Superalgebra
Encyclopedia
In mathematics
and theoretical physics
, a superalgebra is a Z2-graded algebra
. That is, it is an algebra
over a commutative ring
or field
with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
The prefix super- comes from the theory of supersymmetry
in theoretical physics. Superalgebras and their representations, supermodule
s, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry
where they enter into the definitions of graded manifold
s, supermanifold
s and superschemes.
. In most applications, K is a field
such as R or C.
A superalgebra over K is a K-module
A with a direct sum
decomposition
together with a bilinear
multiplication A × A → A such that
where the subscripts are read modulo
2.
A superring, or Z2-graded ring, is a superalgebra over the ring of integer
s Z.
The elements of Ai are said to be homogeneous. The parity of a homogeneous element x, denoted by |x|, is 0 or 1 according to whether it is in A0 or A1. Elements of parity 0 are said to be even and those of parity 1 to be odd. If x and y are both homogeneous then so is the product xy and
An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative identity element
. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital.
A commutative superalgebra is one which satisfies a graded version of commutativity
. Specifically, A is commutative if
for all homogeneous elements x and y of A.
grading (“even” and “odd” elements) such that (i) the bracket of two generators is always antisymmetric except for two odd elements where it is symmetric and (ii) the
Jacobi identities are satisfied.
The first of these three identities says that the 0 form a representation of the ordinary Lie algebra spanned by E (Consider the 0 as vectors on which the E act.) The second is equivalent to the first if the Killing form is nonsingular. The last identity restricts the possible representations 0 of the ordinary Lie algebra. This relation is the reason that not every ordinary Lie algebra can be extended to a superalgebra.
of A, naturally called the even subalgebra. It forms an ordinary algebra
over K.
The set of all odd elements A1 is an A0-bimodule
whose scalar multiplication is just multiplication in A. The product in A equips A1 with a bilinear form
such that
for all x, y, and z in A1. This follows from the associativity of the product in A.
on any superalgebra called the grade involution. It is given on homogeneous elements by
and on arbitrary elements by
where xi are the homogeneous parts of x. If A has no 2-torsion (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of A:
on homogeneous elements. This can be extended to all of A by linearity. Elements x and y of A are said to supercommute if [x, y] = 0.
The supercenter of A is the set of all elements of A which supercommute with all elements of A:
The supercenter of A is, in general, different than the center of A as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of A.
of two superalgebras may be regarded as a superalgebra with a multiplication rule determined by:
Let R be a commutative superring. A superalgebra over R is a R-supermodule
A with a R-bilinear multiplication A × A → A that respects the grading. Bilinearity here means that
for all homogeneous elements r ∈ R and x, y ∈ A.
Equivalently, one may define a superalgebra over R as a superring A together with an superring homomorphism R → A whose image lies in the supercenter of A.
One may also define superalgebras categorically
. The category
of all R-supermodules forms a monoidal category
under the super tensor product with R serving as the unit object. An associative, unital superalgebra over R can then be defined as a monoid
in the category of R-supermodules. That is, a superalgebra is an R-supermodule A with two (even) morphisms
for which the usual diagrams commute.
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...
and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a superalgebra is a Z2-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 ....
. That is, it is an algebra
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
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....
or 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...
with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
The prefix super- comes from the theory of supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
in theoretical physics. Superalgebras and their representations, supermodule
Supermodule
In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics....
s, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry
Supergeometry
Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, BRST theory, or supergravity.Supergeometry is...
where they enter into the definitions of graded manifold
Graded manifold
Graded manifolds are extensions of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra. Graded manifolds are not supermanifolds though there is a certain correspondence between the graded manifolds and the DeWitt supermanifolds. Both graded manifolds and...
s, supermanifold
Supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.- Physics :...
s and superschemes.
Formal definition
Let K be a fixed commutative ringCommutative 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....
. In most applications, K is 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...
such as R or C.
A superalgebra over K is a K-module
Module (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...
A with a direct sum
Direct 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
together with a bilinear
Bilinear
Bilinear may refer to:* Bilinear sampling, a method in computer graphics for choosing the color of a texture* Bilinear form* Bilinear interpolation* Bilinear map, a type of mathematical function between vector spaces...
multiplication A × A → A such that
where the subscripts are read modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
2.
A superring, or Z2-graded ring, is a superalgebra over the ring of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s Z.
The elements of Ai are said to be homogeneous. The parity of a homogeneous element x, denoted by |x|, is 0 or 1 according to whether it is in A0 or A1. Elements of parity 0 are said to be even and those of parity 1 to be odd. If x and y are both homogeneous then so is the product xy and
An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
. The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital.
A commutative superalgebra is one which satisfies a graded version of commutativity
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
. Specifically, A is commutative if
for all homogeneous elements x and y of A.
Examples
- Any algebra over a commutative ring K may be regarded as a purely even superalgebra over K; that is, by taking A1 to be trivial.
- Any Z or N-graded algebraGraded algebraIn mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as tensor algebraTensor algebraIn 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...
s and polynomial ringPolynomial ringIn 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...
s over K. - In particular, any 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...
over K is a superalgebra. The exterior algebra is the standard example of a supercommutative algebraSupercommutative algebraIn mathematics, a supercommutative algebra is a superalgebra such that for any two homogeneous elements x, y we haveyx = ^In mathematics, a supercommutative algebra is a superalgebra In mathematics, a supercommutative algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two...
. - The symmetric polynomials and alternating polynomials together form a superalgebra, being the even and odd parts, respectively. Note that this is a different grading from the grading by degree.
- 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...
s are superalgebras. They are generally noncommutative. - The set of all endomorphismEndomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
s (both even and odd) of a super vector spaceSuper vector spaceIn mathematics, a super vector space is another name for a Z2-graded vector space, that is, a vector space over a field K with a given decompositionV=V_0\oplus V_1....
forms a superalgebra under composition. - The set of all square supermatrices with entries in K forms a superalgebra denoted by Mp|q(K). This algebra may be identified with the algebra of endomorphisms of a free supermodule over K of rank p|q.
- Lie superalgebraLie superalgebraIn mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
s are a graded analog of Lie algebraLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s. Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a universal enveloping algebraUniversal enveloping algebraIn 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 a Lie superalgebra which is a unital, associative superalgebra.
Further definitions and constructions
A superalgebra is an algebra with a grading (“even” and “odd” elements) such that (i) the bracket of two generators is always antisymmetric except for two odd elements where it is symmetric and (ii) theJacobi identities are satisfied.
The first of these three identities says that the 0 form a representation of the ordinary Lie algebra spanned by E (Consider the 0 as vectors on which the E act.) The second is equivalent to the first if the Killing form is nonsingular. The last identity restricts the possible representations 0 of the ordinary Lie algebra. This relation is the reason that not every ordinary Lie algebra can be extended to a superalgebra.
Even subalgebra
Let A be a superalgebra over a commutative ring K. The submodule A0, consisting of all even elements, is closed under multiplication and contains the identity of A and therefore forms a subalgebraSubalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
of A, naturally called the even subalgebra. It forms an ordinary algebra
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
over K.
The set of all odd elements A1 is an A0-bimodule
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...
whose scalar multiplication is just multiplication in A. The product in A equips A1 with a bilinear form
such that
for all x, y, and z in A1. This follows from the associativity of the product in A.
Grade involution
There is a canonical involutive automorphismAutomorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
on any superalgebra called the grade involution. It is given on homogeneous elements by
and on arbitrary elements by
where xi are the homogeneous parts of x. If A has no 2-torsion (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of A:
Supercommutativity
The supercommutator on A is the binary operator given byon homogeneous elements. This can be extended to all of A by linearity. Elements x and y of A are said to supercommute if [x, y] = 0.
The supercenter of A is the set of all elements of A which supercommute with all elements of A:
The supercenter of A is, in general, different than the center of A as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of A.
Super tensor product
The graded tensor productTensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of two superalgebras may be regarded as a superalgebra with a multiplication rule determined by:
Generalizations and categorical definition
One can easily generalize the definition of superalgebras to include superalgebras over a commutative superring. The definition given above is then a specialization to the case where the base ring is purely even.Let R be a commutative superring. A superalgebra over R is a R-supermodule
Supermodule
In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics....
A with a R-bilinear multiplication A × A → A that respects the grading. Bilinearity here means that
for all homogeneous elements r ∈ R and x, y ∈ A.
Equivalently, one may define a superalgebra over R as a superring A together with an superring homomorphism R → A whose image lies in the supercenter of A.
One may also define superalgebras categorically
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. The category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of all R-supermodules forms a monoidal category
Monoidal category
In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
under the super tensor product with R serving as the unit object. An associative, unital superalgebra over R can then be defined as a monoid
Monoid (category theory)
In category theory, a monoid in a monoidal category is an object M together with two morphisms* \mu : M\otimes M\to M called multiplication,* and \eta : I\to M called unit,...
in the category of R-supermodules. That is, a superalgebra is an R-supermodule A with two (even) morphisms
for which the usual diagrams commute.