(a\).
- Relation to the exterior algebra. If \(2\) is invertible in \(R\), \(\mathrm{Cl}(M, Q)\) is isomorphic as an \(R\)-module to the exterior algebra of \(M\). In general this is not an isomorphism of algebras: for \(M = \mathbb{R}\) and \(Q(x) = -x^2\) it matches \(\mathbb{C}\) with the dual numbers, which are not isomorphic as rings.
- Orthogonal sums. For quadratic forms \(Q_1\) on \(M_1\) and \(Q_2\) on \(M_2\), let \(Q_1 \oplus Q_2\) be the form \((m_1, m_2) \mapsto Q_1(m_1) + Q_2(m_2)\). Then \(\mathrm{Cl}(M_1 \oplus M_2, Q_1 \oplus Q_2)\) is isomorphic to the graded tensor product of \(\mathrm{Cl}(M_1, Q_1)\) and \(\mathrm{Cl}(M_2, Q_2)\), in which odd elements of the two factors anticommute.
Related concepts
The invertible products of vectors in a Clifford algebra act on \(M\) by twisted conjugation through isometries, which leads to the pin and spin groups. For a definite form on \(\mathbb{R}^n\) the pin group, generated by the unit vectors, and the spin group, its even part, are double covers of the orthogonal group \(\mathrm{O}(n)\) and the special orthogonal group \(\mathrm{SO}(n)\), and their representations give the spinors. In Mathlib the pin group consists of the elements of the group generated by invertible vectors that are unitary for Clifford conjugation (the composite of reversion and the grade involution), and the spin group is its intersection with the even subalgebra. A vector \(v\) is unitary in this sense only when \(Q(v) = -1\), so this definition matches the textbook pin group for a negative definite form; for a positive definite form it keeps only products of an even number of vectors, and the pin group coincides with the spin group. The Clifford algebra is a quotient of the tensor algebra, which imposes no relations, and specializes to the exterior algebra when \(Q = 0\).
In Mathlib
Mathlib defines the Clifford algebra for any commutative ring \(R\) and any module \(M\), with no assumption that \(2\) is invertible. A quadratic form there is a map \(Q\) with \(Q(av) = a^2 Q(v)\) whose polar form is bilinear; it need not come from a symmetric bilinear form, which matters in characteristic \(2\). The algebra is built as the quotient of the tensor algebra by the smallest congruence that identifies the square of each vector with \(Q(v)\), which is the same as the quotient by the two-sided ideal above. The universal property is packaged as a single bijection rather than as separate existence and uniqueness statements, and its target may be any \(R\)-algebra. Mathlib asks only that the target be a semiring with an \(R\)-algebra structure; negation then comes for free, as multiplication by the image of \(-1\). The exterior algebra is not built separately: it is the Clifford algebra of the zero form, so results about Clifford algebras apply to it directly. Results that use a hypothesis on \(2\) state it: the module isomorphism with the exterior algebra, and Mathlib's proof that the algebra is nontrivial over a nontrivial ring, both assume that \(2\) is invertible.
Written for Trough, under CC BY-SA 4.0.
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Uses 5
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