- group, ∞-group
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- abelian group, spectrum
- super abelian group
- group action, ∞-action
- representation, ∞-representation
- progroup
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**Classical groups**

**Finite groups**

**Group schemes**

**Topological groups**

**Lie groups**

**Super-Lie groups**

**Higher groups**

**Cohomology and Extensions**

**Related concepts**

**algebraic quantum field theory** (perturbative, on curved spacetimes, homotopical)

**quantum mechanical system**, **quantum probability**

**interacting field quantization**

A **one-parameter group** (of unitary operators in a Hilbert space) is a homomorphism of groups

$\mathbf{R} \to U(H),$

where $H$ is a Hilbert spaces, $U(H)$ denotes its group of unitary operators and $\mathbf{R}$ the additive group of real numbers.

More generally, one can define **one-parameter semigroups** of operators in a Banach space $X$ as homomomorphisms of monoids

$\mathbf{R}_{\ge0} \to B(X),$

where $B(X)$ denotes the semigroup of bounded operators $X\to X$.

Typically, we also require a continuity condition such as continuity in the strong topology.

Strongly continuous one-parameter unitary groups $(U_t)_{t\ge0}$ of operators in a Hilbert space $H$ are in bijection with self-adjoint unbounded operators $A$ on $H$:

This bijection sends

$A
\mapsto
\big(
t\mapsto \exp(itA)
\big)
\,.$

The operator $A$ is bounded if and only if $U$ is norm-continuous.

Strongly continuous one-parameter semigroups $T$ of bounded operators on a Banach space $X$ (alias **$C_0$-semigroups**) satisfying $\|T(t)\|\le M\exp(\omega t)$ are in bijection with closed operators $A\colon X\to X$ with dense domain such that any $\lambda\gt \omega$ belongs to the resolvent set of $A$ and for any $\lambda\gt\omega$ we have

$\|(\lambda I-A)^{-n}\|\le M (\lambda-\omega)^{-n}.$

[…]

See also

- Wikipedia,
*$C_0$ semigroup*

Last revised on June 21, 2022 at 07:35:23. See the history of this page for a list of all contributions to it.