# Self-involution

## Definitions

• Webster's Revised Unabridged Dictionary
• n Self-involution Involution in one's self; hence, abstraction of thought; reverie.
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Chambers's Twentieth Century Dictionary
• n Self-involution mental abstraction
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## Etymology

Chambers's Twentieth Century Dictionary
A.S. self, seolf, sylf; Dut. zelf, Ger. selbe, Goth. silba.

## Usage

### In literature:

Merely to admit the need for all this involution of ambiguity and double-dealing grievously affronted self-esteem.
"Linda Lee, Incorporated" by Louis Joseph Vance
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### In science:

In the case in which, following the recipe of section 8, we have a self-dual measure µ∗ (B ), the ﬁxed point of the involution for the electric charge e is at the value 2e2 = 1.
General duality for abelian-group-valued statistical-mechanics models
Indeed, this condition means that two different points of Σ , connected by the antiholomorphic involution, are stuck together on the curve Γ , which means self-intersection.
Random matrices and Laplacian growth
As usual, this extends to an exact contravariant and involutive self-equivalence ⋆ on the category X.
Blocks and modules for Whittaker pairs
Then X becomes a self-dual vector space with respect to Φ and the algebra F (X, X ) of continuous linear transformations on X has an involution a 7→ a∗ given by Φ(ax, y ) = Φ(x, a∗y ), for all x, y ∈ X .
Inner Ideals of Simple Locally Finite Lie Algebras
Since A is simple with non-zero socle, by [22, 4.9, 4.12], A = F (X, X ) where X is a self-dual vector space over F with respect to a nondegenerate symmetric or skew-symmetric form Φ and the involution a 7→ a∗ of A is given by Φ(ax, y ) = Φ(x, a∗y ), for all x, y ∈ X .
Inner Ideals of Simple Locally Finite Lie Algebras
The −1 eigenspace of the involution multiplied with ı then produces a set of self-adjoint matrices which forms the corresponding H Class of Hamiltonians.
Lyapunov spectra for all symmetry classes of quasi-one-dimensional disordered systems
Restricting g onto one factor S n−1×∗ (where ∗ is a base point) gives a self map of S n−1 which commutes with the antipodal involution and hence has an odd degree (this is a theorem of Borsuk, see , pages 483 - 485).
Topological robotics: motion planning in projective spaces
An involution † on A is equivalent to ∗ if there exists an involutive isomorphism from (A, †) to (A, ∗). A subalgebra A0 of (A, ∗) is involutive (or self-adjoint) if {x∗ : x ∈ A0} ⊂ A0 .
Algebra with indefinite involution and its representation in Krein space
Then there exists a self-adjoint unitary η on H for the GNS representation (H, π) by ω such that (H, π , (·|·)) is an involutive representation of (A, †) with respect to (·|·) in (2.2).
Algebra with indefinite involution and its representation in Krein space
The noncommutative nature of the discrete space F is given by a spectral triple (A, H, D), where A is an involution of operators on the ﬁnite-dimensional Hilbert space H of Euclidean fermions, and D is a self-adjoint unbounded operator in H.
Highlights of Noncommutative Spectral Geometry
As we showed above, if χ1 is invariant under the involution σ , then the operator L4 is self-adjoint. S.P.
Commuting higher rank ordinary differential operators
An element ﬁxed by this involution is called self-dual.
A Path Algorithm for Affine Kazhdan-Lusztig Polynomials
Let B be a unital associative involutive Banach algebra, I a contractive B-bimodule and E a total ly ordered set of self-adjoint idempotents in B such that 0, 1 ∈ E .
Functional analytic background for a theory of infinite-dimensional reductive Lie groups
The involution ∗ of M extends to an anti-linear isometry J : Lp(M, τ ) → Lp (M, τ ), and we will consider the linear space of self-adjoint elements in Lp (M, τ ), that is Lp h = Lp(M, τ )h = {x ∈ Lp (M, τ ) : J x = x}.
Spaces of nonpositive curvature arising from a finite algebra
Consequently, the set of partial isometries of an involutive semigroup is self-adjoint.
A descent homomorphism for semimultiplicative sets
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