The theory of finite-dimensional algebras is one of the oldest branches of modern algebra. The first part consists of chapters 1-3; here the basic concepts of the theory of algebras are discussed and the classical theory of semisimple algebras and radicals is explained.
Introduction
Basic Concepts. Examples
An element e of an algebra A is called the identity of the algebra if ae = ea = a for an arbitrary element a E A. The set of all elements c of an algebra A which together with all elements of the algebra commuting, i.
Isomorphisms and Homomorphisms. Division Algebras
Let A be finite dimensional and let E A be an element that is not a left divisor of zero. If an element a E A is a left divisor of zero, then, given a), it cannot be a right divisor of the identity.
Representations and Modules
Obviously, the correspondence a I t (T(a)) is a homomorphism of the algebra A with the matrix algebra Mn(K), where n is the dimension of the representation T. Assigning to each a the operator T(a) (or its matrix in relation to a basis), we obtain a representation of the algebra A corresponding to the module M.
Submodules and Factor Modules. Ideals and Quotient Algebras
Consequently, we can define on the set of congruence classes the structure of an A-module which defines. We call this algebra the quotient algebra of the algebra A with ideal I and denote it by A/I.
The Jordan-Holder Theorem
The factor modules M;/Mi+l are called the factors of this series and their number s is the length of the series. The length of a compositional series is called the length of the module M and is denoted by e(M), and the factors of a compositional series are called the simple factors of the module M.
Direct Sums
If the above conditions are satisfied, we say that M is decomposable into the direct sum of its submodules Na and L, and we write M = N EB L. In this case, the submodule L is called the complement of the submodule N (and vice versa).
Endomorphisms. The Peirce Decomposition
In this way, the presentation of the algebra EA(M) ensures a faithful representation (more precisely, anti-representation because we agreed to write the endomorphisms on the left and the linear transformations on the right of the elements). Compute a regular matrix representation of the Jordan algebra In(I<); find all ideals of In(I<).
Semisimple Algebras
- Schur's Lemma
- Semisimple Modules and Algebras
- Vector Spaces and Matrices
- The Wedderburn-Artin Theorem
- Uniqueness of the Decomposition
- Representations of Semisimple Algebras
The simple submodules of the regular right (left) A-module are called minimal right (left) ideals of the algebra A. Every simple I< -algebra is isomorphic to an algebra of the form Mn(D), where D is a division. algebra.
The Radical
The Radical of a Module and of an Algebra
A radical of a regular algebra module A is called a radical of the algebra. The radical of the algebra is a unique zero ideal, so the quotient algebra in question is semisimple.
Lifting of Idempotents. Principal Modules
In what follows we will often make use of the next characteristic of the radical. Since all elements of the radical are non-invertible, radical R is, as required, just the set of all non-invertible elements.
Projective Modules and Projective Covers'
If a module 1\.1 has a generator set consisting of n elements, then M is isomorphic to a factor module of the free module of rank n. In view of the fact that there is a bijective correspondence between the decompositions of the identity and the decompositions of the module (theorem 1.7.2), is ei + R = ei as required. island.
The Krull-Schmidt Theorem
EEl N m are two decompositions of the module M into a direct sum of nondecomposable modules, then 11 = m and, after a suitable reindexing, Mi ~ Ni for all i. From Theorem 1.7.2, we have two decompositions of the algebra identity E = EA(M) corresponding to the two decompositions of M into a direct sum of indecomposable modules: 1 = el+ e2+···+ en = h+h+ .
The Radical of an Endomorphism Algebra
EEl Ps is a decomposition of a regular module into a direct sum of principal modules, then Pj 'I-Pj for i i= j. EEl nsP is a decomposition of a regular A-module into a direct sum of principal modules, where Pi 'I- Pj for i i= j. A is isomorphic to the endomorphism algebra of the projective module P over the basic algebra B. to isomorphism) uniquely and is isomorphic to the basic algebra of the algebra A.
Diagram of an Algebra
If an arrow CT of diagram V connects point i to point j, then i is called the tail (origin) and j is called the head (top) of the arrow CT. The set of all elements {v,} (over all arrows of the diagram V) is an aggregate of the module V. A basis of the space K(D) is formed by all possible paths of the diagram and by the symbols {cd (indexed by the points of D).
Hereditary Algebras
Prove that a minimal real ideal is either a principal module or contained in the radical of the algebra. Let A be an algebra over the real JR consisting of 2 x 2 complex matrices of the form. Denote by M
Central Simple Algebras
- Bimodules
- Tensor Products
- Central Simple Algebras
- Fundamental Theorems of the Theory of Division Algebras
- Subfields of Division Algebras. Splitting Fields
- Brauer Group. The Frobenius Theorem
We will apply Theorem 4.3.1 to investigate the structure of the algebra A x B, where A is a central simple and B is an arbitrary K-algebra. Each ideal of the algebra A0B, where A is a central simple algebra, has the form A 0 I, where I is an ideal of the algebra B. Finally, Theorem 4.3.1 shows that the opposite algebra A ° is an inverse of the algebra A in the sense of this operation.
Galois Theory
Elements of Field Theory
A field L :::> K is called a splitting field of the polynomial f( x) over K if f( x) decomposes into linear factors over L, and does not decompose into linear factors over any proper subfield of L containing . The existence of a splitting field follows from the argument above; we can take L to be the subfield of Kn generated by K and the roots of f(x). Since L is a subfield of a field that can be obtained from K by constructing a chain of finite extensions, our theorem is a specific example of the following result.
Finite Fields. The Wedderburn Theorem
Thus, if G is the multiplicative group of the division ring D and H the multiplicative group of a maximal field L, then G = UgHg-1, where 9 runs through all of G.
Separable Extensions
Let Land F be finite extensions of the field K and assume that F = K[a], where the minimal polynomial of a over K is p(x). An irreducible polynomial is called separable if p( x) has no more roots in any extension of the field K. The Galois group 91 An element a in the field L is called an invariant of automorphism (J if (J(a) = a.
The Fundamental Theorem of Galois Theory
We can define the scale of the field Inv Hover 1< using the isomorphism of the left 1 Assume that the characteristic of the body [{ is not 2. Show that every quadratic extension L of the field K, i. Let K be a field with characteristic p > 0, F = K[aJ a finite monogenic extension of K and m( x) the minimal polynomial of the element a over K. Then prove that L = K[a], and the minimal polynomial of the element a has the form xn - a. According to Corollary 5.6.2, there is a separable extension F of the field Ci such that Ai 0c; F ~ Mk(F). In view of Corollary 3.1.8, the radical of a regular bimodule coincides with the radical of the algebra. If the L-algebra AL is separable for some extension L of the field f{, then A is separable. By comparing the coefficients of the basis vectors ae, we obtain a system of linear equations in x ke. Since the coefficients of the equations are the structure constants, they do not change under ground field expansions. If a 8y8tem of linear equation8 with coefficient8 of a field K has a 80lution in an extension of the field, then it has a 801ution in K. In general, the coefficients of XM,a(X), and in particular, TrM(a), and NM(a) are elements of the field L. A K -algebra A is separable if and only if there is a non- degenerate exists AL-module M for some extension L of the field K. Then, for any splitting field L, F®L ~ Ln, where n = [F : I<], and the leading polynomial coincides with the characteristic polynomial of the regular module. The square matrix X = (Xii) is called the character table of the group G over the field K. The elements of the matrix (T 0 S)(g) corresponding to 9 in this representation12 are thus all possible products of the elements of T(g) and S(g ). Denote by G the set of all irreducible representations of the group G over the field K (these are the characters in Gover K). The product gf in the category Co is defined as the product fg in the category C. This functor is called the forgetful functor from the category of groups to the category of sets. V are called contravariant functors from the category C to the category V (and to emphasize that it preserves the direction of the arrows, the ordinary functors from C to V are called covariant functors). In the following, all categories and functors will be assumed to be linear, and we will often use Corollary 8.1.2 without any reference. A module P is projective if and only if every diagram. whose row is correct can be completed in a commutative diagram. recall that exactness means that 9 is an epimorphism and commutativity means that f = 9 j). In view of proposition 1.6.2, it suffices to require the existence of only 1. or g); in this case, M can be identified with the direct sum N EB L, f is the canonical inclusion N -+ N EB L (the mapping of x E N to (x, 0)) and g' the canonical projection of N EB L in the second addition . It is not difficult to see that the isomorphisms constructed in Proposition 8.3.3 and 8.3.4 are actually isomorphisms of the corresponding functors. Denote by Tensor Algebras and Hereditary Algebras 153 to a quotient algebra of the algebra T(V) according to an admissible ideal, where V = RjR2. Prove that the top row of the diagram is exact (this is called a cancellation of the given exact series along 'P'). G is isomorphic to a direct summation of a representation of the form ResH(T), where T is a decomposable representation of G. b). We have already seen the importance of projective and especially of principle modules in the study of the structure of algebras. The existence and properties of injective hulls, given the duality, follow immediately from the respective results on projective hedges. We will formulate the facts that will be necessary in the sequel in the following statement, the proof of which is left to the reader. It follows that nM ~ W ffi X and thus, by the Krull-Schmidt theorem, every nondecomposable direct summation of W is isomorphic to one of the M;s. Let W be an indecomposable module A such that every module A-M is of the form Ml ffi k W, where Ml is a module over a proper coefficient algebra B of the algebra A. Every coprincipal B-module is either a coprincipal A-module or a maximal submodule W2 of the W-module (unique by Corollary 9.2.5). Using Proposition 9.2.6 and the definition of the diagram of an algebra we get the following statement. This means that the lattice of the sub-modules of P is anti-isomorphic to the lattice of the sub-modules of A = Since it is isomorphic to a factor module of a principal module, M contains a unique maximal submodule MI = radM (Corollary 3.2.8). A is serial if and only if all submodules of main right and left A-modules contain a unique maximal submodule. In turn, rad PI is a factor module of a main A-module P2 and thus contains a unique maximal submodule Ma = radM2. To describe all diagrams of right serial algebras, we introduce the following definitions. Then the algebra A is properly serial if and only if, for any minimal idempotent e E B, the proper B-module eV is simple (or zero). Thus, to describe right serial algebras of separable type, it is sufficient to show all admissible ideals of the tensor algebra T = T(V) of a right serial bimodule V over a separable algebra B. A is isomorphic to an algebra of triangular matrices via a division algebra. es is a minimal decomposition of the identity of the. Prove that an algebra A is correctly serial if and only if there is a diagram of the form. Prove that the tensor algebra T(V) is isomorphic to the algebra of matrices of the form. Complexes and Homology Resolutions and Derived Functors Ext and Tor. Extensions Homological Dimensions Duality Almost Split Sequences Preliminaries. Standard and Costandard Modules Trace Filtrations. The Categories F(Ll) and F(\7) Basic Properties Characterization of the Category F(Ll) Final RemarksCrossed Products
Separable Algebras
Bimodules over Separable Algebras
The Wedderburn-Malcev Theorem
Trace, Norm, Discriminant
Representations of Finite Groups
The Morita Theorem
Categories and Functors
Exact Sequences
Tensor Products
Tensor Algebras and Hereditary Algebras
Quasi-Frobenius Algebras
Duality. Injective Modules
Lemma on Separation
Quasi-Frobenius Algebras
Serial Algebras
The N akayama-Skornjakov Theorem
Right Serial Algebras
The Structure of Serial Algebras
Elements of Homological Algebra
