In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the tensor algebra of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'', denoted ''T''(''V'') or ''T''(''V''), is the
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
of
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s on ''V'' (of any rank) with multiplication being the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
. It is the
free algebra on ''V'', in the sense of being
left adjoint to the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
(see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
).
The tensor algebra is important because many other algebras arise as
quotient algebras of ''T''(''V''). These include the
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
, the
symmetric algebra,
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
s, the
Weyl algebra and
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
s.
The tensor algebra also has two
coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a
cofree coalgebra, and a more complicated one, which yields a
bialgebra, and can be extended by giving an antipode to create a
Hopf algebra Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
* Heinz Hopf (actor) (1934–2001), Swe ...
structure.
''Note'': In this article, all algebras are assumed to be
unital and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
. The unit is explicitly required to define the coproduct.
Construction
Let ''V'' be a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a
field ''K''. For any nonnegative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''k'', we define the ''k''th tensor power of ''V'' to be the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of ''V'' with itself ''k'' times:
:
That is, ''T''
''k''''V'' consists of all tensors on ''V'' of
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
''k''. By convention ''T''
0''V'' is the
ground field ''K'' (as a one-dimensional vector space over itself).
We then construct ''T''(''V'') as the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of ''T''
''k''''V'' for ''k'' = 0,1,2,…
:
The multiplication in ''T''(''V'') is determined by the canonical isomorphism
:
given by the tensor product, which is then extended by linearity to all of ''T''(''V''). This multiplication rule implies that the tensor algebra ''T''(''V'') is naturally a
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
with ''T''
''k''''V'' serving as the grade-''k'' subspace. This grading can be extended to a Z grading by appending subspaces
for negative integers ''k''.
The construction generalizes in a straightforward manner to the tensor algebra of any
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
''M'' over a
''commutative'' ring. If ''R'' is a
non-commutative ring, one can still perform the construction for any ''R''-''R''
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...
''M''. (It does not work for ordinary ''R''-modules because the iterated tensor products cannot be formed.)
Adjunction and universal property
The tensor algebra is also called the
free algebra on the vector space , and is
functorial; this means that the map
extends to
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s for forming a ''functor'' from the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of -vector spaces to the category of
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
. Similarly with other
free constructions, the functor is
left adjoint to the
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
that sends each associative -algebra to its underlying vector space.
Explicitly, the tensor algebra satisfies the following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
, which formally expresses the statement that it is the most general algebra containing ''V'':
: Any
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
from to an associative algebra over can be uniquely extended to an
algebra homomorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in ,
* F(kx) = kF( ...
from to as indicated by the following
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
:
Here is the
canonical inclusion of into . As for other universal properties, the tensor algebra can be defined as the unique algebra satisfying this property (specifically, it is unique
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
a unique isomorphism), but this definition requires to prove that an object satisfying this property exists.
The above universal property implies that is a
functor from the
category of vector spaces over , to the category of -algebras. This means that any linear map between -vector spaces and extends uniquely to a -algebra homomorphism from to .
Non-commutative polynomials
If ''V'' has finite dimension ''n'', another way of looking at the tensor algebra is as the "algebra of polynomials over ''K'' in ''n'' non-commuting variables". If we take
basis vectors for ''V'', those become non-commuting variables (or
''indeterminates'') in ''T''(''V''), subject to no constraints beyond
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
, the
distributive law and ''K''-linearity.
Note that the algebra of polynomials on ''V'' is not
, but rather
: a (homogeneous) linear function on ''V'' is an element of
for example coordinates
on a vector space are
covectors, as they take in a vector and give out a scalar (the given coordinate of the vector).
Quotients
Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i.e. by constructing certain
quotient algebras of ''T''(''V''). Examples of this are the
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
, the
symmetric algebra,
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperco ...
s, the
Weyl algebra and
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
s.
Coalgebra
The tensor algebra has two different
coalgebra structures. One is compatible with the tensor product, and thus can be extended to a
bialgebra, and can be further be extended with an antipode to a
Hopf algebra Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
* Heinz Hopf (actor) (1934–2001), Swe ...
structure. The other structure, although simpler, cannot be extended to a bialgebra. The first structure is developed immediately below; the second structure is given in the section on the
cofree coalgebra, further down.
The development provided below can be equally well applied to the
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
, using the wedge symbol
in place of the tensor symbol
; a sign must also be kept track of, when permuting elements of the exterior algebra. This correspondence also lasts through the definition of the bialgebra, and on to the definition of a Hopf algebra. That is, the exterior algebra can also be given a Hopf algebra structure.
Similarly, the
symmetric algebra can also be given the structure of a Hopf algebra, in exactly the same fashion, by replacing everywhere the tensor product
by the symmetrized tensor product
, i.e. that product where
In each case, this is possible because the alternating product
and the symmetric product
obey the required consistency conditions for the definition of a bialgebra and Hopf algebra; this can be explicitly checked in the manner below. Whenever one has a product obeying these consistency conditions, the construction goes through; insofar as such a product gave rise to a quotient space, the quotient space inherits the Hopf algebra structure.
In the language of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, one says that there is a
functor from the category of -vector spaces to the category of -associate algebras. But there is also a functor taking vector spaces to the category of exterior algebras, and a functor taking vector spaces to symmetric algebras. There is a
natural map from to each of these. Verifying that quotienting preserves the Hopf algebra structure is the same as verifying that the maps are indeed natural.
Coproduct
The coalgebra is obtained by defining a
coproduct or diagonal operator
:
Here,
is used as a short-hand for
to avoid an explosion of parentheses. The
symbol is used to denote the "external" tensor product, needed for the definition of a coalgebra. It is being used to distinguish it from the "internal" tensor product
, which is already being used to denote multiplication in the tensor algebra (see the section ''Multiplication'', below, for further clarification on this issue). In order to avoid confusion between these two symbols, most texts will replace
by a plain dot, or even drop it altogether, with the understanding that it is implied from context. This then allows the
symbol to be used in place of the
symbol. This is not done below, and the two symbols are used independently and explicitly, so as to show the proper location of each. The result is a bit more verbose, but should be easier to comprehend.
The definition of the operator
is most easily built up in stages, first by defining it for elements
and then by homomorphically extending it to the whole algebra. A suitable choice for the coproduct is then
:
and
:
where
is the unit of the field
. By linearity, one obviously has
:
for all
It is straightforward to verify that this definition satisfies the axioms of a coalgebra: that is, that
:
where
is the identity map on
. Indeed, one gets
:
and likewise for the other side. At this point, one could invoke a lemma, and say that
extends trivially, by linearity, to all of
, because
is a
free object and
is a
generator of the free algebra, and
is a homomorphism. However, it is insightful to provide explicit expressions. So, for
, one has (by definition) the homomorphism
:
Expanding, one has
:
In the above expansion, there is no need to ever write
as this is just plain-old scalar multiplication in the algebra; that is, one trivially has that
The extension above preserves the algebra grading. That is,
:
Continuing in this fashion, one can obtain an explicit expression for the coproduct acting on a homogenous element of order ''m'':
:
where the
symbol, which should appear as ш, the sha, denotes the
shuffle product. This is expressed in the second summation, which is taken over all
(''p'', ''m'' − ''p'')-shuffles. The shuffle is
:
By convention, one takes that Sh(''m,''0) and Sh(0,''m'') equals . It is also convenient to take the pure tensor products
and
to equal 1 for ''p'' = 0 and ''p'' = ''m'', respectively (the empty product in
). The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements
is ''preserved'' in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.
Equivalently,
:
where the products are in
, and where the sum is over all subsets of
.
As before, the algebra grading is preserved:
:
Counit
The counit
is given by the projection of the field component out from the algebra. This can be written as
for
and
for
. By homomorphism under the tensor product
, this extends to
:
for all
It is a straightforward matter to verify that this counit satisfies the needed axiom for the coalgebra:
:
Working this explicitly, one has
:
where, for the last step, one has made use of the isomorphism
, as is appropriate for the defining axiom of the counit.
Bialgebra
A
bialgebra defines both multiplication, and comultiplication, and requires them to be compatible.
Multiplication
Multiplication is given by an operator
:
which, in this case, was already given as the "internal" tensor product. That is,
:
That is,
The above should make it clear why the
symbol needs to be used: the
was actually one and the same thing as
; and notational sloppiness here would lead to utter chaos. To strengthen this: the tensor product
of the tensor algebra corresponds to the multiplication
used in the definition of an algebra, whereas the tensor product
is the one required in the definition of comultiplication in a coalgebra. These two tensor products are ''not'' the same thing!
Unit
The unit for the algebra
:
is just the embedding, so that
:
That the unit is compatible with the tensor product
is "trivial": it is just part of the standard definition of the tensor product of vector spaces. That is,
for field element ''k'' and any
More verbosely, the axioms for an
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
require the two homomorphisms (or commuting diagrams):
:
on
, and that symmetrically, on
, that
:
where the right-hand side of these equations should be understood as the scalar product.
Compatibility
The unit and counit, and multiplication and comultiplication, all have to satisfy compatibility conditions. It is straightforward to see that
:
Similarly, the unit is compatible with comultiplication:
:
The above requires the use of the isomorphism
in order to work; without this, one loses linearity. Component-wise,
:
with the right-hand side making use of the isomorphism.
Multiplication and the counit are compatible:
:
whenever ''x'' or ''y'' are not elements of
, and otherwise, one has scalar multiplication on the field:
The most difficult to verify is the compatibility of multiplication and comultiplication:
:
where
exchanges elements. The compatibility condition only needs to be verified on
; the full compatibility follows as a homomorphic extension to all of
The verification is verbose but straightforward; it is not given here, except for the final result:
:
For
an explicit expression for this was given in the coalgebra section, above.
Hopf algebra
The
Hopf algebra Hopf is a German surname. Notable people with the surname include:
* Eberhard Hopf (1902–1983), Austrian mathematician
* Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
* Heinz Hopf (actor) (1934–2001), Swe ...
adds an antipode to the bialgebra axioms. The antipode
on
is given by
:
This is sometimes called the "anti-identity". The antipode on
is given by
:
and on
by
:
This extends homomorphically to
:
Compatibility
Compatibility of the antipode with multiplication and comultiplication requires that
:
This is straightforward to verify componentwise on
:
:
Similarly, on
:
:
Recall that
:
and that
:
for any
that is ''not'' in
One may proceed in a similar manner, by homomorphism, verifying that the antipode inserts the appropriate cancellative signs in the shuffle, starting with the compatibility condition on
and proceeding by induction.
Cofree cocomplete coalgebra
One may define a different coproduct on the tensor algebra, simpler than the one given above. It is given by
:
Here, as before, one uses the notational trick
(recalling that
trivially).
This coproduct gives rise to a coalgebra. It describes a coalgebra that is
dual to the algebra structure on ''T''(''V''
∗), where ''V''
∗ denotes the
dual vector space of linear maps ''V'' → F. In the same way that the tensor algebra is a
free algebra, the corresponding coalgebra is termed cocomplete co-free. With the usual product this is not a bialgebra. It ''can'' be turned into a bialgebra with the product
where ''(i,j)'' denotes the binomial coefficient for
. This bialgebra is known as the
divided power Hopf algebra.
The difference between this, and the other coalgebra is most easily seen in the
term. Here, one has that
:
for
, which is clearly missing a shuffled term, as compared to before.
See also
*
Braided vector space In mathematics, a braided vectorspace \;V is a vector space together with an additional structure map \tau symbolizing interchanging of two vector tensor copies:
::\tau:\; V\otimes V\longrightarrow V\otimes V
such that the Yang–Baxter equation ...
*
Braided Hopf algebra In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra ''H'', particularly the Nichols algebra of a braided vect ...
*
Monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
*
Multilinear algebra
*
Stanisław Lem's ''Love and Tensor Algebra''
*
Fock space
References
* ''(See Chapter 3 §5)''
*
{{Tensors
Algebras
Multilinear algebra
Tensors
Hopf algebras