Misplaced Pages

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67-459: One can define L -groups for any ring with involution R : the quadratic L -groups L ∗ ( R ) {\displaystyle L_{*}(R)} (Wall) and the symmetric L -groups L ∗ ( R ) {\displaystyle L^{*}(R)} (Mishchenko, Ranicki). The even-dimensional L -groups L 2 k ( R ) {\displaystyle L_{2k}(R)} are defined as

134-479: A {\displaystyle a} commutes with a basis of K [ G ] {\displaystyle K[G]} , and therefore And we see that a , b {\displaystyle a,b} are not zero, which shows K [ G ] {\displaystyle K[G]} is not prime. This shows the original statement. Group algebras occur naturally in the theory of group representations of finite groups . The group algebra K [ G ] over

201-473: A 2 = ∑ h ∈ H h a = | H | a {\displaystyle a^{2}=\sum _{h\in H}ha=|H|a} . Taking b = | H | 1 − a {\displaystyle b=|H|\,1-a} , we have a b = 0 {\displaystyle ab=0} . By normality of H {\displaystyle H} ,

268-597: A chain duality , as in Ranicki (section 1). The simply connected L -groups are also the L -groups of the integers, as L ( e ) := L ( Z [ e ] ) = L ( Z ) {\displaystyle L(e):=L(\mathbf {Z} [e])=L(\mathbf {Z} )} for both L {\displaystyle L} = L ∗ {\displaystyle L^{*}} or L ∗ . {\displaystyle L_{*}.} For quadratic L -groups, these are

335-472: A finite group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of finite sums, corresponds to functions on the group that vanish for cofinitely many points; topologically (using the discrete topology ), these correspond to functions with compact support . However, the group algebra K [ G ] and the space of functions K  := Hom( G , K ) are dual: given an element of

402-431: A matrix T . Every matrix has a transpose , obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwise complex conjugation is an independent involution, the conjugate transpose or Hermitian adjoint is also an involution. The definition of involution extends readily to modules . Given a module M over a ring R , an R endomorphism f of M

469-548: A =0 and b =1 then f 4 ( x ) := ( f 1 ∘ f 2 ) ( x ) = ( f 2 ∘ f 1 ) ( x ) = − 1 x {\displaystyle f_{4}(x):=(f_{1}\circ f_{2})(x)=(f_{2}\circ f_{1})(x)=-{\frac {1}{x}}} is an involution, and more generally the function g ( x ) = x + b c x − 1 {\displaystyle g(x)={\frac {x+b}{cx-1}}}

536-444: A basis for a vector space V is chosen, and that e 1 and e 2 are basis elements. There exists a linear transformation f that sends e 1 to e 2 , and sends e 2 to e 1 , and that is the identity on all other basis vectors. It can be checked that f ( f ( x )) = x for all x in V . That is, f is an involution of V . For a specific basis, any linear operator can be represented by

603-584: A central role in the surgery classification of the homotopy types of n {\displaystyle n} -dimensional manifolds of dimension n > 4 {\displaystyle n>4} , and in the formulation of the Novikov conjecture . The distinction between symmetric L -groups and quadratic L -groups, indicated by upper and lower indices, reflects the usage in group homology and cohomology. The group cohomology H ∗ {\displaystyle H^{*}} of

670-458: A color value stored as integers in the form ( R , G , B ) , could exchange R and B , resulting in the form ( B , G , R ) : f ( f (RGB)) = RGB, f ( f (BGR)) = BGR . Legendre transformation , which converts between the Lagrangian and Hamiltonian , is an involutive operation. Integrability , a central notion of physics and in particular the subfield of integrable systems ,

737-413: A different element s as s = w 0 1 G + w 1 a + w 2 a 2 {\displaystyle s=w_{0}1_{G}+w_{1}a+w_{2}a^{2}} , their sum is and their product is Notice that the identity element 1 G of G induces a canonical embedding of the coefficient ring (in this case C ) into C [ G ]; however strictly speaking

SECTION 10

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804-402: A field K is essentially the group ring, with the field K taking the place of the ring. As a set and vector space, it is the free vector space on G over the field K . That is, for x in K [ G ], The algebra structure on the vector space is defined using the multiplication in the group: where on the left, g and h indicate elements of the group algebra, while the multiplication on

871-625: A field has a further structure of a Hopf algebra ; in this case, it is thus called a group Hopf algebra . The apparatus of group rings is especially useful in the theory of group representations . Let G {\displaystyle G} be a group, written multiplicatively, and let R {\displaystyle R} be a ring. The group ring of G {\displaystyle G} over R {\displaystyle R} , which we will denote by R [ G ] {\displaystyle R[G]} , or simply R G {\displaystyle RG} ,

938-420: A finite set and its number of elements have the same parity . Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an odd number of elements has at least one fixed point . This can be used to prove Fermat's two squares theorem . The graph of an involution (on the real numbers) is symmetric across the line y = x . This

1005-407: A given value for one parameter is an involution on the other parameter. XOR masks in some instances were used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state. Two special cases of this, which are also involutions, are the bitwise NOT operation which is XOR with an all-ones value, and stream cipher encryption , which

1072-514: A group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups . An element x of a group G is called strongly real if there is an involution t with  x = x (where  x = x = t ⋅ x ⋅ t ). Coxeter groups are groups generated by a set S of involutions subject only to relations involving powers of pairs of elements of S . Coxeter groups can be used, among other things, to describe

1139-505: A mapping f {\displaystyle f} is defined as the mapping x ↦ α ⋅ f ( x ) {\displaystyle x\mapsto \alpha \cdot f(x)} , and the module group sum of two mappings f {\displaystyle f} and g {\displaystyle g} is defined as the mapping x ↦ f ( x ) + g ( x ) {\displaystyle x\mapsto f(x)+g(x)} . To turn

1206-463: A negation is realized as an involution on the algebra of truth values . Examples of logics that have involutive negation are Kleene and Bochvar three-valued logics , Łukasiewicz many-valued logic , the fuzzy logic ' involutive monoidal t-norm logic ' (IMTL), etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in t-norm fuzzy logics . The involutiveness of negation

1273-755: A real vector space , while the skew field of quaternions has dimension 4 as a real vector space . 4. Another example of a non-abelian group ring is Z [ S 3 ] {\displaystyle \mathbb {Z} [\mathbb {S} _{3}]} where S 3 {\displaystyle \mathbb {S} _{3}} is the symmetric group on 3 letters. This is not an integral domain since we have [ 1 − ( 12 ) ] ∗ [ 1 + ( 12 ) ] = 1 − ( 12 ) + ( 12 ) − ( 12 ) ( 12 ) = 1 − 1 = 0 {\displaystyle [1-(12)]*[1+(12)]=1-(12)+(12)-(12)(12)=1-1=0} where

1340-413: A subring isomorphic to R , and its group of invertible elements contains a subgroup isomorphic to G . For considering the indicator function of {1 G }, which is the vector f defined by the set of all scalar multiples of f is a subring of R [ G ] isomorphic to R . And if we map each element s of G to the indicator function of { s }, which is the vector f defined by the resulting mapping

1407-490: A transformation x ↦ f ( x ) {\displaystyle x\mapsto f(x)} then it is an involution if An anti-involution does not obey the last axiom but instead This former law is sometimes called antidistributive . It also appears in groups as ( xy ) = ( y ) ( x ) . Taken as an axiom, it leads to the notion of semigroup with involution , of which there are natural examples that are not groups, for example square matrix multiplication (i.e.

SECTION 20

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1474-503: Is [ − ψ ] {\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 π {\displaystyle \pi } are the L -groups L ∗ ( Z [ π ] ) {\displaystyle L_{*}(\mathbf {Z} [\pi ])} of

1541-463: Is a bijection . The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation ( x ↦ − x ), reciprocation ( x ↦ 1/ x ), and complex conjugation ( z ↦ z ) in arithmetic ; reflection , half-turn rotation , and circle inversion in geometry ; complementation in set theory ; and reciprocal ciphers such as the ROT13 transformation and

1608-443: Is a polarity that is a correlation of period 2. In linear algebra, an involution is a linear operator T on a vector space, such that T = I . Except for in characteristic 2, such operators are diagonalizable for a given basis with just 1 s and −1 s on the diagonal of the corresponding matrix. If the operator is orthogonal (an orthogonal involution ), it is orthonormally diagonalizable. For example, suppose that

1675-538: Is a nonidentity finite normal subgroup of G {\displaystyle G} . Take a = ∑ h ∈ H h {\displaystyle a=\sum _{h\in H}h} . Since h H = H {\displaystyle hH=H} for any h ∈ H {\displaystyle h\in H} , we know h a = a {\displaystyle ha=a} , therefore

1742-416: Is a zero divisor: For example, consider the group ring Z [ S 3 ] and the element of order 3 g =(123). In this case, A related result: If the group ring K [ G ] {\displaystyle K[G]} is prime , then G has no nonidentity finite normal subgroup (in particular, G must be infinite). Proof: Considering the contrapositive , suppose H {\displaystyle H}

1809-435: Is an XOR with a secret keystream . This predates binary computers; practically all mechanical cipher machines implement a reciprocal cipher , an involution on each typed-in letter. Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way. Another involution used in computers is an order-2 bitwise permutation. For example.

1876-564: Is an important characterization property for logics and the corresponding varieties of algebras . For instance, involutive negation characterizes Boolean algebras among Heyting algebras . Correspondingly, classical Boolean logic arises by adding the law of double negation to intuitionistic logic . The same relationship holds also between MV-algebras and BL-algebras (and so correspondingly between Łukasiewicz logic and fuzzy logic BL ), IMTL and MTL , and other pairs of important varieties of algebras (respectively, corresponding logics). In

1943-543: Is an injective group homomorphism (with respect to multiplication, not addition, in R [ G ]). If R and G are both commutative (i.e., R is commutative and G is an abelian group ), R [ G ] is commutative. If H is a subgroup of G , then R [ H ] is a subring of R [ G ]. Similarly, if S is a subring of R , S [ G ] is a subring of R [ G ]. If G is a finite group of order greater than 1, then R [ G ] always has zero divisors . For example, consider an element g of G of order | g | = m > 1. Then 1 - g

2010-2025: Is an involution for constants b and c which satisfy bc ≠ −1 . (This is the self-inverse subset of Möbius transformations with a = − d , then normalized to a = 1 .) Other nonlinear examples can be constructed by wrapping an involution g in an arbitrary function h and its inverse, producing f := h − 1 ∘ g ∘ h {\displaystyle f:=h^{-1}\circ g\circ h} , such as: f ( x ) = 1 − x 2 g ( x ) = 1 − x h ( x ) = x 2 , f ( x ) = ln ⁡ ( e x + 1 e x − 1 ) g ( x ) = x + 1 x − 1 h ( x ) = e x , f ( x ) = exp ⁡ ( 1 ln ⁡ x ) g ( x ) = 1 x h ( x ) = ln ⁡ x , f ( x ) = x x 2 − 1 g ( x ) = x x − 1 h ( x ) = x 2 . {\displaystyle {\begin{alignedat}{3}f(x)&={\sqrt {1-x^{2}}}&g(x)&=1-x&h(x)&=x^{2},\\f(x)&=\ln \left({\frac {e^{x}+1}{e^{x}-1}}\right)&g(x)&={\frac {x+1}{x-1}}&h(x)&=e^{x},\\f(x)&=\exp \left({\frac {1}{\ln x}}\right)&g(x)&={\frac {1}{x}}&h(x)&=\ln x,\\f(x)&={\frac {x}{\sqrt {x^{2}-1}}}&\qquad g(x)&={\frac {x}{x-1}}&\quad h(x)&=x^{2}.\end{alignedat}}} Other elementary involutions are useful in solving functional equations . A simple example of an involution of

2077-409: Is called an involution if f is the identity homomorphism on M . Involutions are related to idempotents ; if 2 is invertible then they correspond in a one-to-one manner. In functional analysis , Banach *-algebras and C*-algebras are special types of Banach algebras with involutions. In a quaternion algebra , an (anti-)involution is defined by the following axioms: if we consider

L-theory - Misplaced Pages Continue

2144-401: Is closely related to involution, for example in context of Kramers–Wannier duality . Group ring In algebra , a group ring is a free module and at the same time a ring , constructed in a natural way from any given ring and any given group . As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law

2211-825: Is due to the fact that the inverse of any general function will be its reflection over the line y = x . This can be seen by "swapping" x with y . If, in particular, the function is an involution , then its graph is its own reflection. Some basic examples of involutions include the functions f 1 ( x ) = a − x , f 2 ( x ) = b x , f 3 ( x ) = x c x − 1 , {\displaystyle {\begin{alignedat}{1}f_{1}(x)&=a-x,\\f_{2}(x)&={\frac {b}{x}},\\f_{3}(x)&={\frac {x}{cx-1}},\\\end{alignedat}}} These may be composed in various ways to produce additional involutions. For example, if

2278-486: Is in fact a field K {\displaystyle K} , then the module structure of the group ring R G {\displaystyle RG} is in fact a vector space over K {\displaystyle K} . 1. Let G = C 3 , the cyclic group of order 3, with generator a {\displaystyle a} and identity element 1 G . An element r of C [ G ] can be written as where z 0 , z 1 and z 2 are in C ,

2345-538: Is isomorphic to the general linear group of invertible matrices: A u t ( V ) ≅ G L d ( K ) {\displaystyle \mathrm {Aut} (V)\cong \mathrm {GL} _{d}(K)} . Any such representation induces an algebra representation simply by letting ρ ~ ( e g ) = ρ ( g ) {\displaystyle {\tilde {\rho }}(e_{g})=\rho (g)} and extending linearly. Thus, representations of

2412-401: Is isomorphic to the ring of d × d matrices: E n d ( V ) ≅ M d ( K ) {\displaystyle \mathrm {End} (V)\cong M_{d}(K)} . Equivalently, this is a left K [ G ]-module over the abelian group V . Correspondingly, a group representation is a group homomorphism from G to the group of linear automorphisms of V , which

2479-417: Is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra , for it is indeed an algebra over the given ring. A group algebra over

2546-466: Is the identity element . Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; that is, group was taken to mean permutation group . By the end of the 19th century, group was defined more broadly, and accordingly so was involution . A permutation is an involution if and only if it can be written as a finite product of disjoint transpositions . The involutions of

2613-979: Is the dimension of V k . The subalgebra of C [ G ] corresponding to End( V k ) is the two-sided ideal generated by the idempotent where χ k ( g ) = t r ρ k ( g ) {\displaystyle \chi _{k}(g)=\mathrm {tr} \,\rho _{k}(g)} is the character of V k . These form a complete system of orthogonal idempotents, so that ϵ k 2 = ϵ k {\displaystyle \epsilon _{k}^{2}=\epsilon _{k}} , ϵ j ϵ k = 0 {\displaystyle \epsilon _{j}\epsilon _{k}=0} for j ≠ k , and 1 = ϵ 1 + ⋯ + ϵ m {\displaystyle 1=\epsilon _{1}+\cdots +\epsilon _{m}} . The isomorphism ρ ~ {\displaystyle {\tilde {\rho }}}

2680-516: Is the set of mappings f : G → R {\displaystyle f\colon G\to R} of finite support ( f ( g ) {\displaystyle f(g)} is nonzero for only finitely many elements g {\displaystyle g} ), where the module scalar product α f {\displaystyle \alpha f} of a scalar α {\displaystyle \alpha } in R {\displaystyle R} and

2747-431: Is the set of real numbers. An arbitrary element of this group ring is of the form where x i {\displaystyle x_{i}} is a real number. Multiplication, as in any other group ring, is defined based on the group operation. For example, Note that R Q is not the same as the skew field of quaternions over R . This is because the skew field of quaternions satisfies additional relations in

L-theory - Misplaced Pages Continue

2814-531: The Beaufort polyalphabetic cipher . The composition g ∘ f of two involutions f and g is an involution if and only if they commute : g ∘ f = f ∘ g . The number of involutions, including the identity involution, on a set with n = 0, 1, 2, ... elements is given by a recurrence relation found by Heinrich August Rothe in 1800: The first few terms of this sequence are 1 , 1, 2 , 4 , 10 , 26 , 76 , 232 (sequence A000085 in

2881-659: The OEIS ); these numbers are called the telephone numbers , and they also count the number of Young tableaux with a given number of cells. The number a n can also be expressed by non-recursive formulas, such as the sum a n = ∑ m = 0 ⌊ n 2 ⌋ n ! 2 m m ! ( n − 2 m ) ! . {\displaystyle a_{n}=\sum _{m=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n!}{2^{m}m!(n-2m)!}}.} The number of fixed points of an involution on

2948-606: The Witt groups of ε-quadratic forms over the ring R with ϵ = ( − 1 ) k {\displaystyle \epsilon =(-1)^{k}} . More precisely, is the abelian group of equivalence classes [ ψ ] {\displaystyle [\psi ]} of non-degenerate ε-quadratic forms ψ ∈ Q ϵ ( F ) {\displaystyle \psi \in Q_{\epsilon }(F)} over R, where

3015-437: The complex numbers . This is the same thing as a polynomial ring in variable a {\displaystyle a} such that a 3 = a 0 = 1 {\displaystyle a^{3}=a^{0}=1} i.e. C [ G ] is isomorphic to the ring C [ a {\displaystyle a} ]/ ( a 3 − 1 ) {\displaystyle (a^{3}-1)} . Writing

3082-405: The full linear monoid ) with transpose as the involution. In ring theory , the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. Examples of involutions in common rings: In group theory , an element of a group is an involution if it has order 2; that is, an involution is an element a such that a ≠ e and a = e , where e

3149-536: The group ring Z [ π ] {\displaystyle \mathbf {Z} [\pi ]} . In the applications to topology π {\displaystyle \pi } is the fundamental group π 1 ( X ) {\displaystyle \pi _{1}(X)} of a space X {\displaystyle X} . The quadratic L -groups L ∗ ( Z [ π ] ) {\displaystyle L_{*}(\mathbf {Z} [\pi ])} play

3216-416: The infinite cyclic group Z over R . 3. Let Q be the quaternion group with elements { e , e ¯ , i , i ¯ , j , j ¯ , k , k ¯ } {\displaystyle \{e,{\bar {e}},i,{\bar {i}},j,{\bar {j}},k,{\bar {k}}\}} . Consider the group ring R Q , where R

3283-454: The additive group R [ G ] {\displaystyle R[G]} into a ring, we define the product of f {\displaystyle f} and g {\displaystyle g} to be the mapping The summation is legitimate because f {\displaystyle f} and g {\displaystyle g} are of finite support, and the ring axioms are readily verified. Some variations in

3350-647: The complex irreducible representations of G as V k for k = 1, . . . , m , these correspond to group homomorphisms ρ k : G → A u t ( V k ) {\displaystyle \rho _{k}:G\to \mathrm {Aut} (V_{k})} and hence to algebra homomorphisms ρ ~ k : C [ G ] → E n d ( V k ) {\displaystyle {\tilde {\rho }}_{k}:\mathbb {C} [G]\to \mathrm {End} (V_{k})} . Assembling these mappings gives an algebra isomorphism where d k

3417-769: The cyclic group Z 2 {\displaystyle \mathbf {Z} _{2}} deals with the fixed points of a Z 2 {\displaystyle \mathbf {Z} _{2}} -action, while the group homology H ∗ {\displaystyle H_{*}} deals with the orbits of a Z 2 {\displaystyle \mathbf {Z} _{2}} -action; compare X G {\displaystyle X^{G}} (fixed points) and X G = X / G {\displaystyle X_{G}=X/G} (orbits, quotient) for upper/lower index notation. The quadratic L -groups: L n ( R ) {\displaystyle L_{n}(R)} and

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3484-411: The element ( 12 ) ∈ S 3 {\displaystyle (12)\in \mathbb {S} _{3}} is the transposition that swaps 1 and 2. Therefore the group ring need not be an integral domain even when the underlying ring is an integral domain. Using 1 to denote the multiplicative identity of the ring R , and denoting the group unit by 1 G , the ring R [ G ] contains

3551-430: The group algebra and a function on the group f  : G → K these pair to give an element of K via which is a well-defined sum because it is finite. Taking K [ G ] to be an abstract algebra, one may ask for representations of the algebra acting on a K- vector space V of dimension d . Such a representation is an algebra homomorphism from the group algebra to the algebra of endomorphisms of V , which

3618-557: The group correspond exactly to representations of the algebra, and the two theories are essentially equivalent. The group algebra is an algebra over itself; under the correspondence of representations over R and R [ G ] modules, it is the regular representation of the group. Written as a representation, it is the representation g ↦ ρ g with the action given by ρ ( g ) ⋅ e h = e g h {\displaystyle \rho (g)\cdot e_{h}=e_{gh}} , or The dimension of

3685-403: The multiplicative identity element of C [ G ] is 1⋅1 G where the first 1 comes from C and the second from G . The additive identity element is zero. When G is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms. 2. The ring of Laurent polynomials over a ring R is the group ring of

3752-432: The notation and terminology are in use. In particular, the mappings such as f : G → R {\displaystyle f:G\to R} are sometimes written as what are called "formal linear combinations of elements of G {\displaystyle G} with coefficients in R {\displaystyle R} ": or simply Note that if the ring R {\displaystyle R}

3819-470: The possible regular polyhedra and their generalizations to higher dimensions . The operation of complement in Boolean algebras is an involution. Accordingly, negation in classical logic satisfies the law of double negation : ¬¬ A is equivalent to A . Generally in non-classical logics , negation that satisfies the law of double negation is called involutive . In algebraic semantics , such

3886-403: The quadratic L -groups, detect the signature; in dimension (4 k +1), the L -groups detect the de Rham invariant . Ring with involution In mathematics , an involution , involutory function , or self-inverse function is a function f that is its own inverse , for all x in the domain of f . Equivalently, applying f twice produces the original value. Any involution

3953-401: The right is the group operation (denoted by juxtaposition). Because the above multiplication can be confusing, one can also write the basis vectors of K [ G ] as e g (instead of g ), in which case the multiplication is written as: Thinking of the free vector space as K -valued functions on G , the algebra multiplication is convolution of functions. While the group algebra of

4020-432: The ring, such as − 1 ⋅ i = − i {\displaystyle -1\cdot i=-i} , whereas in the group ring R Q , − 1 ⋅ i {\displaystyle -1\cdot i} is not equal to 1 ⋅ i ¯ {\displaystyle 1\cdot {\bar {i}}} . To be more specific, the group ring R Q has dimension 8 as

4087-403: The study of binary relations , every relation has a converse relation . Since the converse of the converse is the original relation, the conversion operation is an involution on the category of relations . Binary relations are ordered through inclusion . While this ordering is reversed with the complementation involution, it is preserved under conversion. The XOR bitwise operation with

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4154-530: The surgery obstructions to simply connected surgery. The quadratic L -groups of the integers are: In doubly even dimension (4 k ), the quadratic L -groups detect the signature ; in singly even dimension (4 k +2), the L -groups detect the Arf invariant (topologically the Kervaire invariant ). The symmetric L -groups of the integers are: In doubly even dimension (4 k ), the symmetric L -groups, as with

4221-815: The symmetric L -groups are 4-fold periodic (the comment of Ranicki, page 12, on the non-periodicity of the symmetric L -groups refers to another type of L -groups, defined using "short complexes"). In view of the applications to the classification of manifolds there are extensive calculations of the quadratic L {\displaystyle L} -groups L ∗ ( Z [ π ] ) {\displaystyle L_{*}(\mathbf {Z} [\pi ])} . For finite π {\displaystyle \pi } algebraic methods are used, and mostly geometric methods (e.g. controlled topology) are used for infinite π {\displaystyle \pi } . More generally, one can define L -groups for any additive category with

4288-400: The symmetric L -groups: L n ( R ) {\displaystyle L^{n}(R)} are related by a symmetrization map L n ( R ) → L n ( R ) {\displaystyle L_{n}(R)\to L^{n}(R)} which is an isomorphism modulo 2-torsion, and which corresponds to the polarization identities . The quadratic and

4355-510: The three-dimensional Euclidean space is reflection through a plane . Performing a reflection twice brings a point back to its original coordinates. Another involution is reflection through the origin ; not a reflection in the above sense, and so, a distinct example. These transformations are examples of affine involutions . An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. Another type of involution occurring in projective geometry

4422-646: 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 L 2 k ( R ) {\displaystyle L_{2k}(R)} is defined by The zero element is represented by H ( − 1 ) k ( R ) n {\displaystyle H_{(-1)^{k}}(R)^{n}} for any n ∈ N 0 {\displaystyle n\in {\mathbb {N} }_{0}} . The inverse of [ ψ ] {\displaystyle [\psi ]}

4489-465: The vector space K [ G ] is just equal to the number of elements in the group. The field K is commonly taken to be the complex numbers C or the reals R , so that one discusses the group algebras C [ G ] or R [ G ]. The group algebra C [ G ] of a finite group over the complex numbers is a semisimple ring . This result, Maschke's theorem , allows us to understand C [ G ] as a finite product of matrix rings with entries in C . Indeed, if we list

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