By using some details and facts from [2, 3], Lodder [4] has proved that the structure of the Leibniz homology of [g.sub.n] is determined by the exterior algebra of the forms [mathematical expression not reproducible] are the unit vector fields parallel to [x.sub.i] and [y.sub.i] axes, respectively, and the Lie algebra homology [H.sup.Lie.sub.*]([g.sub.n]) has been proved to have an isomorphic vector space as follows: [mathematical expression not reproducible] is the singular homology of the real symplectic Lie algebra s[p.sub.n] and [mathematical expression not reproducible].

The result shows that the image is the tensor of a real number with the exterior algebra [[conjunction].sup.*] ([w.sub.n]).

There is a canonical projection T([g.sub.n]) [right arrow] [[conjunction].sup.*]([g.sub.n]), where T([g.sub.n]) is the tensor algebra of [g.sub.n] and [[conjunction].sup.*]([g.sub.n]) is the exterior algebra of [g.sub.n], which is naturally defined by [mathematical expression not reproducible] Thus, the map n induces a k-linear map on homology

For the affine symplectic Lie algebra [g.sub.n], the image of H[L.sub.*]([g.sub.n]) in the Hochschild homology [H.sup.Lie.sub.*- 1](U([g.sub.n])) can be identified injectively as the exterior algebra [[conjunction].sup.*]([w.sub.n]).

(C) There is an isomorphism between the Clifford algebra Cl(d) in d-dimensions and the exterior algebra [LAMBDA] x M of cotangent bundle [T.sup.*] M over M [9, 10] (the space of the Clifford algebra Cl(d) is isomorphic, as a vector space, to the vector space of the exterior algebra [LAMBDA] x M.

[[GAMMA].sup.d] in the Clifford algebra Cl(d) corresponds to the Hodge-dual operator * : [[omega].sup.k](M) [right arrow] [[omega].sup.d-k] (M) in the exterior algebra [LAMBDA] * M where [[GAMMA].sup.A] (A = 1, ..., d) are d-dimensional Dirac matrices obeying the Dirac algebra

It is amusing to note that the Clifford algebra from a modern viewpoint can be thought of as a quantization of the exterior algebra [10], in the same way that the Weyl algebra is a quantization of the symmetric algebra.

As was pointed out in (41), the Clifford algebra (40) is isomorphic to the exterior algebra [LAMBDA] * M as vector spaces, so the 't Hooft symbol in (59) has a one-to-one correspondence with the basis of two forms in [[OMEGA].sup.2.sub.[+ or -]](M) = [[OMEGA].sup.2](M) [direct sum] * [[OMEGA].sup.4] (M) depending on the chirality for a given orientation.

An N-graded commutative K-ring [[LAMBDA].sup.*] is called the Grassmann-graded K-ring if it is finitely generated in degree 1 (Definition 25) so that it is the exterior algebra of [[LAMBDA].sup.*] = [disjunction] [[LAMBDA].sup.1] of a K-module [[LAMBDA].sup.1] (Example 9).

Its quotient [disjucntion]Q with respect to an ideal generated by elements q [cross product] q +q [cross product] q, q,q [member of] Q, is an N-graded commutative algebra, called the exterior algebra of an A-module Q.

For instance, the exterior algebra [disjunction] Q of a K-module Q in Example 9 is a [Z.sub.2]-graded commutative ring.

is the exterior algebra of a [Z.sub.2]-graded module [P.sub.*] with respect to the graded exterior product