“*-algebra”的意思、由来-开放百科全书

词条 *-algebra

释义

Terminology
*-ring *-algebra Philosophy of the *-operation Notation
Examples
Non-Example
Additional structures
Skew structures
See also
Notes
References

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings {{mvar|R}} and {{mvar|A}}, where {{mvar|R}} is commutative and {{mvar|A}} has the structure of an associative algebra over {{mvar|R}}. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.

However, it may happen that an algebra admits no involution at all.

Terminology

*-ring

In mathematics, a *-ring is a ring with a map {{math|* : A → A}} that is an antiautomorphism and an involution.

More precisely, {{math|*}} is required to satisfy the following properties:^[1]

{{math|size=120%|1=(x + y) = x + y}}
{{math|size=120%|1=(x y) = y x}}
{{math|size=120%|1=1 = 1}}
{{math|size=120%|1=(x) = x}}

for all {{math|x, y}} in {{mvar|A}}.

This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply {{math|1*}} is also a multiplicative identity, and identities are unique.

Elements such that {{math|1=x* = x}} are called self-adjoint.^[2]

Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear form over any *-ring.

{{anchor|*-objects}}Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: {{math|x ∈ I ⇒ x* ∈ I}} and so on.

*-algebra

A *-algebra {{mvar|A}} is a *-ring,{{efn|Most definitions do not require a *-algebra to have the unity, i.e. a *-algebra is allowed to be a *-rng only.}} with involution * that is an associative algebra over a commutative *-ring {{mvar|R}} with involution {{mvar|′}}, such that {{math|1=(r x)* = r′ x* ∀r ∈ R, x ∈ A}}.^[3]

The base *-ring {{mvar|R}} is usually the complex numbers (with * acting as complex conjugation) and is commutative with {{mvar|A}} such that {{mvar|A}} is both left and right algebra.{{what|reason=The concepts of left and right algebra are not needed here, since A is commutative. Nor have they been defined.|date=September 2015}}

Since {{mvar|R}} is central in {{mvar|A}}, that is,{{what|reason=But R is not a subset of A until you clarify that it is.|date=September 2015}}

{{math|size=120%|1=rx = xr ∀r ∈ R, x ∈ A}}

the * on {{mvar|A}} is conjugate-linear in {{mvar|R}}, meaning{{what|reason=This follows from the basic relation (rx)*=r'x* without needing R to be embedded in A.|date=September 2015}}

{{math|size=120%|1=(λ x + μ y)* = λ′ x* + μ′ y*}}

for {{math|λ, μ ∈ R, x, y ∈ A}}.

A *-homomorphism {{math|f : A → B}} is an algebra homomorphism that is compatible with the involutions of {{mvar|A}} and {{mvar|B}}, i.e.,

{{math|size=120%|1=f(a) = f(a)}} for all {{mvar|a}} in {{mvar|A}}.^[2]

Philosophy of the *-operation

The *-operation on a *-ring is analogous to complex conjugation on the complex numbers. The *-operation on a *-algebra is analogous to taking adjoints in {{math|GL_n(C)}}.

Notation

The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:

{{math|size=120%|x ↦ x*}}, or

{{math|size=120%|x ↦ x^∗}} (TeX: x^*),

but not as "{{math|x∗}}"; see the asterisk article for details.

Examples

Any commutative ring becomes a -ring with the trivial (identical) involution.
The most familiar example of a -ring and a -algebra over reals is the field of complex numbers {{math|C}} where is just complex conjugation.
More generally, a field extension made by adjunction of a square root (such as the imaginary unit {{sqrt|−1}}) is a -algebra over the original field, considered as a trivially--ring. The flips the sign of that square root.
A quadratic integer ring (for some {{mvar|D}}) is a commutative -ring with the defined in the similar way; quadratic fields are -algebras over appropriate quadratic integer rings.
Quaternions, split-complex numbers, dual numbers, and possibly other hypercomplex number systems form -rings (with their built-in conjugation operation) and -algebras over reals (where is trivial). Note that neither of the three is a complex algebra.
Hurwitz quaternions form a non-commutative -ring with the quaternion conjugation.
The matrix algebra of {{math|n × n }}matrices over R with given by the transposition.
The matrix algebra of {{math|n × n }}matrices over C with given by the conjugate transpose.
Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a -algebra.
The polynomial ring {{math|R[x]}} over a commutative trivially--ring {{mvar|R}} is a -algebra over {{mvar|R}} with {{math|1=P (x) = P (−x)}}.
If {{math|(A, +, ×, )}} is simultaneously a -ring, an algebra over a ring {{mvar|R}} (commutative), and {{math|1=(r x) = r (x) ∀r ∈ R, x ∈ A}}, then {{mvar|A}} is a -algebra over {{mvar|R}} (where is trivial).
- As a partial case, any -ring is a -algebra over integers.
Any commutative -ring is a -algebra over itself and, more generally, over any its -subring.
For a commutative -ring {{mvar|R}}, its quotient by any its -ideal is a -algebra over {{mvar|R}}.
- For example, any commutative trivially--ring is a -algebra over its dual numbers ring, a -ring with non-trivial , because the quotient by {{math|1=ε = 0}} makes the original ring.
- The same about a commutative ring {{mvar|K}} and its polynomial ring {{math|K[x]}}: the quotient by {{math|1=x = 0}} restores {{mvar|K}}.
In Hecke algebra, an involution is important to the Kazhdan–Lusztig polynomial.
The endomorphism ring of an elliptic curve becomes a -algebra over the integers, where the involution is given by taking the dual isogeny. A similar construction works for abelian varieties with a polarization, in which case it is called the Rosati involution (see Milne's lecture notes on abelian varieties).

Involutive Hopf algebras are important examples of *-algebras (with the additional structure of a compatible comultiplication); the most familiar example being:

The group Hopf algebra: a group ring, with involution given by {{math|g ↦ g⁻¹}}.

Non-Example

Not every algebra admits an involution:

Regard the 2x2 matrices over the complex numbers.

Consider the following subalgebra:

Any nontrivial antiautomorphism necessarily has the form:

for any complex number .

It follows that any nontrivial antiautomorphism fails to be idempotent:

Concluding that the subalgebra admits no involution.

Additional structures

Many properties of the transpose hold for general *-algebras:

The Hermitian elements form a Jordan algebra;
The skew Hermitian elements form a Lie algebra;
If 2 is invertible in the -ring, then {{sfrac|1|2}}{{math|(1 + )}} and {{sfrac|1|2}}{{math|(1 − )}} are orthogonal idempotents,^[2] called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of modules (vector spaces if the -ring is a field) of symmetric and anti-symmetric (Hermitian and skew Hermitian) elements. These spaces do not, generally, form associative algebras, because the idempotents are operators, not elements of the algebra.

Skew structures

Given a *-ring, there is also the map {{math|−* : x ↦ −x*}}.

It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as {{math|1 ↦ −1}}, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where {{math|size=120%|x ↦ x*}}.

Elements fixed by this map (i.e., such that {{math|1=a = −a*}}) are called skew Hermitian.

For the complex numbers with complex conjugation, the real numbers are the Hermitian elements, and the imaginary numbers are the skew Hermitian.

Notes

References

1. ^{{Cite web |url=http://mathworld.wolfram.com/C-Star-Algebra.html |title=C-Star Algebra |website = Wolfram MathWorld |date=2015 |first=Eric W. |last=Weisstein}}
2. ^¹²{{Cite web|url=http://math.ucr.edu/home/baez/octonions/node5.html |title=Octonions |date=2015 |accessdate=27 January 2015 |website=Department of Mathematics |publisher=University of California, Riverside |last=Baez |first=John |archiveurl=https://www.webcitation.org/6XHaTVlBr?url=http://math.ucr.edu/home/baez/octonions/node5.html |archivedate=25 March 2015 |deadurl=no |df= }}
3. ^{{nlab|id=star-algebra}}

2 : Algebras|Ring theory

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