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Lambda-Rings LAMBDA-RINGS ZEHUI CHEN Abstract. We give a brief introduction to the theory of λ-rings accessible to undergraduate students. We begin by introducing the concepts of tensor product, exterior power, and graded vector spaces. Then we give the definition of λ-rings and discuss several examples. Finally, we explain how direct sums, tensor products, and exterior powers of (graded) vector spaces induce λ-ring structures on the Grothendieck groups of these categories. Contents 1. Introduction1 2. Tensor Product2 3. Exterior Power4 4. Graded Vector Spaces6 5. λ-Rings8 6. Direct Sum, Tensor Product, Exterior Power and λ-Ring Structure 12 References 14 1. Introduction In algebra, a λ-ring is a commutative ring equipped with a λ-operation. The study of λ-rings is mainly motivated by the behaviour of tensor products and exterior powers over vector spaces. The categories of vector spaces and graded vector spaces, with direct sum, tensor product, and exterior power, induce the addition, multiplication, and λ-operation in λ-rings of integers Z and Laurent polynomials. The purpose of this paper is to give a brief introduction to the subject accessible to under- graduate students with basic knowledge of linear algebra and group theory. In Section2, we provide the definition of the tensor product and show its basic properties, i.e. the universal property, its dimension, and a kind of distributivity. In Section3, we define the exterior power of vector spaces and prove the theorem of dimensions of exterior powers. After that, we explain graded vector spaces and graded dimension of tensor products and exterior powers in Section4. Then, in Section5, the axioms of λ-rings are given as well as some simple exam- ples of λ-rings, including the integers Z, the ring of power series, and the ring of symmetric functions. Finally, we explain how the operations of direct sum, tensor product, and exterior power of (graded) vector spaces, induce λ-ring structures on the integers and on the ring of Laurent polynomials, when we identify these rings with the Grothendieck groups of the categories of finite-dimensional vector spaces and finite-dimensional graded vector spaces. 2 ZEHUI CHEN Acknowledgements. This project was completed as part of an Undergraduate Research Project at the University of Ottawa. I would like to thank Professor Alistair Savage at the University of Ottawa for his advice and support. 2. Tensor Product Definition 2.1. (Bilinear map). Let U, V , and W be three vector spaces over the same field F . Then a bilinear map b: U × V ! W is a mapping such that (a) b(u1 + u2; v) = b(u1; v) + b(u2; v) for all u1; u2 2 U, v 2 V (b) b(u; v1 + v2) = b(u; v1) + b(u; v2) for all u 2 U, v1; v2 2 V (c) b(αu; v) = αb(u; v) = b(u; αv) for all u 2 U, v 2 V , α 2 F In other words, b(·; v) is linear from U to W for all v 2 V , and b(u; ·) is linear from V to W for all u 2 U. Example 2.2. Let M be an m × n complex matrix, U be the vector space Cm, and V be the vector space Cn. Define the map b: U × V ! C by b(u; v) = uT Mv for all u 2 U and v 2 V where uT is the transpose of vector u. Then, the function b is a bilinear map. Definition 2.3. (Tensor product). Let U; V be vector spaces. The tensor product of U and V , denoted U ⊗ V , is a vector space together with a bilinear map ⊗: U × V ! U ⊗ V such that for any bilinear map b: U × V ! W , there exists a unique linear map `: U ⊗ V ! W such that b = ` ◦ ⊗, that is, such that the following diagram commutes: ⊗ U × V U ⊗ V ` b W Definition 2.4. (Free vector space). Let S be a set and F be a field. The free vector space F (S) generated by S is defined as F (S) = ff : S ! F j f(s) = 0 for all s 2 S n S0 where S0 is some finite subset of Sg where the operations on F (S) are defined as pointwise addition and scalar multiplication. Definition 2.5. (Quotient space). Let V be a vector space and W ⊆ V be a subspace of V . Define ∼ to be the equivalence relation such that v1 ∼ v2 if and only if v1 − v2 2 W for all v1; v2 2 V . Then, the quotient space V=W is defined as V= ∼ and the elements in V=W are equivalence classes [v] for v 2 V . Definition 2.6. (Tensor product). Let U and V be vector spaces over field F . Define W ⊆ F (U × V ) to be W = f(u1 + u2; v) − (u1; v) − (u2; v) j u1; u2 2 U; v 2 V g [ f(u; v1 + v2) − (u; v1) − (u; v2) j u 2 U; v1; v2 2 V g [ fα(u; v) − (αu; v) j u 2 U; v 2 V; α 2 F g [ fα(u; v) − (u; αv) j u 2 U; v 2 V; α 2 F g: LAMBDA-RINGS 3 Then, the tensor product of U and V , denoted U ⊗V , is defined as U ⊗V = F (U ×V )=W . For u 2 U and v 2 V , the tensor product of u and v, denoted u ⊗ v, is defined as u ⊗ v = [(u; v)]. It follows from the definition of the quotient space that the tensor product satisfies the following properties: (a)( u1 + u2) ⊗ v = u1 ⊗ v + u2 ⊗ v for all u1; u2 2 U and v 2 V . (b) u ⊗ (v1 + v2) = u ⊗ v1 + u ⊗ v2 for all u 2 U and v1; v2 2 V . (c) αu ⊗ v = α(u ⊗ v) = u ⊗ αv for all u 2 U; v 2 V , and α 2 F Remark 2.7. One can show that the tensor product as defined in Definition2 :6 satisfies the universal property of Definition2 :3. Theorem 2.8. Let U and V be two finite-dimensional vector spaces over field F . Then, dim(U ⊗ V ) = dim U dim V: Proof. Let dim U = m and dim V = n. Then, U has a basis S1 = fu1; u2; : : : ; umg and V has a basis S2 = fv1; v2; : : : ; vng. Now, define S = fui ⊗ vj : ui 2 S1; vj 2 S2g. The set S is a subset of U ⊗ V with mn elements, so it is enough to prove that S is a basis of U ⊗ V . For any element that can be written as u ⊗ v 2 U ⊗ V , let u = α1u1 + α2u2 + ::: + αmum and v = β1v1 + β2v2 + ::: + βnvn for αi; βj 2 F . Then, it becomes m n X X u ⊗ v = (α1u1 + α2u2 + ::: + αmum) ⊗ (β1v1 + β2v2 + ::: + βnvn) = αiβj(ui ⊗ vj): i=1 j=1 Then, any arbitrary element x = u ⊗ v + u0 ⊗ v0 + ::: can be written as m n X X x = cij(ui ⊗ vj): i=1 j=1 It follows x 2 Span(S), so U ⊗ V ⊆ Span(S). Obviously, S ⊆ U ⊗ V , so Span(S) ⊆ U ⊗ V . Thus, we have U ⊗ V = Span(S). Let 1; : : : ; m be the dual basis to S1 and let φ1; : : : ; φn be the dual basis to S2. Then define the maps bi;j(u; v) = i(u)φj(v) for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. When v is fixed, φj(v) becomes a fixed scalar c 2 F , then bi;j(u; v) = c i(u). The dual basis i(u) is a linear transformation, so bi;j(·; v) is linear from U to F for all v 2 V . Similarly, bi;j(u; ·) is linear from V to F for all u 2 U. Hence, bi;j(u; v) = i(u)φj(v) are bilinear maps. By the universal property, these induce maps on the tensor product, i.e. for all 1 ≤ k ≤ m; 1 ≤ l ≤ n, there exist `k;l such that `k;l(u ⊗ v) = bk;l(u; v). Now, fix 1 ≤ k ≤ m; 1 ≤ l ≤ n, suppose m n X X ci;jui ⊗ vj = 0 for some ci;j 2 F; 1 ≤ i ≤ m; 1 ≤ j ≤ n: i=1 j=1 Then, `k;l(u ⊗ v) = `k;l(0) = 0, which means m n X X ci;jbk;l(ui; vj) = ck;l = 0 for 1 ≤ i ≤ m; 1 ≤ j ≤ n: i=1 j=1 Thus, ck;l = 0 for all 1 ≤ k ≤ m; 1 ≤ l ≤ n. Consequently, S is linearly independent. Therefore, S is a basis of U ⊗V . So, we have dim(U ⊗V ) = jSj = mn = dim U dim V . 4 ZEHUI CHEN Proposition 2.9 (Distributive law of tensor product over direct sum). Let U; V; W be finite- dimensional vector spaces, then U ⊗ (V ⊕ W ) ∼= (U ⊗ V ) ⊕ (U ⊗ W ): Proof. Let U be an l-dimensional vector space, V be an m-dimensional vector space and W be an n-dimensional vector space. Then, by Theorem 2.8, the dimension of U ⊗ (V ⊕ W ) is dim(U ⊗ (V ⊕ W )) = dim(U) dim(V ⊕ W ) = dim(U)(dim(V ) + dim(W )) = dim(U) dim(V ) + dim(U) dim(W ): On the other hand, we have dim((U ⊗ V ) ⊕ (U ⊗ W )) = dim(U ⊗ V ) + dim(U ⊗ W ) = dim(U) dim(V ) + dim(U) dim(W ): Thus, U ⊗(V ⊕W ) and (U ⊗V )⊕(U ⊗W ) have the same dimension, so they are isomorphic. 3. Exterior Power If we generalize the idea of tensor product in Definition2 :3, we can replace the vector spaces U; V with vector spaces V1;V2;:::Vk and change the bilinear map to multilinear map.
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