Noncommutative and nonassociative algebras

We choose the abelian group ($\bbZ_2^n, +$) where $\bbZ_2 = \bbZ / 2 \bbZ$ and define a $\bbZ_2^n$-graded vector space
\[
E = \bigoplus_{x \in \bbZ_2^n} E_x
\]
together with a multiplication $ \cdot :E \times E \longrightarrow E$
respecting the grading
\[
E_x \cdot E_y \subset E_{x+y} \quad \forall x,y \in \bbZ_2^n.
\]
This is called a $\bbZ_2^n$-graded algebra.
We are interested in particular $\bbZ_2^n$-graded algebras where the product in noncommutative and nonassociative.

This talk consists of two parts.
The first one is the study of a series of $\bbZ_2^n$-graded algebras of finite dimension ($2^n$) where $n \geq 3$.
This series of real noncommutative and nonassociative algebras, denoted $\bbO_{p,q}$ ($p+q=n$), generalizes the algebra of octonion numbers $\bbO$. This generalization is similar to the one of the algebra of quaternion numbers in Clifford algebras.
The first \emph{question} is to classify these algebras up to isomorphisms.
The classification table of $\bbO_{p,q}$ is quite similar to that of
the real Clifford algebras $\cC l_{p,q}$.
The second \emph{question} is to find a periodicity between these algebras.
The periodicity for the algebras $\bbO_{p,q}$ is analogous to the periodicity for the Clifford algebras $\cC l_{p,q}$.

In the second part we study $\bbZ_2$-graded algebras ($n=0$, ``superalgebras'') that can be of infinite dimension.
We consider two kind of superalgebras $\cL_{g,N}$ and $\cJ_{g,N}$ that are noncommutative and nonassociative\footnote{The construction coming from spaces on a compact Riemann surface of genus $g$ with $N$ punctures}.
Nevertheless, these superalgebras link together the classical Lie algebras and the classical commutative and associative algebras.
The two last \emph{questions} are can we ``extend'' the algebras $\cL_{g,N}$ and $\cJ_{g,N}$?
The first answer is yes (for $\cL_{g,N}$), while the second one is no (for $\cJ_{g,N}$). However, we can ``extend'' module $\cJ_{g,N}^*$.