Symmetric algebra
Encyclopedia
In mathematics
, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space
V over a field
K is the free
commutative unital associative algebra
over K containing V.
It corresponds to polynomials with indeterminates in V, without choosing coordinates.
The dual,
corresponds to polynomials on V.
It should not be confused with symmetric tensors in V. A Frobenius algebra
whose bilinear form is symmetric
is also called a symmetric algebra, but is not discussed here.
, over K, in indeterminates that are basis vectors for V. Therefore this construction only brings something extra when the "naturality" of looking at polynomials this way has some advantage.
It is possible to use the tensor algebra
T(V) to describe the symmetric algebra S(V). In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative; if elements of V commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra
of T(V) by the ideal
generated by all differences of products
for v and w in V.
decomposition of S(V) as a graded algebra
, into summands
which consist of the linear span of the monomial
s in vectors of V of degree k, for k = 0, 1, 2, ... (with S0(V) = K and S1(V)=V). The K-vector space Sk(V) is the k-th symmetric power of V. The case k = 2, for example, is the symmetric square and denoted Sym2(V). It has a universal property with respect to symmetric multilinear operators defined on Vk.
The symmetric algebra must be a quotient to satisfy its universal property
(since every symmetric algebra is an algebra, the tensor algebra maps to the symmetric algebra).
Conversely, symmetric tensors are defined as invariants: given the natural action of the symmetric group
on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially. Note that under the tensor product, symmetric tensors are not a subalgebra: given vectors v and w, they are trivially symmetric 1-tensors, but
is not a symmetric 2-tensor.
The grade 2 part of this distinction is the difference between symmetric bilinear form
s (symmetric 2-tensors) and quadratic form
s (elements of the symmetric square), as described in ε-quadratic forms.
In characteristic 0 symmetric tensors and the symmetric algebra can be identified. In any characteristic, there is a symmetrization
map from the symmetric algebra to the symmetric tensors, given by:
The composition with the inclusion of the symmetric tensors in the tensor algebra and the quotient to the symmetric algebra is multiplication by
on the kth graded component.
Thus in characteristic 0, the symmetrization map is an isomorphism of graded vector spaces, and one can identify symmetric tensors with elements of the symmetric algebra. One divides by
to make this a section
of the quotient map:
For instance,
.
This is related to the representation theory
of the symmetric group:
in characteristic 0, over an algebraically closed field, the group algebra
is semisimple
, so every representation splits into a direct sum of irreducible representations, and if
, one can identify S as either a subspace of T or as the quotient T/V.
, the symmetric algebra of the dual space: a polynomial on a space evaluates vectors on the space, via the pairing 
.
For instance, given the plane with a basis
, 
, the (homogeneous) linear polynomials on 
are generated by the coordinate functional
s x and y. These coordinates are covectors: given a vector, they evaluate to their coordinate, for instance:
Given monomials of higher degree, these are elements of various symmetric powers, and a general polynomial is an element of the symmetric algebra. Without a choice of basis for the vector space, the same holds, but one has a polynomial algebra without choice of basis.
Conversely, the symmetric algebra of the vector space itself can be interpreted, not as polynomials on the vector space (since one cannot evaluate an element of the symmetric algebra of a vector space against a vector in that space: there is no pairing between
and 
), but polynomials in the vectors, such as 
.
(or its dual, which corresponds to polynomials on that affine space).
The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered algebra
: one can determine the degree of a polynomial on an affine space, but not its homogeneous parts.
For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0. On an affine space, there is no distinguished point, so one cannot do this (choosing a point turns an affine space into a vector space).
in the category of commutative unital associative algebras (in the sequel, "commutative algebras").
Formally, the map that sends a vector space to its symmetric algebra is a functor
from vector spaces over K to commutative algebras over K,
and is a free object, meaning that it is left adjoint
to the
forgetful functor
that sends a commutative algebra to its underlying vector space.
The unit of the adjunction is the map
that embeds a vector space in its symmetric algebra.
Commutative algebras are a reflective subcategory
of algebras;
given an algebra A, one can quotient out by its commutator ideal generated by
, obtaining a commutative algebra, analogously to abelianization of a group. The construction of the symmetric algebra as a quotient of the tensor algebra is, as functors, a composition of the free algebra functor with this reflection.
s comparable to the exterior powers; here, though, the dimension grows with k; it is given by
where n is the dimension of V. This binomial coefficient is the number of n-variable monomials of degree k.
M over a commutative ring
. If M is a free module
over the ring R, then its symmetric algebra is isomorphic to the polynomial algebra over R whose indeterminates are a basis of M, just like the symmetric algebra of a vector space. However, that is not true if M is not free; then S(M) is more complicated.
of an abelian Lie algebra, i.e. one in which the Lie bracket is identically 0.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K is the free
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....
commutative unital associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
over K containing V.
It corresponds to polynomials with indeterminates in V, without choosing coordinates.
The dual,
corresponds to polynomials on V.It should not be confused with symmetric tensors in V. A Frobenius algebra
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in...
whose bilinear form is symmetric
Symmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
is also called a symmetric algebra, but is not discussed here.
Construction
It turns out that S(V) is in effect the same as the polynomial ringPolynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
, over K, in indeterminates that are basis vectors for V. Therefore this construction only brings something extra when the "naturality" of looking at polynomials this way has some advantage.
It is possible to use the tensor algebra
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
T(V) to describe the symmetric algebra S(V). In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative; if elements of V commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra
Quotient algebra
In mathematics, a quotient algebra, , also called a factor algebra is obtained by partitioning the elements of an algebra in equivalence classes given by a congruence, that is an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense...
of T(V) by the ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
generated by all differences of products
for v and w in V.
Grading
Just as with a polynomial ring, there is a direct sumDirect sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
decomposition of S(V) as a graded algebra
Graded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
, into summands
- Sk(V)
which consist of the linear span of the monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
s in vectors of V of degree k, for k = 0, 1, 2, ... (with S0(V) = K and S1(V)=V). The K-vector space Sk(V) is the k-th symmetric power of V. The case k = 2, for example, is the symmetric square and denoted Sym2(V). It has a universal property with respect to symmetric multilinear operators defined on Vk.
Distinction with symmetric tensors
The symmetric algebra and symmetric tensors are easily confused: the symmetric algebra is a quotient of the tensor algebra, while the symmetric tensors are a subspace of the tensor algebra.The symmetric algebra must be a quotient to satisfy its universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
(since every symmetric algebra is an algebra, the tensor algebra maps to the symmetric algebra).
Conversely, symmetric tensors are defined as invariants: given the natural action of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially. Note that under the tensor product, symmetric tensors are not a subalgebra: given vectors v and w, they are trivially symmetric 1-tensors, but
is not a symmetric 2-tensor.The grade 2 part of this distinction is the difference between symmetric bilinear form
Symmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
s (symmetric 2-tensors) and quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s (elements of the symmetric square), as described in ε-quadratic forms.
In characteristic 0 symmetric tensors and the symmetric algebra can be identified. In any characteristic, there is a symmetrization
Symmetrization
In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables.Conversely, anti-symmetrization converts any function in n variables into an antisymmetric function.-2 variables:...
map from the symmetric algebra to the symmetric tensors, given by:
The composition with the inclusion of the symmetric tensors in the tensor algebra and the quotient to the symmetric algebra is multiplication by
on the kth graded component.Thus in characteristic 0, the symmetrization map is an isomorphism of graded vector spaces, and one can identify symmetric tensors with elements of the symmetric algebra. One divides by
to make this a sectionSection (category theory)
In category theory, a branch of mathematics, a section is a right inverse of a morphism. Dually, a retraction is a left inverse...
of the quotient map:
For instance,
.This is related to the representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
of the symmetric group:
in characteristic 0, over an algebraically closed field, the group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
is semisimple
Semisimple algebra
In ring theory, a semisimple algebra is an associative algebra which has trivial Jacobson radical...
, so every representation splits into a direct sum of irreducible representations, and if
, one can identify S as either a subspace of T or as the quotient T/V.Interpretation as polynomials
Given a vector space V, the polynomials on this space are, the symmetric algebra of the dual space: a polynomial on a space evaluates vectors on the space, via the pairing .For instance, given the plane with a basis
, , the (homogeneous) linear polynomials on are generated by the coordinate functionalFunctional
Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
s x and y. These coordinates are covectors: given a vector, they evaluate to their coordinate, for instance:
Given monomials of higher degree, these are elements of various symmetric powers, and a general polynomial is an element of the symmetric algebra. Without a choice of basis for the vector space, the same holds, but one has a polynomial algebra without choice of basis.
Conversely, the symmetric algebra of the vector space itself can be interpreted, not as polynomials on the vector space (since one cannot evaluate an element of the symmetric algebra of a vector space against a vector in that space: there is no pairing between
and ), but polynomials in the vectors, such as .Symmetric algebra of an affine space
One can analogously construct the symmetric algebra on an affine spaceAffine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
(or its dual, which corresponds to polynomials on that affine space).
The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered algebra
Filtered algebra
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory....
: one can determine the degree of a polynomial on an affine space, but not its homogeneous parts.
For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0. On an affine space, there is no distinguished point, so one cannot do this (choosing a point turns an affine space into a vector space).
Categorical properties
The symmetric algebra on a vector space is a free objectFree object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....
in the category of commutative unital associative algebras (in the sequel, "commutative algebras").
Formally, the map that sends a vector space to its symmetric algebra is a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from vector spaces over K to commutative algebras over K,
and is a free object, meaning that it is left adjoint
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
to the
forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...
that sends a commutative algebra to its underlying vector space.
The unit of the adjunction is the map
that embeds a vector space in its symmetric algebra.Commutative algebras are a reflective subcategory
Reflective subcategory
In mathematics, a subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector...
of algebras;
given an algebra A, one can quotient out by its commutator ideal generated by
, obtaining a commutative algebra, analogously to abelianization of a group. The construction of the symmetric algebra as a quotient of the tensor algebra is, as functors, a composition of the free algebra functor with this reflection.Analogy with exterior algebra
The Sk are functorFunctor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
s comparable to the exterior powers; here, though, the dimension grows with k; it is given by
where n is the dimension of V. This binomial coefficient is the number of n-variable monomials of degree k.
Module analog
The construction of the symmetric algebra generalizes to the symmetric algebra S(M) of a moduleModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
M over a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
. If M is a free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
over the ring R, then its symmetric algebra is isomorphic to the polynomial algebra over R whose indeterminates are a basis of M, just like the symmetric algebra of a vector space. However, that is not true if M is not free; then S(M) is more complicated.
As a universal enveloping algebra
The symmetric algebra S(V) is the universal enveloping algebraUniversal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
of an abelian Lie algebra, i.e. one in which the Lie bracket is identically 0.
See also
- exterior algebraExterior algebraIn mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
, the anti-symmetric analog - Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form
- Clifford algebraClifford algebraIn mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
, a quantum deformationQuantization (physics)In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of the exterior algebra by a quadratic formQuadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....