nLab En-algebra

Definition

An $E_n$-algebra is an ∞-algebra over the E-k operad.

Special cases

$E_1$-algebras

$E_1$-algebras are often called A-∞ algebras. See also algebra in an (∞,1)-category.

An $E_1$ algebra in the symmetric monoidal (∞,1)-category Spec of spectra is a ring spectrum.

$E_2$-algebras

The homology of an $E_2$-algebra in chain complexes is a Gerstenhaber algebra.

$E_\infty$-algebra

See E-∞ algebra.

Examples

• universal enveloping E-n algebra

Properties

Relation to Poisson $n$-algebras

The homology of an $E_n$-algebra for $n \geq 2$ is a Poisson n-algebra.

Moreover, in chain complexes over a field of characteristic 0 the E-n operad is formal, hence equivalent to its homology, and so in this context $E_n$-algebras are equivalent to Poisson n-algebras.

See there for more.

Related concepts

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

• May recognition theorem

• monoidal groupoid, braided monoidal groupoid, symmetric monoidal groupoid

• 2-group, braided 2-group, symmetric 2-group

• k-tuply monoidal n-groupoid

• k-tuply monoidal (n,r)-category

• k-tuply groupal n-groupoid

• A-n algebra

• nonunital Ek-algebra

References

Section 5 of

• Jacob Lurie, Higher Algebra,

some summary of which is at Ek-Algebras.

Discussion of derived noncommutative geometry over formal duals of $E_n$-algebras is in

• John Francis, Derived algebraic geometry over $\mathcal{E}_n$-Rings (pdf)

• John Francis, The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings (pdf)