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In [[mathematics]] and [[theoretical physics]], a '''superalgebra''' is a '''Z'''<sub>2</sub>-[[graded algebra]].<ref>Kac, Martinez & Zelmanov (2001), {{Google books quote|id=jTCNZz2Tk4cC|page=3|text=superalgebra|p. 3}}.</ref> That is, it is an [[algebra (ring theory)|algebra]] over a [[commutative ring]] or [[field (mathematics)|field]] with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.


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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 [[superscheme]]s.
 
==Formal definition==
 
Let ''K'' be a fixed [[commutative ring]]. In most applications, ''K'' is a [[field (mathematics)|field]] such as '''R''' or '''C'''.
 
A '''superalgebra''' over ''K'' is a [[module (mathematics)|''K''-module]] ''A'' with a [[direct sum of modules|direct sum]] decomposition
:<math>A = A_0\oplus A_1</math>
together with a [[bilinear map|bilinear]] multiplication ''A'' &times; ''A'' &rarr; ''A'' such that
:<math>A_iA_j \sube A_{i+j}</math>
where the subscripts are read [[Modular arithmetic|modulo]] 2.
 
A '''superring''', or '''Z'''<sub>2</sub>-[[graded ring]], is a superalgebra over the ring of [[integer]]s '''Z'''.
 
The elements of ''A''<sub>''i''</sub> 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 ''A''<sub>0</sub> or ''A''<sub>1</sub>. 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 <math>|xy| = |x| + |y|.</math>
 
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
:<math>yx = (-1)^{|x||y|}xy\,</math>
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 ''A''<sub>1</sub> to be trivial.
*Any '''Z''' or '''N'''-[[graded algebra]] may be regarded as superalgebra by reading the grading modulo 2. This includes examples such as [[tensor algebra]]s and [[polynomial ring]]s over ''K''.
*In particular, any [[exterior algebra]] over ''K'' is a superalgebra. The exterior algebra is the standard example of a [[supercommutative algebra]].
*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 algebra]]s are superalgebras. They are generally noncommutative.
*The set of all [[endomorphism]]s (both even and odd) of a [[super vector space]] forms a superalgebra under composition.
*The set of all square [[supermatrices]] with entries in ''K'' forms a superalgebra denoted by ''M''<sub>''p''|''q''</sub>(''K''). This algebra may be identified with the algebra of endomorphisms of a free supermodule over ''K'' of rank ''p''|''q''.
*[[Lie superalgebra]]s are a graded analog of [[Lie algebra]]s. 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.
 
==Further definitions and constructions==
 
===Even subalgebra===
 
Let ''A'' be a superalgebra over a commutative ring ''K''. The [[submodule]] ''A''<sub>0</sub>, 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'''. It forms an ordinary [[algebra (ring theory)|algebra]] over ''K''.
 
The set of all odd elements ''A''<sub>1</sub> is an ''A''<sub>0</sub>-[[bimodule]] whose scalar multiplication is just multiplication in ''A''. The product in ''A'' equips ''A''<sub>1</sub> with a [[bilinear form]]
:<math>\mu:A_1\otimes_{A_0}A_1 \to A_0</math>
such that
:<math>\mu(x\otimes y)\cdot z = x\cdot\mu(y\otimes z)</math>
for all ''x'', ''y'', and ''z'' in ''A''<sub>1</sub>. This follows from the associativity of the product in ''A''.
 
===Grade involution===
 
There is a canonical [[Involution (mathematics)|involutive]] [[automorphism]] on any superalgebra called the '''grade involution'''. It is given on homogeneous elements by
:<math>\hat x = (-1)^{|x|}x</math>
and on arbitrary elements by
:<math>\hat x = x_0 - x_1</math>
where ''x''<sub>''i''</sub> are the homogeneous parts of ''x''. If ''A'' has no [[torsion (algebra)|2-torsion]] (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of ''A'':
:<math>A_i = \{x \in A : \hat x = (-1)^i x\}.</math>
 
===Supercommutativity===
 
The '''[[supercommutator]]''' on ''A'' is the binary operator given by
:<math>[x,y] = xy - (-1)^{|x||y|}yx\,</math>
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'':
:<math>Z(A) = \{a\in A : [a,x]=0 \text{ for all } x\in A\}.</math>
The supercenter of ''A'' is, in general, different than the [[center of an algebra|center]] of ''A'' as an ungraded algebra. A commutative superalgebra is one whose supercenter is all of ''A''.
 
===Super tensor product===
 
The graded [[tensor product of algebras|tensor product]] of two superalgebras ''A'' and ''B'' may be regarded as a superalgebra ''A'' &otimes; ''B'' with a multiplication rule determined by:
:<math>(a_1\otimes b_1)(a_2\otimes b_2) = (-1)^{|b_1||a_2|}(a_1a_2\otimes b_1b_2).</math>
If either ''A'' or ''B'' is purely even, this is equivalent to the ordinary ungraded tensor product (except that the result is graded). However, in general, the super tensor product is distinct from the tensor product of ''A'' and ''B'' regarded as ordinary, ungraded algebras.
 
==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 [[supermodule|''R''-supermodule]] ''A'' with a ''R''-bilinear multiplication ''A'' &times; ''A'' &rarr; ''A'' that respects the grading. Bilinearity here means that
:<math>r\cdot(xy) = (r\cdot x)y = (-1)^{|r||x|}x(r\cdot y)</math>
for all homogeneous elements ''r'' &isin; ''R'' and ''x'', ''y'' &isin; ''A''.
 
Equivalently, one may define a superalgebra over ''R'' as a superring ''A'' together with an superring homomorphism ''R'' &rarr; ''A'' whose image lies in the supercenter of ''A''.
 
One may also define superalgebras [[category theory|categorically]]. The [[category (mathematics)|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 (category theory)|monoid]] in the category of ''R''-supermodules. That is, a superalgebra is an ''R''-supermodule ''A'' with two (even) morphisms
:<math>\begin{align}\mu &: A\otimes A \to A\\ \eta &: R\to A\end{align}</math>
for which the usual diagrams commute.
 
== Notes ==
<references/>
 
==References==
 
*{{cite conference | authorlink=Pierre Deligne | first = Pierre | last = Deligne | coauthors = John W. Morgan | title = Notes on Supersymmetry (following Joseph Bernstein) | booktitle = Quantum Fields and Strings: A Course for Mathematicians | volume = 1 | pages = 41–97 | publisher = American Mathematical Society | year = 1999 | id = ISBN 0-8218-2012-5}}
*{{cite book | last = Manin | first = Y. I. | title = Gauge Field Theory and Complex Geometry | publisher = Springer | location = Berlin | year = 1997 | edition = (2nd ed.) | isbn = 3-540-61378-1}}
*{{cite book | first = V. S. | last = Varadarajan | year = 2004 | title = Supersymmetry for Mathematicians: An Introduction | series = Courant Lecture Notes in Mathematics '''11''' | publisher = American Mathematical Society | isbn = 0-8218-3574-2}}
*{{cite book | first1 = Victor G. | last1 = Kac | first2 = Consuelo | last2 = Martinez | first3 = Efim | last3 = Zelmanov | year = 2001 | title = Graded simple Jordan superalgebras of growth one | series = Memoirs of the AMS Series | volume = 711 | publisher = AMS Bookstore | isbn = 978-0-8218-2645-4}}
 
[[Category:Algebras]]
[[Category:Super linear algebra]]

Latest revision as of 22:03, 8 November 2013

In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra.[1] 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, supermodules, 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 manifolds, supermanifolds and superschemes.

Formal definition

Let K be a fixed commutative ring. 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 × AA such that

where the subscripts are read modulo 2.

A superring, or Z2-graded ring, is a superalgebra over the ring of integers 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.

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. This includes examples such as tensor algebras and polynomial rings over K.
  • In particular, any exterior algebra over K is a superalgebra. The exterior algebra is the standard example of a supercommutative algebra.
  • 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 algebras are superalgebras. They are generally noncommutative.
  • The set of all endomorphisms (both even and odd) of a super vector space 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 superalgebras are a graded analog of Lie algebras. 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.

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. 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.

Grade involution

There is a canonical involutive 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 by

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.

Super tensor product

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

If either A or B is purely even, this is equivalent to the ordinary ungraded tensor product (except that the result is graded). However, in general, the super tensor product is distinct from the tensor product of A and B regarded as ordinary, ungraded algebras.

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 A with a R-bilinear multiplication A × AA that respects the grading. Bilinearity here means that

for all homogeneous elements rR and x, yA.

Equivalently, one may define a superalgebra over R as a superring A together with an superring homomorphism RA 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.

Notes

  1. Kac, Martinez & Zelmanov (2001), Template:Google books quote.

References

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