Superalgebra

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 commutative ring. In most applications, K is a field of characteristic 0, such as R or C.

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

${\displaystyle A=A_{0}\oplus A_{1}}$

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

${\displaystyle A_{i}A_{j}\subseteq A_{i+j}}$

where the subscripts are read modulo 2, i.e. they are thought of as elements of Z₂.

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

The elements of each of the 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 ${\displaystyle xy = x + y }$.

연관성 초자연적 초자연적 초자연적 초자연적 초자연적 요소인 곱셈이 연상되는 것이다. 단일 초자연적 초자연적 요소에서 정체성 요소는 반드시 균등하다. 달리 명시되지 않는 한, 이 글의 모든 슈퍼걸브라는 연관성이 있고 단일하다고 가정한다.

역행 초특수학(또는 초특수 대수학)은 등급이 매겨진 동급수학 버전을 만족하는 것이다. 구체적으로 A는 다음과 같은 경우에 대응된다.

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

A의 모든 동질 원소 x 및 y에 대해. 일반적인 의미로는 상통하지만, 슈퍼걸브라의 의미로는 상통하지 않은 슈퍼걸브라가 있다. 이러한 이유로, 교감형 초알제브라는 혼동을 피하기 위해 종종 초코메트리로 불린다.^[2]

예

• 교호 고리 K에 대한 대수학은 A를₁ 사소한 것으로 간주함으로써 K에 대한 순전히 심지어 초거대상으로 간주될 수 있다.
• 어떤 Z- 또는 N-graded 대수학도 채점모듈로 2를 읽음으로써 초계수학으로 간주할 수 있다. 여기에는 K 위에 있는 텐서 알헤브라와 다항식 링과 같은 예가 포함된다.
• 특히 K 이상의 어떤 외부 대수학도 초특수학이다. 외부대수는 초역대수의 표준적인 예다.
• 대칭 다항식 및 교번 다항식은 각각 짝수 부분과 홀수 부분을 이루는 초거형을 형성한다. 이것은 등급별 등급과는 다른 점수에 유의하십시오.
• 클리포드 알헤브라는 슈퍼알제브라다. 그들은 일반적으로 타협하지 않는다.
• The set of all endomorphisms (denoted ${\displaystyle \mathbf {End} (V)\equiv \mathbf {Hom} (V,V)}$, where the boldface ${\displaystyle \mathrm {Hom} }$ is referred to as internal ${\displaystyle \mathrm {Hom} }$, composed of all linear maps) of a super vector space forms a superal작문 중인 거브라
• The set of all square supermatrices with entries in K forms a superalgebra denoted by M_{p q}(K). This algebra may be identified with the algebra of endomorphisms of a free supermodule over K of rank p q and is the internal Hom of above for this space.
• 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

${\displaystyle \mu :A_{1}\otimes _{A_{0}}A_{1}\to A_{0}}$

such that

${\displaystyle \mu (x\otimes y)\cdot z=x\cdot \mu (y\otimes z)}$

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

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

and on arbitrary elements by

${\displaystyle {\hat {x}}=x_{0}-x_{1}}$

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:

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

Supercommutativity

The supercommutator on A is the binary operator given by

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

on homogeneous elements, 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:

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

The supercenter of A is, in general, different than the center of A as an ungraded algebra. 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 A ⊗ B with a multiplication rule determined by:

${\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=(-1)^{ b_{1} a_{2} }(a_{1}a_{2}\otimes b_{1}b_{2}).}$

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 × A → A that respects the grading. Bilinearity here means that

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

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. 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

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

for which the usual diagrams commute.

메모들

참조

• Deligne, P.; Morgan, J. W. (1999). "Notes on Supersymmetry (following Joseph Bernstein)". Quantum Fields and Strings: A Course for Mathematicians. 1. American Mathematical Society. pp. 41–97. ISBN 0-8218-2012-5.

• Kac, V. G.; Martinez, C.; Zelmanov, E. (2001). Graded simple Jordan superalgebras of growth one. Memoirs of the AMS Series. 711. AMS Bookstore. ISBN 978-0-8218-2645-4.

• Manin, Y. I. (1997). Gauge Field Theory and Complex Geometry ((2nd ed.) 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 978-0-8218-3574-6.