In mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as "Hermitian K-theory", is important in surgery theory.

One can define L-groups for any ring with involution R: the quadratic L-groups ${\displaystyle L_{*}(R)}$ (Wall) and the symmetric L-groups ${\displaystyle L^{*}(R)}$ (Mishchenko, Ranicki).

The even-dimensional L-groups ${\displaystyle L_{2k}(R)}$ are defined as the Witt groups of ε-quadratic forms over the ring R with ${\displaystyle \epsilon =(-1)^{k}}$. More precisely,

is the abelian group of equivalence classes ${\displaystyle [\psi ]}$ of non-degenerate ε-quadratic forms ${\displaystyle \psi \in Q_{\epsilon }(F)}$ over R, where the underlying R-modules F are finitely generated free. The equivalence relation is given by stabilization with respect to hyperbolic ε-quadratic forms:

The addition in ${\displaystyle L_{2k}(R)}$ is defined by

The zero element is represented by ${\displaystyle H_{(-1)^{k}}(R)^{n}}$ for any ${\displaystyle n\in {\mathbb {N} }_{0}}$. The inverse of ${\displaystyle [\psi ]}$ is ${\displaystyle [-\psi ]}$.

Defining odd-dimensional L-groups is more complicated; further details and the definition of the odd-dimensional L-groups can be found in the references mentioned below.

The L-groups of a group ${\displaystyle \pi }$ are the L-groups ${\displaystyle L_{*}(\mathbf {Z} [\pi ])}$ of the group ring ${\displaystyle \mathbf {Z} [\pi ]}$. In the applications to topology ${\displaystyle \pi }$ is the fundamental group ${\displaystyle \pi _{1}(X)}$ of a space ${\displaystyle X}$. The quadratic L-groups ${\displaystyle L_{*}(\mathbf {Z} [\pi ])}$ play a central role in the surgery classification of the homotopy types of ${\displaystyle n}$-dimensional manifolds of dimension ${\displaystyle n>4}$, and in the formulation of the Novikov conjecture.