Supriya Ghosh (Editor)

* algebra

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In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over 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.

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*-ring

In mathematics, a *-ring is a ring with a map * : AA that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:

  • (x + y)* = x* + y*
  • (x y)* = y* x*
  • 1* = 1
  • (x*)* = x
  • for all x, y in 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 1* is also a multiplicative identity, and identities are unique.

    Elements such that x* = x are called self-adjoint.

    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.

    Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: xIx* ∈ I and so on.

    *-algebra

    A *-algebra A is a *-ring, with involution * that is an associative algebra over a commutative *-ring R with involution , such that (r x)* = r′x*  ∀rR, xA.

    The base *-ring R is usually the complex numbers (with acting as complex conjugation) and is commutative with A such that A is both left and right algebra.

    Since R is central in A, that is,

    rx = xr   ∀rR, xA

    the * on A is conjugate-linear in R, meaning

    (λ x + μy)* = λ′x* + μ′y*

    for λ, μR, x, yA.

    A *-homomorphism f : AB is an algebra homomorphism that is compatible with the involutions of A and B, i.e.,

  • f(a*) = f(a)* for all a in A.
  • 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 GLn(C).

    Notation

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

    xx*, or xx (TeX: x^*),

    but not as "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 C where * is just complex conjugation.
  • More generally, a field extension made by adjunction of a square root (such as the imaginary unit −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 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 n × n matrices over R with * given by the transposition.
  • The matrix algebra of 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 R[x] over a commutative trivially-*-ring R is a *-algebra over R with P *(x) = P (−x).
  • If (A, +, ×, *) is simultaneously a *-ring, an algebra over a ring R (commutative), and (r x)* = r (x*)  ∀rR, xA, then A is a *-algebra over 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 R, its quotient by any its *-ideal is a *-algebra over R.
  • For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by ε = 0 makes the original ring.
  • The same about a commutative ring K and its polynomial ring K[x]: the quotient by x = 0 restores 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 gg−1.
  • Non-Example

    Not every algebra admits an involution:

    Regard the 2x2 matrices over the complex numbers.
    Consider the following subalgebra:

    A := { ( a b 0 0 ) : a , b C }

    Any nontrivial antiautomorphism necessarily has the form:

    φ z [ ( 1 0 0 0 ) ] = ( 1 z 0 0 ) φ z [ ( 0 1 0 0 ) ] = ( 0 0 0 0 )

    for any complex number z C .
    It follows that any nontrivial antiautomorphism fails to be idempotent:

    φ z 2 [ ( 0 1 0 0 ) ] = ( 0 0 0 0 ) ( 0 1 0 0 )

    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 1/2(1 + *) and 1/2(1 − *) are orthogonal idempotents, 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 −* : x ↦ −x*. It does not define a *-ring structure (unless the characteristic is 2, in which case −* is identical to the original *), as 1 ↦ −1, neither is it antimultiplicative, but it satisfies the other axioms (linear, involution) and hence is quite similar to *-algebra where xx*.

    Elements fixed by this map (i.e., such that 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.

    References

    *-algebra Wikipedia


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