It is then a ring with the multiplication, denoted by , such that determined by this condition and the ring axioms. The additive identity is 1 and the multiplicative identity is . Then has a canonical structure of a -algebra given by the ring homomorphism On the other hand, if A is a λ-ring, then there is a ring homomorphism giving a structure of an A-algebra. Given a module M over a commutative ring R, the direct sum of modules has a structure of an R-algebra by thinking M consists of infinitesimal elements; i.e., the multiplication is given as The notion is sometimes called the algebra of dual numbers. Representation theory The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra. If G is a group and R is a commutative ring, the set of all functions from G to R with finite support form an R-algebra with the convolution as multiplication. It is called the group algebra of G. The construction is the starting point for the application to the study of (discrete) groups. If G is an algebraic group (e.g., semisimple complex Lie group), then the coordinate ring of G is the Hopf algebra A corresponding to G. Many structures of G translate to those of A. A quiver algebra (or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph. Analysis Given any Banach space X, the continuous linear operators A : X → X form an associative algebra (using composition of operators as multiplication); this is a Banach algebra. Given any topological space X, the continuous real- or complex-valued functions on X form a real or complex associative algebra; here the functions are added and multiplied pointwise. The set of semimartingales defined on the filtered probability space (Ω, F, (Ft)t ≥ 0, P) forms a ring under stochastic integration. The Weyl algebra An Azumaya algebra Geometry and combinatorics The Clifford algebras, which are useful in geometry and physics. Incidence algebras of locally finite partially ordered sets are associative algebras considered in combinatorics. The partition algebra and its subalgebras, including the Brauer algebra and the Temperley-Lieb algebra. Mathematical physics A Poisson algebra is a commutative associative algebra over a field together with a structure of a Lie algebra so that the Lie bracket