Superalgebra - Citizendia

In mathematics and theoretical physics, a superalgebra is a Z₂-graded algebra. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure 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. 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 In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

The prefix super- comes from the theory of supersymmetry in theoretical physics. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. In Mathematics, a supermodule is a Z 2- Graded module over a Superring or Superalgebra. 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 supermanifolds and superschemes. In Physics and Mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from Supersymmetry.

1 Formal definition
2 Examples
3 Further definitions and constructions
4 Generalizations and categorical definition
5 References

Formal definition

Let K be a fixed commutative ring. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In most applications, K is a field such as R or C. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

A superalgebra over K is an K-module A with a direct sum decomposition

$A = A_0\oplus A_1$

together with a bilinear multiplication A × A → A such that

$A_iA_j \sube A_{i+j}$

where the subscripts are read modulo 2. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure

A superring, or Z₂-graded ring, is a superalgebra over the ring of integers Z. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

The elements of A_i 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 $| x y | = | x | + | y | .$

An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative identity element. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that 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. In Mathematics, a supercommutative algebra is a Superalgebra (i In Mathematics, commutativity is the ability to change the order of something without changing the end result Specifically, A is commutative if

$yx = (-1)^{|x||y|}xy\,$

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 algebra may be regarded as superalgebra by reading the grading modulo 2. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure This includes examples such as tensor algebras and polynomial rings over K. In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra 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
In particular, any exterior algebra over K is a superalgebra. The exterior algebra is the standard example of a supercommutative algebra. In Mathematics, a supercommutative algebra is a Superalgebra (i
Clifford algebras are superalgebras. In Mathematics, Clifford algebras are a type of Associative algebra. They are generally noncommutative.
The set of all endomorphisms (both even and odd) of a super vector space forms a superalgebra under composition. In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself In Mathematics, a super vector space is another name for a Z 2- Graded vector space, that is a Vector space over a field
The set of all square supermatrices with entries in K forms a superalgebra denoted by M_p|q(K). In Mathematics and Theoretical physics, a supermatrix is a Z 2-graded analog of an ordinary matrix. This algebra may be identified with the algebra of endomorphisms of a free supermodule over K of rank p|q.
Lie superalgebras are a graded analog of Lie algebras. In Mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z 2- grading. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Lie superalgebras are nonunital and nonassociative; however, one may construct the analog of a universal enveloping algebra of a Lie superalgebra which is a unital, associative superalgebra. In Mathematics, for any Lie algebra L one can construct its universal enveloping algebra U ( L)

Further definitions and constructions

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 subalgebra of A, naturally called the even subalgebra. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation It forms an ordinary algebra over K. 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

The set of all odd elements A₁ is a A₀-bimodule whose scalar multiplication is just multiplication in A. 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 The product in A equips A₁ with a bilinear form

$\mu:A_1\otimes_{A_0}A_1 \to A_0$

such that

$\mu(x\otimes y)\cdot z = x\cdot\mu(y\otimes z)$

for all x, y, and z in A₁. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V × V → F, where This follows from the associativity of the product in A.

Grade involution

The is a canonical involutive automorphism on any superalgebra called the grade involution. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself It is given on homogeneous elements by

$\hat x = (-1)^{|x|}x$

and on arbitrary elements by

$\hat x = x_0 - x_1$

where x_i 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:

$A_i = \{x \in A : \hat x = (-1)^i x\}.$

Supercommutativity

The supercommutator on A is the binary operator given by

$[x,y] = xy - (-1)^{|x||y|}yx\,$

on homogeneous elements. In Abstract algebra, the term torsion refers to a number of concepts related to elements of finite order in groups and to the failure of modules to be In Mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z 2- grading. 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:

$Z(A) = \{a\in A : [a,x]=0 \text{ for all } x\in A\}.$

The supercenter of A is, in general, different than the center of A as an ungraded algebra. The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements A commutative superalgebra is one whose supercenter is all of A.

Super tensor product

The graded tensor product of two superalgebras may be regarded as a superalgebra with a multiplication rule determined by:

$(a_1\otimes b_1)(a_2\otimes b_2) = (-1)^{|b_1||a_2|}(a_1a_2\otimes b_1b_2).$

Generalizations and categorical definition

One can easily generalize the definition of superalgebras to include superalgebras over a commutative superring. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector 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 A with a R-bilinear multiplication A × A → A that respects the grading. In Mathematics, a supermodule is a Z 2- Graded module over a Superring or Superalgebra. Bilinearity here means that

$r\cdot(xy) = (r\cdot x)y = (-1)^{|r||x|}x(r\cdot y)$

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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets The category of all R-supermodules forms a monoidal category under the super tensor product with R serving as the unit object. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a monoidal category (or tensor category) is a category C equipped with a Bifunctor &otimes: C An associative, unital superalgebra over R can then be defined as a monoid in the category of R-supermodules. In Category theory, a monoid (or monoid object) (M\mu\eta in a Monoidal category C is an object M together with two That is, a superalgebra is an R-supermodule A with two (even) morphisms

$\begin{align}\mu &: A\otimes A \to A\\ \eta &: R\to A\end{align}$

for which the usual diagrams commute.

References

Deligne, Pierre; John W. Pierre René Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian Mathematician. Morgan (1999). "Notes on Supersymmetry (following Joseph Bernstein)". Quantum Fields and Strings: A Course for Mathematicians 1: 41–97, American Mathematical Society. ISBN 0-8218-2012-5.
Manin, Y. I. (1997). Gauge Field Theory and Complex Geometry, (2nd ed. ), Berlin: Springer. ISBN 3-540-61378-1.
Varadarajan, V. S. (2004). Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes in Mathematics 11. American Mathematical Society. ISBN 0-8218-3574-2.

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